Question

3. The following frequency table summarizes the distances in miles of 120 patients from a regional hospital. Distance Frequency 0-4 40 4-8 30 8-12 20 12-16 20 16-20 10 Calculate the sample variance and standard deviation for this data (since it is a case of grouped data- use group or class midpoints in the formula in place of X values, and first calculate the sample mean).

Answer 1

Solution:

The formulas for sample mean and sample standard deviation for grouped data are given as below:

Sample mean = Xbar = ∑xf/∑f

Sample variance = S^2 = ∑(x - mean)^2*f / (∑f – 1)

Sample standard deviation = sqrt[∑(x - mean)^2*f / (∑f – 1)]

Calculation table is given as below:

Class	Midpoint X	f	xf	(x - mean)^2*f
0 to 4	2	40	80	1284.459556
4 to 8	6	30	180	83.3366667
8 to 12	10	20	200	108.8857778
12 to 16	14	20	280	802.2137778
16 to 20	18	10	180	1067.770889
Total		120	920	3346.666667

Sample mean = Xbar = 920/120 = 7.666666667

Sample variance = S^2 = ∑(x - mean)^2*f / (∑f – 1)

Sample variance = S^2 = 3346.666667/ (120 – 1)

Sample variance = S^2 = 3346.666667/119

Sample variance = S^2 = 28.1232493

Sample standard deviation = sqrt[∑(x - mean)^2*f / (∑f – 1)]

Sample standard deviation = sqrt(28.1232493)

Sample standard deviation = 5.303135799