A personal trainer is studying whether a new stretching routine improves flexibility. She records the forward reach (in cm) of 6 clients before and after a 4-week program. Calculate the difference (after - before) for each client, the mean difference, and standard deviation.

• A. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2$; and Standard Deviation = $1.91$
• B. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2.5$; and Standard Deviation = $1.91$
• C. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2.5$; and Standard Deviation = $2.10$
• D. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2$; and Standard Deviation = $2.10$

Answer: (D)

Verified step by step guidance

1. Step 1: Calculate the difference (After - Before) for each client. Use the values from the table: Client A: 26 - 22 = 4, Client B: 23 - 19 = 4, Client C: 27 - 24 = 3, Client D: 20 - 21 = -1, Client E: 18 - 18 = 0, Client F: 25 - 23 = 2.
2. Step 2: Compute the mean difference. Add all the differences calculated in Step 1 (4 + 4 + 3 - 1 + 0 + 2) and divide by the number of clients (6). The formula for mean difference is: $\frac{\sum d}{n}$, where d represents the differences and n is the number of clients.
3. Step 3: Calculate the squared deviations from the mean for each difference. For each client, subtract the mean difference from their individual difference, then square the result. The formula is: ${(}^d$.
4. Step 4: Compute the variance. Add all the squared deviations calculated in Step 3 and divide by the number of clients minus 1 (n - 1). The formula for variance is: $\frac{\sum {(}^d}{n-1}$.
5. Step 5: Calculate the standard deviation. Take the square root of the variance calculated in Step 4. The formula for standard deviation is: $\sqrt{\frac{\sum {(}^d}{n-1}}$.