Chapter 12.3, Problem 23HP

To determine

To calculate : the mean and standard deviation of the given population of data represented by the histogram.

Answer to Problem 23HP

The mean of the given data is $\underset{\_}{4}$ .

The standard deviation of the data is $\underset{\_}{1.9}$ .

Explanation of Solution

Given information :

Consider the histogram given in the question:

  

Formula used :

Mean, $\overline{x}=\frac{\Sigma f_i{x}_i}{\Sigma f_i}$ , where $f_i$ is the frequency of each class interval and $x_i$ is the mid-point of each class.

Variance, ${\sigma }^2=\frac{\Sigma f_i{(x_i-\overline{x})}^2}{\Sigma f_i}$

Standard deviation is the square root of variance.

Calculation :

As per problem,

From the histogram:

  

Divide the sum of lower limit and upper limit of each class by two to get the mid-point of each class.

To find mean, calculate the sum of the product of mid-point and frequency of each class and the sum of frequencies.

  

Therefore the mean of the given data is $\overline{x}=\frac{\Sigma f_i{x}_i}{\Sigma f_i}=\frac{281}{70}\approx 4.0$

To calculate the variance, find the square of the difference between the mid points of each class and the mean. Then find the product of frequency and the square of the difference. Finally, calculate the sum of the product.

  

The variance of the data is, ${\sigma }^2=\frac{\Sigma f_i{(x_i-\overline{x})}^2}{\Sigma f_i}=\frac{254.5}{70}\approx 3.6$

The standard deviation is the square root of variance.

Therefore, standard deviation, $\sigma =\sqrt{3.6}\approx 1.9$