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Chapter 23: Problem 2

Find the variance and standard deviation of the following frequency distribution: $$ \begin{array}{rr} \hline x & f \\ \hline 6 & 7 \\ 7 & 3 \\ 8 & 2 \\ 9 & 4 \\ 10 & 2 \\ \hline \end{array} $$

Short Answer

Expert verified
Answer: The variance of the given frequency distribution is approximately 2.139, and the standard deviation is approximately 1.462.

Step by step solution

01

Calculate the mean

To calculate the mean, we first need to find the sum of the products of each value (x) and its frequency (f), and then divide by the total frequency. The mean can be found using the formula: $$ \bar{x} = \frac{\sum f \times x}{\sum f} $$ First, calculate the total frequency: $$ \sum f = 7 + 3 + 2 + 4 + 2 = 18 $$ Next, calculate the sum of the products of each value and its frequency: $$ \sum f \times x = (6 \times 7) + (7 \times 3) + (8 \times 2) + (9 \times 4) + (10 \times 2) = 42 + 21 + 16 + 36 + 20 = 135 $$ Now find the mean of the distribution: $$ \bar{x} = \frac{135}{18} = 7.5 $$
02

Find the deviation of each value from the mean

Subtract the mean from each value of x to find the deviations: $$ x - \bar{x} = [6-7.5, 7-7.5, 8-7.5, 9-7.5, 10-7.5] = [-1.5, -0.5, 0.5, 1.5, 2.5] $$
03

Square each deviation

Square each of the deviations to remove the negative values and emphasize larger deviations: $$ (x - \bar{x})^2 = [2.25, 0.25, 0.25, 2.25, 6.25] $$
04

Multiply the squared deviations by their corresponding frequencies

Multiply each squared deviation by its corresponding frequency: $$ f \times (x - \bar{x})^2 = [7 \times 2.25, 3 \times 0.25, 2 \times 0.25, 4 \times 2.25, 2 \times 6.25] = [15.75, 0.75, 0.50, 9, 12.5] $$
05

Sum up the products from step 4

Sum up the products obtained from step 4: $$ \sum f \times (x - \bar{x})^2 = 15.75 + 0.75 + 0.50 + 9 + 12.5 = 38.5 $$
06

Divide the sum by the total frequency to find the variance

Divide the sum obtained in step 5 by the total frequency to find the variance: $$ \text{Variance} = \frac{38.5}{18} \approx 2.139 $$
07

Take the square root of the variance to find the standard deviation

Take the square root of the variance to find the standard deviation: $$ \text{Standard Deviation} = \sqrt{2.139} \approx 1.462 $$ The variance of the given frequency distribution is approximately 2.139, and the standard deviation is approximately 1.462.

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