RBSE Solutions Class 11 Economics Chapter 8 Arithmetic Mean

RBSE Class 11 Economics Chapter 8 Text book Questions

RBSE Class 11 Economics Chapter 8 Multiple Choice Questions

 

Question 1. In order to present the properties of data-items in a concise form, calculation is done by
(a) Statistical method
(b) Statistical mean
(c) Statistical Formulae
(d) Tabulation
Answer: (b) Statistical mean
In simple words: To show data properties simply, we often use statistical mean calculations. It helps condense information into a single representative value.

🎯 Exam Tip: Remember that 'statistical mean' is a key concept for summarizing data and provides a quick overview of the dataset.

 

Question 2. The objective of arithmetic mean is to find-
Answer: The objective of arithmetic mean is to find the central or average value of a given set of data. This helps in understanding the typical value in a dataset.
In simple words: The main goal of arithmetic mean is to find the average number in a group.

🎯 Exam Tip: The arithmetic mean is a fundamental measure of central tendency, used to represent the entire dataset with a single value.

 

Question 3. In which mean is algebraic investigation possible-
(a) Arithmetic mean
(b) Median
(c) Mode
(d) All of the options
Answer: (a) Arithmetic mean
In simple words: You can use arithmetic mean in math calculations and formulas more easily than median or mode. It means it's good for algebraic work.

🎯 Exam Tip: Algebraic treatment is a significant advantage of the arithmetic mean, allowing for further mathematical analysis and manipulation.

 

Question 4. If X\(_{1}\) = 4, X\(_{2}\) = 5, N\(_{1}\) = 10, N\(_{2}\) = 15, then collective mean would be
(a) 4.5
(b) 4.6
(c) 5
(d) 4.8
Answer: (b) 4.6
In simple words: If you have two groups with different averages (X1, X2) and different numbers of items (N1, N2), the combined average of both groups would be 4.6. This is found by summing the products of each mean and its count, then dividing by the total count.

🎯 Exam Tip: To calculate the collective mean, multiply each group's mean by its number of items, sum these products, and then divide by the total number of items from all groups.

 

Question 5. The sum of deviations taken from arithmetic mean for any series is
(a) Maximum Sum
(b) Minimum Sum
(c) Zero
(d) Infinity
Answer: (c) Zero
In simple words: If you subtract the average (arithmetic mean) from each number in a list and then add up all those differences, the total will always be zero. This is a special property of the arithmetic mean.

🎯 Exam Tip: This property is fundamental to the arithmetic mean; it implies that the mean is the "balance point" of the data, where positive and negative deviations cancel each other out.

RBSE Class 11 Economics Chapter 8 Very Short Answer Type Questions

 

Question 1. What is arithmetic mean?
Answer: Equal importance is given to all values when calculating a simple average mean. However, in a weighted average mean, each value is given importance based on its specific weight. This difference highlights how different types of means treat data values.
In simple words: In simple mean, all numbers are treated equally. In weighted mean, some numbers are more important than others.

🎯 Exam Tip: Understand that simple mean treats all data points equally, while weighted mean assigns varying importance, which is crucial when some data points have more influence.

 

Question 3. When is step-deviation method used to determine arithmetic mean?
Answer: The step-deviation method is used to make the shortcut method even simpler. This happens when dealing with continuous data where the class intervals are equal in size and there are many class intervals. This method helps to simplify calculations with large or complex data.
In simple words: Step-deviation is used when there are many groups of data that have the same size, making calculations easier.

🎯 Exam Tip: The step-deviation method is particularly useful for continuous frequency distributions with equal class intervals, as it reduces large numbers into smaller, more manageable ones for calculation.

 

Question 4. Which are means of first order?
Answer: Arithmetic mean is a mean of the first order. It is also known as the simplest and most common average.
In simple words: Arithmetic mean is a basic type of average.

🎯 Exam Tip: "First order" refers to means that are direct calculations from the data, distinguishing them from more complex statistical measures.

 

Question 5. Write the formula to determine collective arithmetic mean.
Answer: The formula to determine collective arithmetic mean is:
\( \overline { X } ={ \overline { X } }_{1,2}=\frac {{N}_{ 1 }{ \overline { X }}_{1}+{ N }_{ 2 }{ \overline {X}}_{2}}{{N}_{1}+{N}_{2}} \)
In simple words: To find the average of combined groups, multiply each group's average by its count, add these results, and then divide by the total count of all items.

🎯 Exam Tip: This formula is crucial for combining the means of two or more distinct groups into a single overall mean, especially useful in data aggregation.

RBSE Class 11 Economics Chapter 8 Short Answer Type Questions

 

Question 1. Prove, using an example, that the sum of deviations of various values from arithmetic mean is always zero.
Answer: Let's use an example to prove that the sum of deviations from the arithmetic mean is always zero. Consider the following values: 4, 6, 8, 10, 12.
First, we calculate the arithmetic mean (\( \overline{X} \)):
\( \overline{X} = \frac{(4+6+8+10+12)}{5} = \frac{40}{5} = 8 \)
Next, we find the deviation of each value from the mean (\( X - \overline{X} \)):
\( X_1 = 4 - 8 = -4 \)
\( X_2 = 6 - 8 = -2 \)
\( X_3 = 8 - 8 = 0 \)
\( X_4 = 10 - 8 = 2 \)
\( X_5 = 12 - 8 = 4 \)
Now, we sum these deviations:
\( \Sigma(X - \overline{X}) = (-4) + (-2) + (0) + (2) + (4) = 0 \)
Therefore, the sum of deviations from the arithmetic mean is always zero. This property is what makes the arithmetic mean the center of the data.
In simple words: We added up numbers and found their average. Then, we subtracted the average from each number. When we added all those differences, the answer was zero. This shows that the average is the central point.

🎯 Exam Tip: This property is a defining characteristic of the arithmetic mean; clearly showing the calculation steps for individual deviations and their sum is vital for proof.

 

Question 2. What would be the effect on the arithmetic mean of a series if a constant (definite) quantity is added to, subtracted from, multiplied into or divides each value of the distribution.
Answer: If a constant quantity (k) is added to or subtracted from every item in a series, the new arithmetic mean will be \( \overline{X} + k \) or \( \overline{X} - k \), respectively. Similarly, if every item in a series is multiplied by a constant (k) or divided by a constant (k), the new arithmetic mean will become \( k\overline{X} \) or \( \overline{X}/k \), respectively. This means the mean changes in the same way the individual values change.
In simple words: If you add, subtract, multiply, or divide every number in a list by the same amount, the average (mean) of the list will also change by that exact same amount.

🎯 Exam Tip: This property, known as the "effect of change of origin and scale," is crucial for understanding how linear transformations impact the arithmetic mean.

 

Question 3. Write any four characteristics of an ideal mean.
Answer: Here are four characteristics of an ideal mean:

• **It should be clearly defined:** The mean should have a very clear definition so that it can only be interpreted in one way.
• **It should be simple to understand and compute:** The mean should be easy for everyone to understand and simple to calculate without complex steps.
• **It should be based on all values:** An ideal mean should consider every single value in the series. If it doesn't, it might not accurately represent the entire dataset.
• **It should be least affected by extreme values (maximum/minimum):** The mean should not be greatly influenced by extremely small or large values in the data series. This helps keep the mean representative.

An ideal mean acts as a reliable summary for the entire dataset.
In simple words: A good average should be clear, easy to calculate, use all the numbers, and not change too much if there are very big or very small numbers.

🎯 Exam Tip: When listing characteristics, ensure clarity and provide a brief explanation for each point to demonstrate full understanding.

 

Question 5. What is the objective behind study of means?
Answer: Statistical means are very useful in real life. They help present complex data in a simple form, representing the entire dataset effectively. Means allow for the comparison of two or more groups easily. They also provide a foundation for other statistical analyses and help in making future policies by giving a clear basis for decision-making.
In simple words: Averages help us understand big, confusing data by making it simple. They let us compare different groups and help make plans for the future.

🎯 Exam Tip: Focus on the roles of simplification, comparison, and policy-making when discussing the objectives of studying statistical means.

RBSE Class 11 Economics Chapter 8 Long Answer Type Questions

 

Question 1. What is meant by measures of central tendency? Explain the characteristics of an ideal mean.
Answer: **Meaning of Measures of Central Tendency:**
In any data series, there is a particular point around which all other data values tend to gather or concentrate. This value is usually found near the center of the series and shows its main features. This value is what we call a measure of central tendency or mean.
According to Simpson and Kafka, "The measure of central tendency is a representative value toward which other values tend to concentrate."

**Following are the four characteristics of an ideal mean:**

• **It should be defined clearly:** The mean must be defined very precisely so that it has only one meaning and interpretation.
• **It should be simple to understand and simple to compute:** The mean should be easy for anyone to understand and straightforward to calculate.
• **It should be based on all the values:** An ideal mean should include every single value in the series in its calculation. If it doesn't, it will not accurately reflect the entire dataset.
• **It should be the least affected by extreme values (maximum/minimum):** Very small and very large values in the data series should have minimal impact on the mean. This ensures the mean remains a fair representation.

These qualities ensure that the mean is a robust and reliable summary of the data.
In simple words: Central tendency means finding a middle value in data where most other values gather. A good average should be clear, easy to calculate, use all numbers, and not be too affected by very high or low numbers.

🎯 Exam Tip: When defining central tendency, emphasize the concept of a single representative value. For ideal mean characteristics, ensure each point is distinct and explains why it's a desirable trait.

 

Question 2. Find the number of children (x) if the arithmetic mean of age in the following series is 11.9 years.
Answer: To find the missing number of children (x), we first calculate the mid-point (X) for each age group and then multiply it by the number of children (f) to get fx.

We use the formula for arithmetic mean (\( \overline{X} \)):
\( \overline{X} = \frac{\Sigma fx}{N} \)
Given \( \overline{X} = 11.9 \), \( N = 30 + X \), and \( \Sigma fx = 335 + 13X \):
\( 11.9 = \frac{335 + 13X}{30 + X} \)
Now, we cross-multiply to solve for X:
\( 11.9 \times (30 + X) = 335 + 13X \)
\( 357 + 11.9X = 335 + 13X \)
Next, we gather the X terms on one side and the constants on the other:
\( 357 - 335 = 13X - 11.9X \)
\( 22 = 1.1X \)
To find X, we divide 22 by 1.1:
\( X = \frac{22}{1.1} \)
\( X = 20 \)
So, the missing number of children (X) is 20. Finding missing frequencies is a common application of the mean formula.
In simple words: We used the given average age (11.9 years) and the numbers we knew to find the missing count of children. We set up an equation with the total sum of ages and total number of children, then solved it like a math puzzle. The missing number was 20.

🎯 Exam Tip: When a frequency is missing, set up the mean formula as an equation and solve for the unknown variable. Ensure all sums (\( \Sigma fx \) and \( N \)) are correctly expressed with the variable.

 

Question 3. Find the arithmetic mean, median and mode in the following frequency distribution.
Answer: We need to compute the Mean, Median, and Mode from the given frequency distribution.

**Computation of Mean, Median & Mode**

**Arithmetic Mean (\( \overline{X} \)):**
\( \overline{X} = \frac{\Sigma fx}{N} = \frac{1090}{50} = 21.8 \)
**Median (M):**
First, find the median class.
\( \frac{N}{2} = \frac{50}{2} = 25 \)
The cumulative frequency (c.f.) just greater than 25 is 25 itself, which corresponds to the class interval 15-20. So, the median class is 15-20.
Using the median formula:
\( M = l_1 + \frac{\frac{N}{2} - c.f.}{f} \times i \)
Here, \( l_1 = 15 \) (lower limit of median class)
\( \frac{N}{2} = 25 \)
\( c.f. = 13 \) (cumulative frequency of class preceding median class)
\( f = 12 \) (frequency of median class)
\( i = 5 \) (class interval size)
\( M = 15 + \frac{25 - 13}{12} \times 5 \)
\( M = 15 + \frac{12}{12} \times 5 \)
\( M = 15 + 1 \times 5 \)
\( M = 15 + 5 = 20 \)
**Mode (Z):**
The highest frequency is 14, which corresponds to the class interval 25-30. So, the modal class is 25-30.
Using the mode formula:
\( Z = l_1 + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i \)
Here, \( l_1 = 25 \) (lower limit of modal class)
\( f_1 = 14 \) (frequency of modal class)
\( f_0 = 0 \) (frequency of class preceding modal class, i.e., 20-25)
\( f_2 = 6 \) (frequency of class succeeding modal class, i.e., 30-35)
\( i = 5 \) (class interval size)
\( Z = 25 + \frac{14 - 0}{(2 \times 14) - 0 - 6} \times 5 \)
\( Z = 25 + \frac{14}{28 - 6} \times 5 \)
\( Z = 25 + \frac{14}{22} \times 5 \)
\( Z = 25 + \frac{70}{22} \)
\( Z = 25 + 3.18 \) (approximately)
\( Z = 28.18 \)
Therefore, the arithmetic mean is 21.8, the median is 20, and the mode is 28.18. Each measure gives a different insight into the central tendency of the data.
In simple words: We calculated three main numbers for the data: the average (mean) which is 21.8, the middle number (median) which is 20, and the most common number (mode) which is about 28.18. We used specific formulas for each calculation.

🎯 Exam Tip: When calculating mean, median, and mode for grouped data, always clearly identify the mid-points, cumulative frequencies, and the correct modal/median class to avoid errors. Ensure you use the right formula for each measure.

 

Question 4. Find the weighted average mean.
Answer: We need to calculate the weighted mean for the given expenditure data.

The formula for weighted arithmetic mean is:
\( \overline{X}_{w} = \frac{\Sigma Xw}{\Sigma w} \)
Substitute the values:
\( \overline{X}_{w} = \frac{8670}{13} \)
\( \overline{X}_{w} = 666.92 \)
The weighted average mean is Rs. 666.92. This type of mean is useful when different items have different levels of importance.
In simple words: We found the average spending by giving different importance (weights) to each type of expense. We multiplied each expense by its importance, added them up, and then divided by the total importance. The weighted average was Rs. 666.92.

🎯 Exam Tip: Remember to multiply each value by its corresponding weight and sum these products (\( \Sigma Xw \)). Then, divide this sum by the total sum of weights (\( \Sigma w \)).

RBSE Economics Chapter 8 Other Important Questions

 

Question 1. (b) Mode
(c) Arithmetic Mean
(d) None of the options
Answer: (c) Arithmetic Mean
In simple words: Out of the given choices, the Arithmetic Mean is the mathematical average used to find the central value of a dataset. It is a common measure of central tendency.

🎯 Exam Tip: Even if the question stem is missing, analyze the options and the selected answer to infer the underlying concept being tested, often a definition or a property of means.

 

Question 2. The algebraic sum of the deviation of a set of values from the arithmetic mean is
(a) -1
(b) 0
(c) 1
(d) None of the options
Answer: (b) 0
In simple words: If you take each number in a list, subtract the average, and then add up all those differences, the total will always be zero. This is a basic rule for the arithmetic mean.

🎯 Exam Tip: This property is fundamental to the arithmetic mean, indicating its position as the balancing point of the data distribution.

 

Question 3. The value obtained by dividing the sum of all the values in a data series, by the number of values is
(a) Median
(b) Mode
(c) Arithmetic Mean
(d) None of the options
Answer: (c) Arithmetic Mean
In simple words: When you add up all the numbers in a list and then divide by how many numbers there are, you get the arithmetic mean. This is how we find a simple average.

🎯 Exam Tip: This is the classic definition of the arithmetic mean; ensure you distinguish it from median (middle value) and mode (most frequent value).

 

Question 4. Which one of the following is not a mathematical mean?
(a) Arithmetic mean
(b) Geometric mean
(c) Mode
(d) Harmonic mean
Answer: (c) Mode
In simple words: Mode is the number that appears most often, but it is not calculated using a math formula like the other averages. Mode is a positional average, not a mathematical one.

🎯 Exam Tip: Mathematical means (Arithmetic, Geometric, Harmonic) are based on specific calculations, while positional averages (Median, Mode) are based on the position or frequency of values.

 

Question 6. Which mean is the most affected by marginal values?
(a) Mode
(b) Arithmetic mean
(c) Median
(d) Geometric mean
Answer: (b) Arithmetic mean
In simple words: The arithmetic mean is most influenced by very big or very small numbers (extreme values) in a list. This is because every number is used in its calculation.

🎯 Exam Tip: The arithmetic mean uses all values in its calculation, making it sensitive to outliers, while the median and mode are more robust to extreme values.

 

Question 7. Each item of the series is included in the calculation
(a) In mode
(b) In median
(c) In arithmetic mean
(d) In all of these
Answer: (c) In arithmetic mean
In simple words: For the arithmetic mean, every single number in the list is used to find the average. It considers all items equally.

🎯 Exam Tip: This property is a key advantage of the arithmetic mean, as it ensures that no data point is ignored in the overall summary, but it also makes it sensitive to extreme values.

 

Question 8. Mid-value of 10-15 is
(a) 10
(b) 12.5
(c) 15
(d) None of the options
Answer: (b) 12.5
In simple words: To find the middle number of a group like 10-15, you add the smallest number and the largest number, then divide by two. This gives you 12.5.

🎯 Exam Tip: Always calculate the mid-value by adding the lower and upper limits of a class interval and dividing by two; this is crucial for calculations in grouped data.

 

Question 9. Measure of central tendency is
Answer: A measure of central tendency is a single value that tries to describe a set of data by identifying the central position within that set. It indicates where most values cluster.
In simple words: A measure of central tendency finds a single number that best shows the middle or typical value of a group of data.

🎯 Exam Tip: The primary measures of central tendency are the mean, median, and mode, each offering a different perspective on the "center" of a dataset.

RBSE Class 11 Economics Chapter 8 Very Short Answer Type Questions

 

Question 1. How many types of means are related to place?
Answer: There are two types of means related to place: Median and Mode. These are known as positional averages because their values depend on their position in the data rather than mathematical calculations.
In simple words: Two types of averages depend on where numbers are placed: Median and Mode.

🎯 Exam Tip: Remember that Median and Mode are positional averages, which means their calculation relies on ordering the data or identifying the most frequent value, not on arithmetic operations across all data points.

 

Question 2. How many types of mathematical means are there?
Answer: There are four types of mathematical means: Arithmetic Mean, Geometric Mean, Harmonic Mean, and Quadratic Mean. Each mean uses different mathematical formulas to find the average.
In simple words: There are four math-based averages: Arithmetic, Geometric, Harmonic, and Quadratic Mean.

🎯 Exam Tip: While Arithmetic Mean is most common, be aware of Geometric and Harmonic Means, which are used for specific types of data, such as growth rates or rates of change.

 

Question 3. Write the names of any two types of mathematical means.
Answer:
1. Arithmetic Mean
2. Geometric Mean
These are two common types of mathematical means, each with its own method of calculation and application.
In simple words: Two types of math averages are the Arithmetic Mean and the Geometric Mean.

🎯 Exam Tip: Arithmetic Mean is the simple average, while Geometric Mean is often used for data that grows exponentially, like investment returns or population growth.

 

Question 4. Generally, the mean used by the common man in daily life is?
Answer: Generally, the mean used by the common man in daily life is the Arithmetic Mean. It is simple to understand and widely used for everyday averages.
In simple words: The average people use most often every day is the Arithmetic Mean.

🎯 Exam Tip: The arithmetic mean's simplicity and intuitive nature make it the most frequently encountered average in everyday contexts, such as calculating average scores or temperatures.

 

Question 6. Of how many types is the arithmetic mean?
Answer: The arithmetic mean is of two types: Simple/Unweighted Arithmetic Mean and Weighted Arithmetic Mean. Each type is used depending on whether all data points have equal importance.
In simple words: There are two kinds of arithmetic mean: simple (where all numbers are equal) and weighted (where some numbers are more important).

🎯 Exam Tip: Differentiate between simple and weighted arithmetic mean based on the assumption of equal or unequal importance of data points, respectively.

 

Question 7. Write the names of the types of arithmetic mean.
Answer:
1. Simple/unweighted Arithmetic Mean
2. Weighted Arithmetic Mean
These two types cover scenarios where data points have uniform or varied significance in calculating the average.
In simple words: The types of arithmetic mean are simple mean and weighted mean.

🎯 Exam Tip: Be able to clearly state the two primary types of arithmetic mean and understand when each is appropriately applied.

 

Question 8. Write the formula of arithmetic mean by the direct method in the individual series.
Answer: The formula for the arithmetic mean by the direct method in an individual series is:
\( \overline{X} = \frac{\Sigma X}{N} \)
Here, \( \Sigma X \) represents the sum of all the values in the series, and \( N \) is the total number of items. This method is straightforward for calculating the average of a simple list of numbers.
In simple words: To find the average directly, add up all the numbers and then divide by how many numbers there are.

🎯 Exam Tip: The direct method for individual series is the most basic way to calculate the arithmetic mean; ensure you correctly sum all values and count the total number of observations.

 

Question 9. What is the use of shortcut method in the individual series?
Answer: The shortcut method in individual series is used when the number of values in a series is high, or the numbers are large or in decimal form. This method simplifies calculations by working with deviations from an assumed mean, which reduces the size of the numbers involved.
In simple words: The shortcut method is used to make it easier to calculate the average when there are many numbers, or if the numbers are big or have decimals.

🎯 Exam Tip: The shortcut method streamlines calculations, especially for large datasets, by using an assumed mean to reduce the complexity of individual deviations.

 

Question 10. Write the formula for the arithmetic mean in shortcut method.
Answer: The formula for the arithmetic mean using the shortcut method is:
\( \overline{X} = A + \frac{\Sigma d}{N} \)
Here, \( A \) is the assumed mean, \( \Sigma d \) is the sum of deviations of values from the assumed mean, and \( N \) is the total number of items. This formula adjusts an initial guess to find the true mean.
In simple words: For the shortcut average, you start with a guessed average, then add the sum of all differences from that guess divided by the total number of items.

🎯 Exam Tip: Remember that \( d \) represents \( X - A \), where \( X \) is each data point and \( A \) is the assumed mean. The shortcut method is effective for simplifying calculations with larger data values.

 

Question 11. What is the sum of deviation of different item values taken from the arithmetic mean?
Answer: The sum of deviations of different item values taken from the arithmetic mean is always zero. This is a fundamental property of the arithmetic mean, meaning the positive and negative deviations perfectly balance each other out.
In simple words: If you subtract the average from each number and then add up all the results, the total will always be zero.

🎯 Exam Tip: This property ( \( \Sigma(X - \overline{X}) = 0 \) ) is crucial and often used as a check for calculations involving the arithmetic mean.

 

Question 13. Write the formula of arithmetic mean, which is calculated by the step deviation method.
Answer: The formula for the arithmetic mean using the step-deviation method is:
\( \overline{X} = A + \frac{\Sigma fd'}{N} \times i \)
Here, \( A \) is the assumed mean, \( \Sigma fd' \) is the sum of frequencies multiplied by step deviations, \( N \) is the total number of observations, and \( i \) is the common factor (class interval size). This method simplifies calculations for grouped data with equal class intervals.
In simple words: To find the average using step deviation, start with a guessed average, then add the sum of frequency times the simplified deviations, divided by the total count, and finally multiply by the group size.

🎯 Exam Tip: The step-deviation method is most efficient for frequency distributions with equal class intervals, as it involves dividing deviations by a common factor (i) to work with smaller numbers.

 

Question 14. Define arithmetic mean.
Answer: The arithmetic mean of a series is a value obtained by dividing the sum of all the values in the series by the total number of items present in it. It represents the central or typical value of the data.
In simple words: Arithmetic mean is found by adding all numbers in a list and then dividing by how many numbers there are.

🎯 Exam Tip: This is the most basic definition of an average; ensure you clearly state both parts: sum of values and division by count.

 

Question 15. Which mean is mostly used?
Answer: The arithmetic mean is mostly used. It is popular because of its simplicity and ease of calculation.
In simple words: Arithmetic mean is the average used most often.

🎯 Exam Tip: Understand that the arithmetic mean's widespread use comes from its straightforward calculation and intuitive interpretation.

 

Question 16. Differentiate between simple arithmetic mean and weighted arithmetic mean.
Answer: In the calculation of a simple arithmetic mean, all the individual items or values are given equal importance. This means each value contributes equally to the average. On the other hand, in a weighted arithmetic mean, each item is given a specific weight based on its significance or importance. The weights reflect how much each value should influence the final average.
In simple words: Simple mean gives same importance to all numbers. Weighted mean gives different importance to different numbers.

🎯 Exam Tip: The key distinction lies in the assumption of importance: equal for simple mean, varied for weighted mean. Use examples like average scores (simple) vs. GPA (weighted) to illustrate the difference.

 

Question 17. Write two characteristics of ideal mean.
Answer:
1. It should be defined clearly.
2. It should be simple to understand and simple to compute.
These characteristics ensure that the mean is both unambiguous and accessible to users.
In simple words: A good average should be easy to understand and simple to calculate.

🎯 Exam Tip: For any statistical measure, clear definition prevents misinterpretation, and ease of computation makes it practical for wide use.

 

Question 18. Write the name of commercial means.
Answer: The commercial mean is the Arithmetic Mean. It is widely used in business and economics for calculating averages like profits, costs, or sales.
In simple words: Arithmetic Mean is used a lot in business.

🎯 Exam Tip: Recognize the arithmetic mean as the most common average used in commercial and financial contexts due to its directness and ease of understanding.

 

Question 19. Write the formula of arithmetic mean in discrete series.
Answer: The formula for the arithmetic mean in a discrete series is given by \( \overline{X} = \frac{\Sigma fX}{N} \), where \( \Sigma fX \) is the sum of the products of each item value (X) and its frequency (f), and \( N \) is the total number of observations (sum of all frequencies). This formula helps to calculate the average when data is grouped with frequencies.
In simple words: To find the average, multiply each number by how many times it appears, add all these results, and then divide by the total count of numbers.

🎯 Exam Tip: Remember that in a discrete series, \( N \) represents the sum of all frequencies \( (\Sigma f) \), which is the total number of items.

 

Question 20. Write two flaws of arithmetic mean.
Answer:
1. The arithmetic mean is heavily influenced by very high or very low values in the data. These extreme values can skew the average, making it less representative. For instance, if most people earn a small salary, but a few earn a huge salary, the average salary will seem much higher than what most people actually get.
2. It cannot be figured out just by looking at the data series. You need to do calculations to find it.
In simple words: First, very large or very small numbers can change the average a lot. Second, you cannot guess the average just by looking at the list of numbers; you must calculate it.

🎯 Exam Tip: When evaluating means, always consider how sensitive they are to outliers, as this is a key limitation of the arithmetic mean.

 

Question 21. Write the two purposes of statistical means.
Answer: Statistical means serve two main purposes:
1. They help to simplify large and complicated sets of data into an easy-to-understand and short form. This makes it easier to grasp the main idea of the data quickly.
2. By turning a lot of data into one single average number, they allow for easy comparisons between different groups or sets of data. For example, you can easily compare the average height of students in two different classes.
In simple words: Statistical averages help to make big, complex information simple. They also help us compare different sets of information using just one number.

🎯 Exam Tip: Focus on 'simplification' and 'comparison' as the core benefits when discussing the purposes of statistical means.

 

Question 22. Write the formula of computing weighted arithmetic mean.
Answer: The formula for calculating the weighted arithmetic mean is \( \overline{X}_w = \frac{\Sigma XW}{\Sigma W} \), where \( \Sigma XW \) is the sum of the products of each value (X) and its assigned weight (W), and \( \Sigma W \) is the sum of all the weights. This method gives more importance to certain values.
In simple words: To find the weighted average, you multiply each number by its importance (weight), add all these results together, and then divide by the total of all the importances.

🎯 Exam Tip: Clearly define each symbol in the formula (\( \overline{X}_w, X, W, \Sigma \)) to ensure full understanding and marks.

 

Question 23. What kind of series is 2, 5, 7, 10, 13, 15?
Answer: This is an individual series. In this type of series, each observation or data point is listed separately without any grouping or frequency mentioned. This means every number is considered a single item.
In simple words: This is an individual series because each number is listed one by one, without being grouped or counted.

🎯 Exam Tip: Recognize an individual series by the raw, ungrouped listing of data points, where each value stands alone.

 

Question 24. What kind of series is 0-10, 10-20, 20-30?
Answer: This is a continuous series. In this series, data is grouped into class intervals, where the upper limit of one class is the lower limit of the next class, representing a continuous range of values. This type is used for data that can take any value within a range.
In simple words: This is a continuous series because the numbers are put into groups (like 0 to 10, 10 to 20), and these groups flow into each other without gaps.

🎯 Exam Tip: Identify a continuous series by its use of class intervals, especially when the end of one interval matches the start of the next.

 

Question 1. Explain the characteristics of an ideal mean.
Answer:
**Meaning of Measures of Central Tendency:**
There is a specific point within any data series where all other data values tend to cluster or focus. This central value is typically found near the middle of the series and represents its most important features. This value is known as the measure of central tendency or mean. For example, in a class of students, the average height is a central tendency measure.

According to Simpson and Kafka:
"The measure of central tendency is such a representative value, towards which other values tend to concentrate."

The following are the four characteristics of an ideal mean:

• **It should be defined clearly:** The mean should be explained very clearly so that it has only one specific meaning and interpretation.


• **It should be simple to understand and compute:** The mean should be easy for anyone to grasp and straightforward to calculate without complex steps.


• **It should be based on all the values:** An ideal mean must include every single data point in its calculation. If any values are left out, the mean may not accurately represent the entire series.


• **It should be the least affected by extreme values (maximum/minimum):** Very small or very large values in a data series should have only a minor impact on the mean. If extreme values significantly distort the mean, it might give a misleading picture.

In simple words: A good average should be very clear, easy to calculate, use all the numbers in the data, and not be changed too much by very big or very small numbers.

🎯 Exam Tip: When listing characteristics, aim for clarity and briefly explain each point to demonstrate full understanding.

 

Question 2. Explain four characteristics of arithmetic mean.
Answer: The four characteristics of the arithmetic mean are:
1. **Its algebraic use is possible:** The arithmetic mean is mathematically sound and can be used in further algebraic calculations. This makes it a foundational tool for other statistical methods.
2. **Its calculation is simple:** It is very straightforward to calculate, requiring only the sum of values and the count of values. This simplicity makes it widely used in various fields.
3. **It keeps all the items in mind:** Every single value in the data set contributes to the arithmetic mean. This ensures it's a comprehensive measure that reflects all data points, which makes it a truly representative average.
4. **It is the most popular medium for comparative study:** It is the most common and preferred method for comparing different sets of data. Comparing averages helps understand differences between groups, for example, comparing the average test scores of two different classes.
In simple words: The arithmetic mean can be used in math, is easy to calculate, uses all the numbers, and is good for comparing different groups of numbers.

🎯 Exam Tip: Highlight the mathematical utility and ease of calculation, as these are primary strengths of the arithmetic mean.

 

Question 4. What is the difference between simple and weighted arithmetic mean?
Answer: The differences between simple and weighted arithmetic mean are:
1. **Importance of Values:** In a simple arithmetic mean, all values are treated equally and given the same importance. However, in a weighted arithmetic mean, each value is assigned a different level of importance (a 'weight') based on its significance.
2. **Representativeness:** The simple arithmetic mean might not always accurately represent the entire data series, especially if some items are more important than others. The weighted arithmetic mean, by accounting for importance, gives a more accurate representation of the series.
3. **Accuracy of Conclusions:** Decisions made using the simple arithmetic mean can sometimes be incorrect or misleading. In contrast, conclusions drawn from the weighted arithmetic mean are generally more reliable and accurate, as they factor in the varying importance of data points. For instance, calculating average marks where some subjects carry more credit points is a good example of weighted mean.
In simple words: Simple mean treats all numbers as equally important, while weighted mean gives more importance to some numbers. Weighted mean gives a truer picture and better conclusions than simple mean.

🎯 Exam Tip: Clearly distinguish between 'equal importance' for simple mean and 'assigned weight/importance' for weighted mean, as this is the core difference.

 

Question 5. Explain weighted arithmetic mean. Write its formula.
Answer: The weighted arithmetic mean is a type of average that gives different levels of importance, called 'weights', to different numbers in a data set. This is used when some values are more significant than others. For example, if you are averaging grades, a final exam might have more weight than a quiz.
The formula for the weighted arithmetic mean is: \( \overline{X}_w = \frac{\Sigma XW}{\Sigma W} \), where \( \overline{X}_w \) is the weighted mean, \( X \) represents each value, \( W \) represents its corresponding weight, \( \Sigma XW \) is the sum of (value multiplied by weight), and \( \Sigma W \) is the sum of all weights.
In simple words: Weighted mean is an average where some numbers count more than others. You multiply each number by its importance, add them up, and then divide by the total importance.

🎯 Exam Tip: When explaining, clearly state the 'why' (varying importance) and the 'how' (assigning weights and using the formula) for the weighted arithmetic mean.

 

Question 6. Explain the characteristics of arithmetic mean.
Answer: The characteristics of arithmetic mean include:
1. **Its algebraic use is possible:** The arithmetic mean is mathematically sound and can be used in further algebraic computations, making it useful in advanced statistical analysis.
2. **Its calculation is simple:** It is easy to understand and calculate, which is why it is widely used in everyday situations.
3. **It keeps all the items in mind:** Its calculation takes into account every single value in the data set, ensuring that no information is left out.
4. **It is the most popular medium for comparative study:** It is a very common tool for comparing different sets of data, allowing easy insights into their average differences. A good example is comparing average test scores across two classes.
In simple words: The arithmetic mean can be used in math, is easy to find, uses all the numbers, and is good for comparing different groups of numbers.

🎯 Exam Tip: Focus on the strengths like ease of calculation and comprehensive inclusion of data when describing the characteristics of the arithmetic mean.

 

Question 1. What do you mean by arithmetic mean? Explain the merits and demerits of arithmetic mean.
Answer:
**Meaning of Arithmetic Mean:**
The arithmetic mean is the most commonly used and important type of average among mathematical means. It is widely used in daily life. The arithmetic mean of a series is found by adding up all the values in the series and then dividing that sum by the total number of items present. For example, to find the average income of a few families, you would add all their incomes and divide by the number of families. This gives a single value that represents the 'typical' income.

According to H.Secrist:
"Arithmetic mean is the amount secured by dividing the sum of values of the items in a series by their number."

**Merits of Arithmetic Mean:**

• **Easy to compute and understand:** It is the simplest average to understand and calculate. Even someone without much statistical knowledge can grasp it.


• **Based on all items of the series:** The calculation of the arithmetic mean uses every single item in the data series. This ensures it is a good representative value because all data points contribute to it.


• **Definitiveness:** It is defined by a precise mathematical formula, meaning that anyone who calculates it using the same data will get the exact same answer. This makes it a reliable measure.


• **Stability:** Compared to other types of averages, the arithmetic mean tends to be quite stable. It does not change significantly if you take different samples from the same group, making it consistent.


• **Suitable for algebraic treatment:** Because it is based on a strict formula, it can be easily used in further algebraic calculations and statistical methods.


• **No need for arranging data:** You do not need to arrange the data in any specific order (like smallest to largest) before calculating the arithmetic mean.


• **Comparative Study:** It is very helpful for comparing two or more different series of data, allowing easy insights into their average differences.


• **Calculation of Unknown Values:** If you know the arithmetic mean, the total number of items, and the sum of items, any one missing value can be calculated using the two known values.

**Demerits of Arithmetic Mean:**

• **Effect of extreme values:** The arithmetic mean is heavily influenced by very small or very large values in the data. An extremely high or low number can drastically change the average, making it less representative of the majority.


• **Calculation difficulties:** In some cases, especially with complex data or if values are unknown, calculating the arithmetic mean can be difficult compared to other averages that can be found by just observing the data. It cannot be found by simply looking at the data, and if even one value is missing, it cannot be calculated.


• **Misleading conclusions:** Sometimes, the arithmetic mean can give conclusions that are not entirely accurate or consistent, especially if the data has a skewed distribution.


• **Not suitable in the study of rate, ratio and percentage:** It is not always appropriate for analyzing rates, ratios, or percentages, as other means might be more suitable for these specific types of data.

In simple words: The arithmetic mean is the basic average we find by adding all numbers and dividing by how many there are. Its good points are that it's easy, uses all numbers, and gives a clear answer. Its bad points are that very big or small numbers can change it a lot, and it's not always simple to find or useful for all types of data.

🎯 Exam Tip: When discussing merits and demerits, provide a brief, clear point for each, emphasizing its practical implication. For example, for "effect of extreme values," mention how outliers can distort the average.

 

Question 2. Clarify the direct and shortcut method of arithmetic mean in individual series.
Answer:
The arithmetic mean for an individual series can be calculated using two main methods: the Direct Method and the Short-cut Method. These methods help find the average of a list of numbers where each number is unique or listed individually.

**Calculation of Arithmetic Mean in Individual Series:**

**(a) Direct Method:**
In this method, all the values in the series (denoted as X) are added together to find their sum (\( \Sigma X \)). This sum is then divided by the total number of items (N) in the series. If you have N numbers like \( X_1, X_2, X_3, \dots, X_n \), their arithmetic mean \( (\overline{X}) \) is found as:
Formula: \( \overline{X} = \frac{\Sigma X}{N} \). This method is straightforward and easily understandable.

**(b) Short-cut Method:**
The short-cut method simplifies calculations, especially when dealing with large numbers. The steps are:
1. **Assume a Mean (A):** Choose any value from the data series (or even a value outside it) as the 'assumed mean' or 'arbitrary average'. It's often helpful to pick a value close to the middle of the series.
2. **Find Deviations (dx):** Calculate the difference between each actual value (X) and the assumed mean (A). This difference is called the deviation, \( dx = X - A \).
3. **Sum Deviations ( \( \Sigma dx \) ):** Add up all these deviations. Remember to keep track of their positive and negative signs.
4. **Apply Formula:** The arithmetic mean \( (\overline{X}) \) is then calculated using the formula: \( \overline{X} = A + \frac{\Sigma dx}{N} \). If \( \Sigma dx \) is positive, it's added to the assumed mean; if it's negative, it's subtracted. This method reduces the size of numbers in calculations, making them easier.

**Example:** Calculate the arithmetic mean by direct and shortcut method for the following data:

Serial no.	1	2	3	4	5	6	7	8	9	10
Obtained Marks	35	28	14	16	21	24	17	13	27	45

Here, \( N = 10 \).
Let's calculate \( \Sigma X \). From the data: \( 35 + 28 + 14 + 16 + 21 + 24 + 17 + 13 + 27 + 45 = 240 \).

**Direct method:** \( \overline{X} = \frac{\Sigma X}{N} = \frac{240}{10} = 24 \).
So, the Arithmetic mean is 24.

**Short-cut Method:** Let's take an assumed mean \( A = 28 \).

X (Marks)	\(dx = X - A\) (where \(A = 28\))
35	\(35 - 28 = 7\)
28	\(28 - 28 = 0\)
14	\(14 - 28 = -14\)
16	\(16 - 28 = -12\)
21	\(21 - 28 = -7\)
24	\(24 - 28 = -4\)
17	\(17 - 28 = -11\)
13	\(13 - 28 = -15\)
27	\(27 - 28 = -1\)
45	\(45 - 28 = 17\)
\( \Sigma X = 240 \)	\( \Sigma dx = -40 \)

Using the formula: \( \overline{X} = A + \frac{\Sigma dx}{N} = 28 + \frac{-40}{10} = 28 - 4 = 24 \).
Both methods consistently yield the same arithmetic mean of 24, demonstrating their accuracy.
In simple words: The direct way means adding all numbers and dividing by how many there are. The shortcut way means picking a middle number, finding how much each number differs from it, adding these differences, and then adjusting the middle number. Both ways give the same average.

🎯 Exam Tip: Always show both the formula and the steps clearly for both methods. Double-check your calculations, especially the signs for deviations in the shortcut method.

 

Question 3. Explain the direct and shortcut method of discrete series for calculating arithmetic mean with example.
Answer:
The arithmetic mean for a discrete series, where data points have associated frequencies, can be found using the Direct and Shortcut methods.

**Calculation of Arithmetic Mean in Discrete Series:**

**(i) Direct Method:**
In this method, the steps are:
1. **Multiply Item Value by Frequency (fx):** Each item value (X) is multiplied by its corresponding frequency (f) to get the product, which is denoted as \( fx \).
2. **Sum the Products ( \( \Sigma fx \) ):** All these products (\( fx \)) are then added together to find their total sum, \( \Sigma fx \).
3. **Apply Formula:** The arithmetic mean \( (\overline{X}) \) is calculated using the formula: \( \overline{X} = \frac{\Sigma fx}{N} \) or \( \overline{X} = \frac{\Sigma fx}{\Sigma f} \), where \( N \) or \( \Sigma f \) is the total sum of frequencies.

**Example for Direct Method:** Calculate arithmetic mean from the following given data:

Item-value (x)	Frequency (f)	Fx
4	5	20
6	9	54
8	11	88
10	8	80
12	4	48
14	3	42
	\( \Sigma f = 40 \)	\( \Sigma fx = 332 \)

Using the Direct Method formula: \( \overline{X} = \frac{\Sigma fx}{N} = \frac{332}{40} = 8.3 \).

**(ii) Short-cut Method:**
This method is particularly useful for discrete series with large values. The steps are:
1. **Assume a Mean (A):** Select any value (size) from the data as the assumed mean (A).
2. **Find Deviations (dx):** Calculate the deviation of each item value (X) from the assumed mean: \( dx = X - A \).
3. **Multiply Deviations by Frequency (fdx):** Multiply each deviation (\( dx \)) by its corresponding frequency (f) to get \( fdx \). Remember to consider the signs.
4. **Sum the Products ( \( \Sigma fdx \) ):** Add up all the \( fdx \) values to get \( \Sigma fdx \).
5. **Apply Formula:** The arithmetic mean \( (\overline{X}) \) is calculated using: \( \overline{X} = A + \frac{\Sigma fdx}{N} \), where \( N \) is \( \Sigma f \).

**Example for Short-cut Method:** Using the same data as above.
Let's assume \( A = 10 \).

X (Item-value)	f (Frequency)	\(dx = X - A\) (where \(A = 10\))	fdx
4	5	\(4 - 10 = -6\)	\(5 \times -6 = -30\)
6	9	\(6 - 10 = -4\)	\(9 \times -4 = -36\)
8	11	\(8 - 10 = -2\)	\(11 \times -2 = -22\)
10	8	\(10 - 10 = 0\)	\(8 \times 0 = 0\)
12	4	\(12 - 10 = 2\)	\(4 \times 2 = 8\)
14	3	\(14 - 10 = 4\)	\(3 \times 4 = 12\)
	\( \Sigma f = 40 \)		\( \Sigma fdx = -68 \)

Using the Short-cut Method formula: \( \overline{X} = A + \frac{\Sigma fdx}{N} = 10 + \frac{-68}{40} = 10 - 1.7 = 8.3 \).
Both methods correctly yield an arithmetic mean of 8.3 for the discrete series, confirming their accuracy.
In simple words: For numbers with counts, the direct way is to multiply each number by its count, add these up, and divide by the total count. The shortcut way involves choosing an assumed average, finding how much each number (times its count) differs from it, adding those differences, and then adjusting the assumed average. Both should give the same result.

🎯 Exam Tip: When using the shortcut method, choose an assumed mean (A) that is close to the middle of the data or a common value to minimize calculation errors and simplify deviations.

 

Question 4. Explain the direct and shortcut methods for calculating the arithmetic mean in a continuous series with an example.
Answer:
In a continuous series, data is presented in class intervals (like 0-5, 5-10). To calculate the arithmetic mean, we first convert the continuous series into a discrete series by finding the mid-point (X) of each class interval. Once the mid-points are found, the process becomes similar to calculating the mean for a discrete series using either the Direct Method or the Shortcut Method.

**Example:** Calculate the arithmetic mean from the following data:

Obtained Marks	0-5	5-10	10-15	15-20	20-25	25-30	30-35
No. of Students	3	4	12	14	8	7	5

**Solution Steps:**
First, we find the mid-points (X) for each class interval. Then we calculate \( fx \) and \( fd \) values as shown in the table below, taking the assumed mean \( A = 17.5 \):

Obtained Marks C.I.	Mid-point(x)	Frequency(f)	Fx	\(d = x - A\) (A = 17.5)	fd
0-5	2.5	3	7.5	-15	-45
5-10	7.5	4	30.0	-10	-40
10-15	12.5	12	150.0	-5	-60
15-20	17.5	14	245.0	0	0
20-25	22.5	8	180.0	5	40
25-30	27.5	7	192.5	10	70
30-35	32.5	5	162.5	15	75
**Total**		\( \Sigma f = N = 53 \)	\( \Sigma fx = 967.5 \)		\( \Sigma fd = 40 \)

**By Direct method:**
The formula for the arithmetic mean is \( \overline{X} = \frac{\Sigma fx}{N} \).
Using the values from the table: \( \overline{X} = \frac{967.5}{53} \approx 18.25 \).
So, the arithmetic mean by the direct method is approximately 18.25 marks.

**By Shortcut method:**
The formula for the arithmetic mean is \( \overline{X} = A + \frac{\Sigma fd}{N} \).
Here, the assumed mean \( A = 17.5 \).
Using the values from the table: \( \overline{X} = 17.5 + \frac{40}{53} \approx 17.5 + 0.75 \approx 18.25 \).
So, the arithmetic mean by the shortcut method is approximately 18.25 marks. The small difference is due to rounding.
In simple words: For groups of numbers with ranges, first find the middle point of each range. Then, use either the direct method (multiply middle point by count, sum up, then divide by total count) or the shortcut method (pick an assumed average, find differences, multiply by count, sum up, then adjust the assumed average). Both methods should give similar results.

🎯 Exam Tip: When dealing with continuous series, remember to always calculate the mid-point of each class interval first. This mid-point then acts as the 'X' value for subsequent calculations.

 

Question 5. Explain the calculation of arithmetic mean in cumulative frequency distribution with an example.
Answer:
In a cumulative frequency distribution, the frequencies are given as 'less than' or 'more than' a certain value, meaning they are already added up. To calculate the arithmetic mean from such a distribution, the first step is to convert it back into a simple frequency distribution by finding the individual frequencies for each class interval. Once converted, the arithmetic mean can be calculated using the standard methods (Direct, Shortcut, or Step-deviation) as for a continuous series. This conversion is crucial for accurate average calculations.

**Example:** Calculate the arithmetic mean from the following cumulative frequency distribution:

Marks (Less Than)	5	10	15	20	25	30
No. of Students	4	16	30	38	45	50

**Solution Steps:**
First, we need to convert the 'Less Than' cumulative frequency distribution into a simple frequency distribution with class intervals and their corresponding frequencies. We assume the class interval size is 5 (e.g., 0-5, 5-10, etc.). We then calculate the mid-points (X) for each class interval, \( fx \) values, deviations \( d \), step-deviations \( d' \), \( fd \) and \( fd' \) values, taking an assumed mean \( A = 12.5 \) and common factor \( i = 5 \).

Obtained Marks (C.I.)	Cumulative Frequency (cf)	Frequency (f)	Mid-Value (X)	Fx	\(d = X - A\) (A = 12.5)	fd	\(d' = d/i\) (i = 5)	fd'
0-5	4	4	2.5	10.0	-10	-40	-2	-8
5-10	16	12	7.5	90.0	-5	-60	-1	-12
10-15	30	14	12.5	175.0	0	0	0	0
15-20	38	8	17.5	140.0	5	40	1	8
20-25	45	7	22.5	157.5	10	70	2	14
25-30	50	5	27.5	137.5	15	75	3	15
**Total**		\( \Sigma f = N = 50 \)		\( \Sigma fx = 710.0 \)		\( \Sigma fd = 85 \)		\( \Sigma fd' = 17 \)

**By Direct method:**
\( \overline{X} = \frac{\Sigma fx}{N} = \frac{710}{50} = 14.2 \).
**By Shortcut method:**
\( \overline{X} = A + \frac{\Sigma fd}{N} = 12.5 + \frac{85}{50} = 12.5 + 1.7 = 14.2 \).
**By Step-deviation method:**
\( \overline{X} = A + \frac{\Sigma fd'}{N} \times i = 12.5 + \frac{17}{50} \times 5 = 12.5 + \frac{85}{50} = 12.5 + 1.7 = 14.2 \).
All three methods consistently yield an arithmetic mean of 14.2, confirming the accuracy of the conversions and calculations.
In simple words: First, change the 'less than' or 'more than' counts into simple counts for each group. Then, find the middle point of each group. After that, you can use any average-finding method (direct, shortcut, or step-deviation) just like for regular grouped data. All methods should give the same average.

🎯 Exam Tip: The crucial first step with cumulative frequency distributions is always to convert them into simple frequency distributions. If this step is incorrect, all subsequent calculations will be wrong.

 

Question 6. What is the step-deviation method of calculating arithmetic mean? When is this method used? Explain it with an example.
Answer:
The step-deviation method is an advanced version of the shortcut method used to calculate the arithmetic mean. It is particularly useful when the deviations (differences from the assumed mean) are large and share a common factor, making calculations simpler. This method reduces the size of numbers involved, which helps prevent calculation errors and saves time.

**When is it used?**
This method is best used when:
1. The values in the series are very large.
2. The class intervals are of equal size, meaning there's a common factor (i) that can divide all the deviations.
3. It simplifies the shortcut method even further, especially in continuous series.

**The various steps are the following:**
1. **Assume a Mean (A):** Take any value (size) as the assumed mean (A). This value can be from within the data set or outside it.
2. **Find Deviations (dx):** Calculate the deviations of each value (X) from the assumed mean: \( dx = X - A \).
3. **Find Step-Deviations (d'x):** Divide each deviation (\( dx \)) by a common factor (i) that is present in all \( dx \) values. This new value is called the step-deviation, \( d'x = dx / i \).
4. **Multiply Step-Deviations by Frequency (fd'x):** Multiply each step-deviation (\( d'x \)) by its corresponding frequency (f) to get \( fd'x \). Pay attention to the positive and negative signs.
5. **Sum the Products ( \( \Sigma fd'x \) ):** Add up all the \( fd'x \) values to get \( \Sigma fd'x \).
6. **Apply Formula:** Finally, apply the formula for the arithmetic mean \( (\overline{X}) \):
\( \overline{X} = A + \frac{\Sigma fd'x}{N} \times i \), where \( N \) is the sum of frequencies \( (\Sigma f) \).

**Example:** Calculate the arithmetic mean by the step-deviation method from the following series:

Wages (In Rs)	0-10	10-20	20-30	30-40	40-50
No. of Worker	8	12	20	6	4

**Solution:**
Let's calculate the mid-value (X) for each class. Assume mean \( A = 25 \) and common factor \( i = 10 \).

Wages (in Rs)	Mid-value(x)	Frequency(f) No. of Workers	\(d = x - A\) (A=25)	\(d' = d/i\) (i=10)	fd'
0-10	5	8	-20	-2	-16
10-20	15	12	-10	-1	-12
20-30	25	20	0	0	0
30-40	35	6	10	1	6
40-50	45	4	20	2	8
**Total**		\( \Sigma f = N = 50 \)			\( \Sigma fd' = -14 \)

Using the Step-deviation formula: \( \overline{X} = A + \frac{\Sigma fd'}{N} \times i = 25 + \frac{-14}{50} \times 10 = 25 + \frac{-140}{50} = 25 - 2.8 = 22.2 \).
The arithmetic mean calculated using the step-deviation method is Rs 22.2. This illustrates how the method simplifies calculations with a common factor.
In simple words: The step-deviation method makes finding the average easier for big numbers. You first pick an assumed average, then find how much each number differs. If these differences can all be divided by a common number, you divide them to make them smaller. Then you use a special formula with these smaller numbers to get the average. It helps when numbers are large or repeated.

🎯 Exam Tip: Ensure that the common factor 'i' you choose actually divides all deviations completely. A wrong common factor will lead to incorrect step-deviations and an inaccurate final mean.

 

Question 7. What is meant by weighted average mean? When is this method used ? Explain it with an example.
Answer:
**Meaning of Weighted Arithmetic Mean:**
The weighted arithmetic mean is a type of average where each value in a data set is given a different level of importance or 'weight'. Unlike the simple arithmetic mean, which assumes all values are equally important, the weighted mean acknowledges that some items contribute more significantly than others. For example, in calculating a student's average grade, the final exam might have a higher weight (importance) than a regular quiz. The mean calculated on the basis of these weights is called the weighted arithmetic mean.

**When is this method used?**
This method is used in situations where:
1. The relative importance of different items in a series is not equal.
2. We need a more accurate representation of the average, considering the varying significance of each data point.
3. Common applications include calculating average prices, average profits, birth rates, death rates, and index numbers.

**Calculating Weighted Arithmetic Mean:**

**A. Direct Method:**
1. **Multiply Value by Weight (XW):** Multiply each item's value (X) by its assigned weight (W) to get the product (XW).
2. **Sum the Products ( \( \Sigma XW \) ):** Add up all these products to get \( \Sigma XW \).
3. **Sum the Weights ( \( \Sigma W \) ):** Add all the weights together to get \( \Sigma W \).
4. **Apply Formula:** Use the formula: \( \overline{X}_w = \frac{\Sigma XW}{\Sigma W} \).

**B. Shortcut Method:**
1. **Assume a Mean (A):** Take any value (size) as an assumed mean (A).
2. **Find Deviations (dx):** Calculate the deviations of each value (X) from the assumed mean: \( dx = X - A \).
3. **Multiply Deviations by Weight (Wdx):** Multiply each deviation (\( dx \)) by its respective weight (W) to get \( Wdx \). Pay attention to signs.
4. **Sum the Products ( \( \Sigma Wdx \) ):** Add up all the \( Wdx \) values to get \( \Sigma Wdx \).
5. **Apply Formula:** The weighted arithmetic mean is: \( \overline{X}_w = A + \frac{\Sigma Wdx}{\Sigma W} \).

**Example:** Find out the weighted average mean by direct and short-cut method with the help of the given series:

Category of Employees	Monthly Salary (In Rs)	No. of Employees (Weight)
Principal	42,000	1
Senior Lecturer	38,000	5
Junior Lecturer	24,000	8
Teacher	17,000	12
Clerk	12,000	5
Peon	7,000	8

**Solution Steps:**
Let's calculate the products (XW) and deviations (dx, Wdx) as shown in the table below. We choose an assumed mean \( A = 24,000 \).

Category of Employees	X (Monthly Salary in Rs)	W (No. of Employees)	XW	\(dx = X - A\) (A=24,000)	Wdx
Principal	42,000	1	42,000	18000	18000
Senior Lecturer	38,000	5	1,90,000	14000	70000
Junior Lecturer	24,000	8	1,92,000	0	0
Teacher	17,000	12	2,04,000	-7000	-84000
Clerk	12,000	5	60,000	-12000	-60000
Peon	7,000	8	56,000	-17000	-136000
**Total**		\( \Sigma W = 39 \)	\( \Sigma XW = 7,44,000 \)		\( \Sigma Wdx = -1,92,000 \)

**Direct Method:**
\( \overline{X}_w = \frac{\Sigma XW}{\Sigma W} = \frac{7,44,000}{39} \approx 19,076.92 \) Rs.

**Short-cut Method:**
\( \overline{X}_w = A + \frac{\Sigma Wdx}{\Sigma W} = 24,000 + \frac{-1,92,000}{39} = 24,000 - 4,923.08 \approx 19,076.92 \) Rs.
Both the direct and shortcut methods give a weighted average mean of approximately Rs 19,076.92, showing that both approaches yield consistent results.
In simple words: Weighted average mean means finding the average where some things are more important than others (like how much each type of employee contributes to total salary). You add up each salary multiplied by its importance (number of employees) and divide by the total importance. It helps get a truer average when items are not equally valuable.

🎯 Exam Tip: When calculating the weighted mean, clearly identify the values (X) and their corresponding weights (W). Ensure that all calculations, especially those involving negative deviations, are meticulously checked.

 

Question 1. Calculate the arithmetic mean from the data obtained in statistics from the 10 students of an exam:

Enrolment no.	1	2	3	4	5	6	7	8	9	10
Obtained Marks	16	24	18	48	36	28	32	41	23	12

Answer:
This is an individual series. To calculate the arithmetic mean, we sum all the obtained marks (\( \Sigma X \)) and divide by the number of students (N).

Enrolment	Obtained Marks (X)
1	16
2	24
3	18
4	48
5	36
6	28
7	32
8	41
9	23
10	12
**Total**	\( \Sigma X = 278 \)

Number of students, \( N = 10 \).
Sum of marks, \( \Sigma X = 278 \).
The formula for arithmetic mean \( \overline{X} = \frac{\Sigma X}{N} \).
\( \overline{X} = \frac{278}{10} = 27.8 \).
Thus, the arithmetic mean of the obtained marks is 27.8.
In simple words: To find the average marks, add up all the marks received by the 10 students, and then divide that total by the number of students, which is 10.

🎯 Exam Tip: For individual series, ensure every single data point is included in the sum. A common mistake is to miss one or two values, leading to an incorrect total.

 

Question 2. Calculate arithmetic mean from the following data:

Item Value	5	15	25	35	45
Frequency	4	7	12	4	3

Answer: To find the arithmetic mean, we first need to calculate the sum of all item values multiplied by their frequencies ( \( \Sigma fx \) ) and the total number of items ( \( N \) or \( \Sigma f \) ).

Item-Value	Frequency	Direct Method	Indirect Method
			\( D = X - A \) (A = 25)	fd
5	4	20	-20	-80
15	7	105	-10	-70
25	12	300	0	0
35	4	140	10	40
45	3	135	20	60
\( N = 30 \)		\( \Sigma fx = 700 \)		\( \Sigma fd = -50 \)

Answer (Continuation):

Using the Direct Method formula:

\( \overline{X} = \frac{\Sigma fx}{N} \)

\( \overline{X} = \frac{700}{30} \)

\( \overline{X} = 23.33 \)

Using the Shortcut Method formula:

\( \overline{X} = A + \frac{\Sigma fd}{N} \)

\( \overline{X} = 25 + \frac{-50}{30} \)

\( \overline{X} = 25 - 1.67 \)

\( \overline{X} = 23.33 \)
In simple words: The arithmetic mean is calculated by multiplying each item value by its frequency, adding these products, and then dividing by the total number of items. This gives the average value for the entire dataset.

🎯 Exam Tip: Remember to choose an assumed mean (A) that simplifies calculations, often a value near the center of your data. This makes working with deviations easier.

 

Question 3. Calculate arithmetic mean from the following series:

Wages (In Rs)	0-10	10-20	20-30	30-40	40-50
No. of Workers	8	12	20	6	4

Answer: To calculate the arithmetic mean for this series, we first find the mid-value for each wage group. Then, we use the direct and indirect methods.

Wages (In Rs)	Mid-Value (X)	No. of Workers (f)	Direct Method	Indirect Method
				\( d = X - A \) (A = 25)	fd
0-10	5	8	40	-20	-160
10-20	15	12	180	-10	-120
20-30	25	20	500	0	0
30-40	35	6	210	10	60
40-50	45	4	180	20	80
		\( \Sigma N = 50 \)	\( \Sigma fx = 1110 \)		\( \Sigma fd = -140 \)

Answer (Continuation):

Arithmetic mean-

Direct Method -
\( \overline{X} = \frac{\Sigma fx}{N} = \frac{1110}{50} = 22.2 \)

Indirect Method -
\( \overline{X} = A + \frac{\Sigma fd}{N} = 25 + \frac{-140}{50} = 25 - 2.8 = 22.2 \)
In simple words: To find the average wage, we sum up all the wages (multiplied by how many workers earn them) and divide by the total number of workers. This method helps us get a central value that represents the wages of all workers.

🎯 Exam Tip: When dealing with grouped data, calculate the mid-point of each class interval first. This mid-point acts as the 'X' value for calculations, representing all values within that interval.

 

Question 4. Calculate arithmetic mean from the following series:

Item Value	1-5	6-10	11-15	16-20	21-25
Frequency	2	5	8	6	4

Answer: The arithmetic mean is calculated for this inclusive series by first finding the mid-point (X) for each class interval. Since the mid-value is similar in both inclusive and exclusive series for these intervals, no conversion is needed. Then, we proceed with the standard calculations.

Class Interval	Mid-Value X	Frequency f	fx
1-5	3	2	6
6-10	8	5	40
11-15	13	8	104
16-20	18	6	108
21-25	23	4	92
		\( \Sigma N = 25 \)	\( \Sigma fx = 350 \)

Answer (Continuation):

\( \overline{X} = \frac{\Sigma fx}{N} = \frac{350}{25} = 14 \)
In simple words: First, find the middle point of each group (like for 1-5, the middle is 3). Then, multiply this middle point by how many times it appears. Add all these results together and divide by the total count of all items to get the average.

🎯 Exam Tip: For inclusive series (e.g., 1-5, 6-10), the mid-point calculation is usually straightforward. If intervals overlap (e.g., 0-5, 5-10), ensure you use the correct class boundaries for mid-points.

 

Question 5. Find out arithmetic mean from the following series:

Obtained Marks (less than)	10	20	30	40	50	60	70	80
No. of Students	25	40	60	75	95	125	190	240

Answer: First, we convert the cumulative frequency distribution ("less than" series) into a simple frequency distribution by finding the frequency for each class interval. Then, we calculate the mid-value (X) for each class and use it to find the arithmetic mean.

Class Interval	Frequency (f)	Mid-Value (X)	fx
0-10	25	5	125
10-20	\( 40 - 25 = 15 \)	15	225
20-30	\( 60 - 40 = 20 \)	25	500
30-40	\( 75 - 60 = 15 \)	35	525
40-50	\( 95 - 75 = 20 \)	45	900
50-60	\( 125 - 95 = 30 \)	55	1650
60-70	\( 190 - 125 = 65 \)	65	4225
70-80	\( 240 - 190 = 50 \)	75	3750
	\( \Sigma f = 240 \)		\( \Sigma fx = 11900 \)

Answer (Continuation):

\( \overline{X} = \frac{\Sigma fx}{N} = \frac{11900}{240} = 49.58 \)
In simple words: When data is given as "less than" a certain mark, we first figure out how many students fall into each specific mark range. Then, we find the middle mark for each range. Multiplying the middle mark by the number of students in that range, adding them all up, and dividing by the total number of students gives us the average mark.

🎯 Exam Tip: Always convert cumulative frequency distributions ("less than" or "more than" series) into simple frequency distributions before calculating the mean. This involves finding the frequency for each specific class interval.

 

Question 6. Find out arithmetic mean by step-deviation method from the following information:

Obtained Marks (More Than)	0	10	20	30	40	50	60
No. of Students	50	42	36	25	16	10	6

Answer: First, we convert the cumulative frequency distribution ("more than" series) into a simple frequency distribution. Then, we use the step-deviation method which involves taking deviations from an assumed mean, dividing by a common factor (class interval), and then applying the formula.

Class Interval	Frequency (f)	Mid-Value (X)	\( d = X - A \) (A = 35)	\( d' = d/i \) (i = 10)	fd'
0-10	\( 50 - 42 = 8 \)	5	-30	-3	-24
10-20	\( 42 - 36 = 6 \)	15	-20	-2	-12
20-30	\( 36 - 25 = 11 \)	25	-10	-1	-11
30-40	\( 25 - 16 = 9 \)	35	0	0	0
40-50	\( 16 - 10 = 6 \)	45	10	1	6
50-60	\( 10 - 6 = 4 \)	55	20	2	8
60-70	6	65	30	3	18
	\( \Sigma f = 50 \)				\( \Sigma fd' = -15 \)

Answer (Continuation):

Formula for arithmetic mean using step-deviation method:

\( \overline{X} = A + \frac{\Sigma fd'}{N} \times i \)

\( \overline{X} = 35 + \frac{-15}{50} \times 10 \)

\( \overline{X} = 35 - 3 \)

\( \overline{X} = 32 \)
In simple words: To find the average marks, first convert the "more than" data into simple ranges and count students in each range. Then, pick a middle number (assumed mean), find how far each group's mid-point is from it, and divide that difference by the range width. Multiply these new values by student counts, add them up, and use the formula to get the final average.

🎯 Exam Tip: When using the step-deviation method, correctly identifying the class interval (i) for division is crucial. Ensure you use the consistent width of the classes in your series.

 

Question 7. Find the arithmetic mean of the following data showing marks obtained by the student in a statistics exam:

Mid Marks (X)	5	10	15	20	25	30	35	40
No. of Students	5	7	9	10	8	6	3	2

Answer: To find the arithmetic mean, we multiply the mid-marks (X) by the number of students (f) for each category. Then, we sum these products ( \( \Sigma fx \) ) and divide by the total number of students ( \( \Sigma f \) ).

Mid Marks (X)	No. of Students (f)	Fx
5	5	25
10	7	70
15	9	135
20	10	200
25	8	200
30	6	180
35	3	105
40	2	80
	\( \Sigma f = 50 \)	\( \Sigma fx = 995 \)

Answer (Continuation):

\( \overline{X} = \frac{\Sigma fx}{N} = \frac{995}{50} = 19.9 \)
In simple words: To find the average marks for students, we multiply each mark by how many students got it. Then, we add all those results together and divide by the total number of students. This gives us the overall average mark.

🎯 Exam Tip: Always make sure to accurately sum \( \Sigma fx \) and \( \Sigma f \) as any error in these sums will lead to an incorrect arithmetic mean. Double-check your multiplication and addition steps.

 

Question 8. Calculate weighted mean from the following given data:

Item- Value (X)	15	18	21	25	32	40
Weight	6	5	4	3	2	1

Answer: To calculate the weighted mean, we multiply each item's value by its assigned weight to get the weighted products (XW). Then, we sum these weighted products ( \( \Sigma XW \) ) and divide by the sum of all weights ( \( \Sigma W \) ).

Item- Value (X)	Weight (W)	XW
15	6	90
18	5	90
21	4	84
25	3	75
32	2	64
40	1	40
	\( \Sigma W = 21 \)	\( \Sigma XW = 443 \)

Answer (Continuation):

Formula for weighted mean:

\( \overline{X}_w = \frac{\Sigma XW}{\Sigma W} \)

\( \overline{X}_w = \frac{443}{21} \)

\( \overline{X}_w \approx 21.095 \)
In simple words: When some items are more important than others, we give them "weights." To find the weighted average, we multiply each item by its weight, add up these weighted values, and then divide by the total of all the weights. This makes sure important items count more.

🎯 Exam Tip: The key to a correct weighted mean is ensuring each item's value is multiplied by its *specific* weight, not just its frequency. This is used when different values have different levels of significance.