Question

Calculate the sample standard deviation and sample variance for the following frequency distribution of heart rates for a sample of American adults. If necessary, round to one more decimal place than the largest number of decimal places given in the data.

Class Frequency
61 - 66 4
67 - 72 8
73 - 78 5
79 - 84 7
85 - 90 13

Answer 1

Formula

The below statistical formulas are employed to find the standard deviation for the frequency distribution table data set

How to calculate grouped data standard deviation?

Step by step calculation:
Follow these below steps using the above formulas to understand how to calculate standard deviation for the frequency table data set

step 1: find the mid-point for each group or range of the frequency table.
step 2: calculate the number of samples of a data set by summing up the frequencies.
step 3: find the mean for the grouped data by dividing the addition of multiplication of each group mid-point and frequency of the data set by the number of samples.
step 4: calculate the variance for the frequency table data by using the above formula.
step 5:estimate standard deviation for the frequency table by taking square root of the variance.

Input:

Range	Frequency
61-66	4
67-72	8
73-78	5
79-84	7
85-90	13

step 1: find the mid-point M for each group
(61+66)/2 = 63.5
(67 + 72)/2 = 69.5
(73 + 78)/2 = 75.5
(79+84)/2 = 81.5
(85+90)/2 = 87.5

step 2: calculate the number of samples n
n = 4 + 8 + 5 + 7 + 13 = 37

step 3: find the grouped data mean μ
μ = (63.5 x 4 + 69.5 x 8 + 75.5 x 5 + 81.5 x 7 + 87.5 x 13 )/n
= 2895.5/37
μ = 78.2568

step 4: find the variance σ²
= ((63.5² x 4 + 69.5² x 8 + 75.5² x 5 + 81.5² x 7 + 87.5² x 13 ) - (37 x 78.26²))/(37 - 1)
= 2706.5604/ 36
σ² = 75.1822

step 5: find standard deviation for the grouped data
Take square root of variance
σ =