Calculate [tex]$\bar{d}$[/tex] and [tex]$s_d$[/tex] for the following set of paired data. Calculate each paired difference by subtracting the values in Group A from the corresponding values in Group B. Round your answer to three decimal places, if necessary.
| Group A | 25 | 27 | 16 | 26 | 29 | 20 | 24 | 20 | 19 | 27 |
| Group B | 26 | 19 | 24 | 21 | 17 | 16 | 22 | 19 | 27 | 28 |
Answer (1)
Calculate the differences between each pair of data points by subtracting Group B from Group A.
Calculate the mean of the differences: d ˉ = n ∑ d i = 10 14 = 1.4 .
Calculate the sample standard deviation of the differences: s d = n − 1 ∑ ( d i − d ˉ ) 2 = 9 364.4 ≈ 6.363 .
The mean of the differences is 1.4 and the sample standard deviation of the differences is approximately 6.363: d ˉ = 1.4 , s d = 6.363 .
Explanation
Understanding the Problem We are given two sets of paired data, Group A and Group B, and we need to calculate the mean of the differences ( d ˉ ) and the sample standard deviation of the differences ( s d ). The paired difference is calculated by subtracting the values in Group B from the corresponding values in Group A.
Calculating the Differences First, we calculate the differences between each pair of data points:
d 1 = 25 − 26 = − 1 d 2 = 27 − 19 = 8 d 3 = 16 − 24 = − 8 d 4 = 26 − 21 = 5 d 5 = 29 − 17 = 12 d 6 = 20 − 16 = 4 d 7 = 24 − 22 = 2 d 8 = 20 − 19 = 1 d 9 = 19 − 27 = − 8 d 10 = 27 − 28 = − 1
So the differences are: -1, 8, -8, 5, 12, 4, 2, 1, -8, -1
Calculating the Mean of the Differences Next, we calculate the mean of the differences, d ˉ .
d ˉ = 10 ∑ i = 1 10 d i = 10 − 1 + 8 − 8 + 5 + 12 + 4 + 2 + 1 − 8 − 1 = 10 14 = 1.4
Thus, d ˉ = 1.4
Calculating the Sample Standard Deviation of the Differences Now, we calculate the sample standard deviation of the differences, s d .
First, we calculate the squared differences from the mean ( d i − d ˉ ) 2 :
( − 1 − 1.4 ) 2 = ( − 2.4 ) 2 = 5.76 ( 8 − 1.4 ) 2 = ( 6.6 ) 2 = 43.56 ( − 8 − 1.4 ) 2 = ( − 9.4 ) 2 = 88.36 ( 5 − 1.4 ) 2 = ( 3.6 ) 2 = 12.96 ( 12 − 1.4 ) 2 = ( 10.6 ) 2 = 112.36 ( 4 − 1.4 ) 2 = ( 2.6 ) 2 = 6.76 ( 2 − 1.4 ) 2 = ( 0.6 ) 2 = 0.36 ( 1 − 1.4 ) 2 = ( − 0.4 ) 2 = 0.16 ( − 8 − 1.4 ) 2 = ( − 9.4 ) 2 = 88.36 ( − 1 − 1.4 ) 2 = ( − 2.4 ) 2 = 5.76
Then, we sum these squared differences:
∑ i = 1 10 ( d i − d ˉ ) 2 = 5.76 + 43.56 + 88.36 + 12.96 + 112.36 + 6.76 + 0.36 + 0.16 + 88.36 + 5.76 = 364.4
Now, we divide by n − 1 = 10 − 1 = 9 :
9 ∑ i = 1 10 ( d i − d ˉ ) 2 = 9 364.4 = 40.4888...
Finally, we take the square root:
s d = 9 ∑ i = 1 10 ( d i − d ˉ ) 2 = 40.4888... ≈ 6.363
Thus, s d ≈ 6.363
Final Answer Therefore, d ˉ = 1.4 and s d ≈ 6.363 .
Examples
Paired data analysis is commonly used in medical studies to compare the effectiveness of a treatment. For example, researchers might measure a patient's blood pressure before and after taking a new medication. By calculating the difference in blood pressure for each patient, they can determine if the medication has a significant effect. The mean difference indicates the average change in blood pressure, while the standard deviation of the differences measures the variability in the medication's effect across the patient population. This helps doctors understand how consistently the medication works.
