Samacheer Kalvi Class 8 Maths Solutions Chapter 6 Statistics Exercise 6.1

 

Question 1. Fill in the blanks:
(i) Data has already been collected by some other person is _______ data.
(ii) The upper limit of the class interval (25-35) is _______.
(iii) The range of the data 200, 15, 20, 103, 3, 196, is _______.
(iv) If a class size is 10 and range is 80 then the number of classes are _______.
(v) Pie chart is a _______ graph.
Answer:
(i) Data that has already been gathered by someone else is called **Secondary** data.
(ii) For the class interval (25-35), the upper limit is **35**. The upper limit is the higher number in the interval.
(iii) The range of the data set (200, 15, 20, 103, 3, 196) is **197**. We find this by subtracting the smallest value (3) from the largest value (200).
(iv) If the class size is 10 and the range is 80, then the number of classes is **8**. This is found by dividing the range by the class size (80 / 10).
(v) A pie chart is a **circular** graph. It uses a circle divided into sectors to show parts of a whole.
In simple words: Secondary data comes from others. The upper limit is the bigger number in a range. Range is the biggest number minus the smallest. To find the number of classes, divide the range by the class size. A pie chart is shaped like a circle.

🎯 Exam Tip: Remember key definitions in statistics: primary vs. secondary data, class limits, and how to calculate range and number of classes for grouping data. Understanding these basics is crucial for higher-level topics.

 

Question 2. Say True or False:
(i) Inclusive series is a continuous series.
(ii) Comparison of parts of a whole may be done by a pie chart.
(iii) Media and business people use pie charts.
(iv) A pie diagram is a circle broken down into component sectors.
Answer:
(i) **False**. An inclusive series means that both the lower and upper limits are included in the class interval, and there are gaps between intervals. A continuous series has no gaps between class intervals. For instance, in a continuous series, the upper limit of one class is the lower limit of the next, like 0-10, 10-20.
(ii) **True**. A pie chart is an excellent tool for showing how different parts contribute to a whole, as each sector represents a proportion of the total.
(iii) **True**. Pie charts are widely used in media and business because they visually represent proportions effectively, making complex data easier to understand at a glance.
(iv) **True**. A pie diagram, or pie chart, is indeed a circle divided into sectors, where each sector's size shows the proportion of a particular category to the total.
In simple words: Inclusive series has gaps, so it's not continuous. Pie charts are good for comparing parts of a whole. Many people in business and media use pie charts. A pie diagram divides a circle into parts to show data.

🎯 Exam Tip: Know the difference between inclusive and exclusive (or continuous) class intervals. For true/false questions, think about the primary purpose of each statistical tool or definition.

 

Question 3. Represent the following data in ungrouped frequency table which gives the number of children in 25 families.
1, 3, 0, 2, 5, 2, 3, 4, 1, 0, 5, 4, 3, 1, 3, 2, 5, 2, 1, 1, 2, 6, 2, 1, 4
Answer:
First, we arrange the given raw data in ascending order. This makes it easier to count frequencies.
Ascending order: 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6
Now, we tabulate this data into an ungrouped frequency distribution table, showing the number of children, tally marks for each occurrence, and the total frequency.

In simple words: First, list all the numbers in order from smallest to biggest. Then, count how many times each number appears. We use tally marks to keep track, and the final count is the frequency. Summing all frequencies should give the total number of families (25).

🎯 Exam Tip: When creating a frequency table, always arrange the raw data first. Use tally marks carefully to avoid errors, and ensure the sum of frequencies matches the total number of observations given in the question.

 

Question 4. Form a continuous frequency distribution table for the marks obtained by 30 students in a public examination.
320, 470, 405, 375, 298, 326, 276, 362, 410, 255, 391, 370, 455, 229, 300, 183, 283, 366, 400, 495, 215, 157, 374, 306, 280, 409, 321, 269, 398, 200.
Answer:
To create a continuous frequency distribution table, we first need to find the range of the data and decide on a class size. We then determine the number of classes and define the class intervals.
Maximum mark obtained = 495
Minimum marks obtained = 157
Range = Maximum value - Minimum value
\( \implies \) Range = 495 - 157
\( \implies \) Range = 338
If we choose the class size as 50, then we can find the number of class intervals possible.
Number of class intervals = \( \frac{\text{Range}}{\text{Class size}} \)
\( \implies \) Number of class intervals = \( \frac{338}{50} \)
\( \implies \) Number of class intervals = 6.76
Since the number of classes must be a whole number, we round this up to 7. This means we will have 7 class intervals.
We will start the first class interval from a number slightly less than the minimum mark (157) and make it an exclusive series so it's continuous.

In simple words: First, find the highest and lowest marks to get the range. Then, choose a class size (like 50) and divide the range by it to see how many groups (classes) you need. Round up if it's not a whole number. After that, make intervals like 150-200, 200-250, and count how many students fall into each group. Remember, for continuous tables, the upper limit of one class becomes the lower limit of the next.

🎯 Exam Tip: When forming a continuous frequency table, ensure the class intervals are mutually exclusive and exhaustive. This means no data point falls into two intervals, and all data points are covered. Always consider rounding the number of classes up to accommodate all data.

 

Question 5. A paint company asked a group of students about their favourite colours and made a pie chart of their findings. Use the information to answer the following questions.
(i) What percentage of the students like red colour?
(ii) How many students liked green colour?
(iii) What fraction of the students liked blue?
(iv) How many students did not like red colour?
(v) How many students liked pink or blue?
(vi) How many students were asked about their favourite colours?
Answer:
Based on the data inferred from the solution, we can determine the number of students. The percentages for favorite colors are: Pink (30%), Red (20%), Blue (25%), Green (15%). The remaining 10% (which equals 50 students) represent other colors. This implies a total of 500 students were surveyed.
Total percentage of students = 100 %
We are given that 50 students represent 10% of the total students. We can use this to find the total number of students.
\( \implies \) Total students = \( \frac{50 \times 100}{10} = 500 \).
So, 500 students were asked about their favourite colors.

(i) What percentage of the students like red colour?
From the implied data, 20% of the students liked red colour.

(ii) How many students liked green colour?
15% of the students liked green colour.
Number of students who liked green = \( \frac{15}{100} \times 500 = 75 \) students.

(iii) What fraction of the students liked blue?
25% of the students liked blue colour.
Fraction of students who liked blue = \( \frac{25}{100} = \frac{1}{4} \).

(iv) How many students did not like red colour?
Percentage of students who did not like red colour = 100% - 20% = 80%.
Number of students who did not like red = \( \frac{80}{100} \times 500 = 400 \) students.

(v) How many students liked pink or blue?
Percentage of students who liked pink = 30%.
Percentage of students who liked blue = 25%.
Total percentage who liked pink or blue = 30% + 25% = 55%.
Number of students who liked pink or blue = \( \frac{55}{100} \times 500 = 275 \) students.

(vi) How many students were asked about their favourite colours?
Total number of students asked = 500 students.
In simple words: First, we figured out that 500 students were asked in total, because 50 students made up 10% of the group. Then, we used the given percentages for each color to answer the questions. For example, 20% liked red, and to find the number for green, we calculated 15% of 500. To find a fraction, we simplified the percentage. If students didn't like red, it's 100% minus the percentage who did.

🎯 Exam Tip: When analyzing pie charts or percentage-based data, always find the total quantity first if it's not directly given. Convert percentages to numbers or fractions carefully for each part of the question.

 

Question 6. A survey gives the following information of food items preferred by people. Draw a Pie chart.

Answer:
To draw a pie chart, we first need to find the total number of people surveyed and then calculate the central angle for each food item. The central angle represents the proportion of each item out of the total 360 degrees in a circle.
Total number of people = 160 + 90 + 80 + 50 + 30 + 40 = 450
The formula for the central angle of a component is: \( \text{Central angle} = \frac{\text{Value of the component}}{\text{Total value}} \times 360^\circ \).
Let's calculate the central angle for each food item:

Now, we can draw the pie chart using these calculated central angles.

In simple words: First, add up all the numbers to get the total. Then, for each item, figure out its "slice" of a 360-degree circle by dividing its number by the total and multiplying by 360. Make sure all your angles add up to 360. Finally, draw a circle and divide it into these slices using a protractor, labeling each slice with its item and angle.

🎯 Exam Tip: Always calculate the central angles accurately. A common mistake is not getting the total angle to exactly 360 degrees. Use a clear legend to identify each sector in your pie chart.

 

Question 7. Income from various sources for Government of India from a rupee is given below. Draw a pie chart.

Answer:
To draw a pie chart for the government's income sources, we first need to calculate the central angle for each source based on its proportion of the total 100 paise (which represents 1 rupee).
Total income = 100 paise.
The formula for the central angle of a component is: \( \text{Central angle} = \frac{\text{Value of the component}}{\text{Total value}} \times 360^\circ \).
Let's calculate the central angle for each income source:

Now, we can draw the pie chart using these calculated central angles.

In simple words: First, find out the total income from all sources. Then, for each source, calculate its part of the full 360-degree circle by taking its income value, dividing it by the total income, and multiplying by 360. Make sure the total of all angles is 360 degrees. Finally, draw a circle and divide it into these calculated slices, labelling each slice with its source and angle.

🎯 Exam Tip: When dealing with percentage or proportion-based pie charts, remember that 100% or the total value corresponds to 360 degrees. Ensure all angles are calculated precisely and that their sum is exactly 360 degrees to avoid errors in the drawing.

 

Question 8. expenditure of Kumaran's family is given below. Draw a suitable Pie chart. Also, answer the following questions:
1. Find the amount spent for education if Kumaran spends Rs. 6000 for Rent.
2. What is the total salary of Kumaran?
3. How much did he spend more for food than education?

Answer:
First, we calculate the central angle for each expenditure category to draw the pie chart. The total expenditure represents 100% or 360 degrees.
The formula for the central angle of a component is: \( \text{Central angle} = \frac{\text{Value of the component}}{\text{Total value}} \times 360^\circ \).
Let's calculate the central angle for each category:

Now, we can draw the pie chart using these calculated central angles.

Now, let's answer the sub-questions:

1. Find the amount spent for education if Kumaran spends Rs. 6000 for Rent.
From the table, Rent is 15% of the total expenditure.
So, 15% of Total Expenditure = Rs. 6000
Total Expenditure = \( \frac{6000 \times 100}{15} \)
\( \implies \) Total Expenditure = Rs. 40,000
Education is 20% of the total expenditure.
Amount spent for education = 20% of Rs. 40,000
\( \implies \) Amount spent for education = \( \frac{20}{100} \times 40,000 \)
\( \implies \) Amount spent for education = Rs. 8000

2. What is the total salary of Kumaran?
Assuming that the total expenditure equals the total salary.
Total salary of Kumaran = Total Expenditure = Rs. 40,000.

3. How much did he spend more for food than education?
Amount spent for food = 50% of Rs. 40,000
\( \implies \) Amount spent for food = \( \frac{50}{100} \times 40,000 \)
\( \implies \) Amount spent for food = Rs. 20,000
Amount spent for education = 20% of Rs. 40,000 = Rs. 8000 (calculated in part 1).
Difference = Amount spent for food - Amount spent for education
\( \implies \) Difference = Rs. 20,000 - Rs. 8000
\( \implies \) Difference = Rs. 12,000
So, Kumaran spent Rs. 12,000 more on food than on education.
In simple words: First, we drew the pie chart by finding each expense's slice of the circle. Then, using the fact that Rent (15%) was Rs. 6000, we found the total money spent was Rs. 40,000. This is also his total salary. To find the money spent on education, we took 20% of the total. Finally, we compared the money spent on food (50% of total) and education to find the difference.

🎯 Exam Tip: When solving problems involving pie charts and specific amounts, always determine the total amount first if it's not given. Then, use percentages to calculate individual amounts and answer specific questions accurately.