Question

When making inferences concerning the mean difference using two dependent samples, it is necessary to calculate the standard deviation of the sample differences.

Calculate the standard deviation of the sample differences using the following information.

Round to 2 decimal places.

College Placement Test Results

Applicant	1	2	3	4	5
Before	51	62	74	64	72
After	68	87	82	75	84

Answer 1

Solution:
First, we will calculate the differences of all applicants which can be calculated as

Applicant	Before	After	Differences(Before - After)
1	51	68	-17
2	62	87	-25
3	74	82	-8
4	64	75	-11
5	72	84	-12

Now we will calculate differences mean which can be calculated as
Dbar = Difference/n = (-17-25-8-11-12)/5 = -73/5 = -14.6
So standard deviation of difference can be calculated as
Standard deviation (Sd) = sqrt(((Di-Dbar)^2)/(n-1)) = sqrt((-17-(-14.6))^2 + (-25-(-14.6))^2 + (-8-(-14.6))^2 + (-11-(-14.6))^2 + (-12-(-14.6))^2)/4)

Applicant	Before	After	Differences(Before - After)	Di-mean	(Di-mean)^2
1	51	68	-17	-2.4	5.76
2	62	87	-25	-10.4	108.16
3	74	82	-8	6.6	43.56
4	64	75	-11	3.6	12.96
5	72	84	-12	2.6	6.76

Sd = sqrt((5.76+108.16+43.56+12.96+6.76)/4) = sqrt(177.2/4) = sqrt(43.3) = 6.66

standard deviation of the sample differences = 6.66