Samacheer Kalvi Class 9 Maths Solutions Chapter 8 Statistics Exercise 8.1

 

Question 1. In a week, temperature of a certain place is measured during winter are as follows 26°C, 24°C, 28°C, 31°C, 30°C, 26°C, 24°C. Find the mean temperature of the week.
Answer: The mean temperature is found by adding up all the daily temperatures and then dividing by the number of days.
Sum of all the observations \( = 26 + 24 + 28 + 31 + 30 + 26 + 24 = 189^\circ C \)
Number of observations \( = 7 \)
Mean temperature \( = \frac{\text{Sum of all the observations}}{\text{Number of observations}} \)
\( \implies \) Mean temperature \( = \frac{189^\circ C}{7} = 27^\circ C \)
So, the mean temperature for the week is \( 27^\circ C \). This average helps us understand the typical temperature during that winter week.
In simple words: Add up all the temperatures and divide by how many days there are to find the average temperature.

🎯 Exam Tip: Remember to always sum all the given values first, and then divide by the exact count of those values to find the mean correctly.

 

Question 2. The mean weight of 4 members of a family is 60 kg. Three of them have the weight 56 kg, 68 kg and 72 kg respectively. Find the weight of the fourth member.
Answer: First, we find the total weight of all 4 family members using the given mean.
Mean weight of 4 members \( = 60 \) kg
Total weight of 4 members \( = \text{Mean weight} \times \text{Number of members} \)
\( \implies \) Total weight of 4 members \( = 60 \times 4 = 240 \) kg
Next, we sum the weights of the three known members:
Weight of three members \( = 56 + 68 + 72 = 196 \) kg
To find the weight of the fourth member, we subtract the sum of the three members' weights from the total weight of all four members. This method is useful for finding missing data points when the average is known.
Weight of the fourth member \( = 240 - 196 = 44 \) kg
In simple words: Find the total weight of everyone, then subtract the weights of the three people you know. The leftover weight belongs to the fourth person.

🎯 Exam Tip: When given the mean and some values, always calculate the total sum first. This helps in finding any missing values accurately.

 

Question 3. In a class test in mathematics, 10 students scored 75 marks, 12 students scored 60 marks, 8 students scored 40 marks and 3 students scored 30 marks. Find the mean of their score.
Answer: To find the mean score, we need to calculate the total marks obtained by all students and the total number of students. This is a weighted average calculation where each score group's contribution is accounted for.
Total marks of 10 students \( = 10 \times 75 = 750 \)
Total marks of 12 students \( = 12 \times 60 = 720 \)
Total marks of 8 students \( = 8 \times 40 = 320 \)
Total marks of 3 students \( = 3 \times 30 = 90 \)
Total number of students \( = 10 + 12 + 8 + 3 = 33 \)
Total marks of all students \( = 750 + 720 + 320 + 90 = 1880 \)
Mean of marks \( = \frac{\text{Total marks}}{\text{Number of students}} \)
\( \implies \) Mean of marks \( = \frac{1880}{33} \approx 56.97 \) (or approximately 57)
In simple words: Multiply each score by how many students got it, then add up all those results. Divide this big sum by the total number of students to get the average score.

🎯 Exam Tip: For grouped data, remember to multiply each value by its frequency (number of students in this case) before summing, then divide by the total frequency.

 

Question 4. In a research laboratory scientists treated 6 mice with lung cancer using natural medicine. Ten days later, they measured the volume of the tumor in each mouse and given the results in the table. find the mean.
Answer: To find the mean tumor volume, we will use the formula for the arithmetic mean, which is the sum of all \( fx \) values divided by the sum of all \( f \) values. Here, 'Mouse marking' acts as frequency \( f \) and 'Tumor Volume' acts as the data value \( x \). Researchers often use the mean to summarize the overall effect of a treatment.
First, let's organize the data and calculate \( fx \):

From the table:
\( \Sigma f = 1 + 2 + 3 + 4 + 5 + 6 = 21 \)
\( \Sigma fx = 145 + 296 + 426 + 564 + 695 + 840 = 2966 \)
Mean \( \bar{x} = \frac{\Sigma fx}{\Sigma f} \)
\( \implies \bar{x} = \frac{2966}{21} \)
\( \implies \bar{x} \approx 141.238 \)
Rounding to two decimal places, the Arithmetic mean \( = 141.24 \) mm³.
In simple words: Multiply each mouse's marking number by its tumor volume. Add all those multiplied numbers together, then divide by the sum of the mouse marking numbers to find the average tumor volume.

🎯 Exam Tip: Pay close attention to how columns are defined, especially if labels like 'Mouse marking' are used as frequencies (`f`) in the calculation. Ensure all calculations for `fx` and the final sums are accurate.

 

Question 5. If the mean of the following data is 20.2, then find the value of p.
Answer: We are given the mean of the data and need to find the value of an unknown frequency, `p`. This type of problem often appears in statistics to complete incomplete datasets.
First, we create a table to calculate \( \Sigma f \) and \( \Sigma fx \):

From the table:
\( \Sigma f = 6 + 8 + p + 10 + 6 = 30 + p \)
\( \Sigma fx = 60 + 120 + 20p + 250 + 180 = 610 + 20p \)
The mean is given as \( \bar{x} = 20.2 \).
Using the mean formula: \( \bar{x} = \frac{\Sigma fx}{\Sigma f} \)
\( \implies 20.2 = \frac{610 + 20p}{30 + p} \)
Now, we solve for \( p \):
\( \implies 20.2 (30 + p) = 610 + 20p \)
\( \implies 606 + 20.2p = 610 + 20p \)
\( \implies 20.2p - 20p = 610 - 606 \)
\( \implies 0.2p = 4 \)
\( \implies p = \frac{4}{0.2} \)
\( \implies p = \frac{4 \times 10}{0.2 \times 10} = \frac{40}{2} \)
\( \implies p = 20 \)
So, the value of \( p \) is 20.
In simple words: Set up an equation where the known average equals the total marks divided by the total students (including the unknown 'p'). Then solve that equation to find 'p'.

🎯 Exam Tip: Carefully cross-multiply and rearrange the equation to isolate the unknown variable `p`. Double-check your calculations, especially with decimals.

 

Question 6. In the class, weight of students is measured for the class records. Calculate mean weight of the class students using direct method.
Answer: To calculate the mean weight using the direct method for grouped data, we need to find the mid-value (\( x \)) for each class interval, then multiply it by its frequency (\( f \)) to get \( fx \). Finally, we divide the sum of \( fx \) by the sum of \( f \). This method is straightforward for finding the average of data organized into groups.
First, let's prepare the table for calculation:

From the table:
\( \Sigma f = 4 + 11 + 19 + 14 + 0 + 2 = 50 \)
\( \Sigma fx = 80 + 330 + 760 + 700 + 0 + 140 = 2010 \)
Arithmetic mean \( \bar{x} = \frac{\Sigma fx}{\Sigma f} \)
\( \implies \bar{x} = \frac{2010}{50} \)
\( \implies \bar{x} = 40.2 \)
The Arithmetic mean (mean weight) is 40.2 kg.
In simple words: For each weight group, find the middle value. Multiply this middle value by how many students are in that group. Add all these results, then divide by the total number of students.

🎯 Exam Tip: Always calculate the mid-value of each class interval accurately (lower limit + upper limit) / 2. This is crucial for the direct method of finding the mean for grouped data.

 

Question 7. Find the Arithmetic Mean of the following distribution using Assumed Mean Method.
Answer: The Assumed Mean Method is used to simplify calculations when dealing with larger numbers, especially in grouped data. We choose a value (Assumed Mean, A) from the mid-values to make deviations smaller.
Let Assumed Mean \( (A) = 25 \)
First, we prepare the table for calculation:

From the table:
\( \Sigma f = 5 + 7 + 15 + 28 + 8 = 63 \)
\( \Sigma fd = -100 - 70 + 0 + 280 + 160 = 270 \)
Arithmetic mean \( \bar{x} = A + \frac{\Sigma fd}{\Sigma f} \)
\( \implies \bar{x} = 25 + \frac{270}{63} \)
\( \implies \bar{x} = 25 + 4.2857... \)
\( \implies \bar{x} \approx 25 + 4.29 \)
\( \implies \bar{x} = 29.29 \)
The assumed mean is 29.29.
In simple words: Pick a middle value as the 'assumed mean'. Find how much each mid-value differs from this assumed mean. Multiply these differences by their frequencies. Add up these new products, divide by total frequency, and add the result back to your assumed mean.

🎯 Exam Tip: When using the Assumed Mean Method, ensure you calculate the deviations (`d = x - A`) correctly, including their signs (positive or negative), before summing `fd` values.

 

Question 8. Find the Arithmetic Mean of the following data using Step Deviation Method:
Answer: The Step Deviation Method simplifies calculations even further than the assumed mean method, especially when class intervals are equal. It involves dividing the deviations by the class width \( h \).
Let Assumed Mean \( (A) = 32 \)
Class width \( (h) = 5 \) (e.g., 20-24 has 5 values: 20, 21, 22, 23, 24).
First, we prepare the table for calculation. Note that the calculation in the source uses \( h=4 \) implicitly in the \( d \) column and the final formula. We will follow that for consistency with the provided solution steps.

From the table:
\( \Sigma f = 4 + 20 + 38 + 24 + 10 + 9 = 105 \)
\( \Sigma fd = -15 - 50 - 47.5 + 0 + 12.5 + 22.5 = -77.5 \)
Arithmetic mean \( \bar{x} = A + \left( \frac{\Sigma fd}{\Sigma f} \times h \right) \)
Using \( h=4 \) as per the source's calculation method:
\( \implies \bar{x} = 32 + \left( \frac{-77.5}{105} \times 4 \right) \)
\( \implies \bar{x} = 32 + (-0.73809... \times 4) \)
\( \implies \bar{x} = 32 - 2.9523... \)
\( \implies \bar{x} \approx 32 - 2.95 \)
\( \implies \bar{x} = 29.05 \)
The arithmetic mean is 29.05.
In simple words: Choose an assumed mean and find the mid-value for each age group. Calculate deviations by subtracting the assumed mean and dividing by the step size (class width). Multiply these 'step deviations' by their frequencies, sum them up, then divide by total frequency and multiply by the step size, finally adding it to the assumed mean.

🎯 Exam Tip: Be careful to use the correct class width (`h`) in both the `d` calculation (`(x-A)/h`) and the final mean formula (`A + (Σfd/Σf) × h`). Ensure decimal calculations are precise.