Data sets A and B are dependent. Find s Subscript d.
Start 2 By 6 Table 1st Row 1st Column Upper A 2nd Column 19 3rd Column 17 4st Column 36 5st Column 32 6st Column 20 2nd Row 1st Column Upper B 2nd Column 17 3rd Column 13 4st Column 14 5st Column 24 6st Column 11 EndTable

To find \( s_d \), which is the standard deviation of the differences between the paired values in datasets \( A \) and \( B \), follow these steps:

### 1. Calculate the Differences (\( D_i = A_i - B_i \)):
Let's subtract each value in \( B \) from the corresponding value in \( A \).

\[
\begin{aligned}
D_1 & = 19 - 17 = 2 \\
D_2 & = 17 - 13 = 4 \\
D_3 & = 36 - 14 = 22 \\
D_4 & = 32 - 24 = 8 \\
D_5 & = 20 - 11 = 9 \\
\end{aligned}
\]

### 2. Find the Mean of the Differences (\( \overline{D} \)):
Sum up the differences and divide by the number of differences.

\[
\overline{D} = \frac{2 + 4 + 22 + 8 + 9}{5} = \frac{45}{5} = 9
\]

### 3. Calculate the Squared Differences from the Mean:
For each difference, subtract the mean \( \overline{D} \) and square the result.

\[
\begin{aligned}
(D_1 - \overline{D})^2 & = (2 - 9)^2 = (-7)^2 = 49 \\
(D_2 - \overline{D})^2 & = (4 - 9)^2 = (-5)^2 = 25 \\
(D_3 - \overline{D})^2 & = (22 - 9)^2 = 13^2 = 169 \\
(D_4 - \overline{D})^2 & = (8 - 9)^2 = (-1)^2 = 1 \\
(D_5 - \overline{D})^2 & = (9 - 9)^2 = 0^2 = 0 \\
\end{aligned}
\]

### 4. Find the Variance of the Differences (\( s_d^2 \)):
The variance is the sum of these squared differences divided by \( n-1 \) (where \( n \) is the number of differences).

\[
s_d^2 = \frac{49 + 25 + 169 + 1 + 0}{4} = \frac{244}{4} = 61
\]

### 5. Find the Standard Deviation of the Differences (\( s_d \)):
Take the square root of the variance.

\[
s_d = \sqrt{61} \approx 7.81
\]

### Final Answer:
The standard deviation of the differences, \( s_d \), is approximately \( 7.81 \).

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Would you like a detailed breakdown of any specific step, or do you have any questions? 

Here are some related questions you might find useful:
1. How do you interpret the standard deviation of differences in paired data?
2. What would happen if datasets \( A \) and \( B \) were independent?
3. How does sample size affect the calculation of \( s_d \)?
4. What is the formula for the variance of differences in paired data?
5. How would you calculate \( s_d \) if you had an additional pair of values?
6. Why do we divide by \( n-1 \) instead of \( n \) when calculating the variance?
7. Can you describe a scenario where you might use this type of calculation in real life?
8. What assumptions must hold for this calculation to be valid?

**Tip:** Always check whether your datasets are dependent or independent before choosing a statistical method.

Learn how to calculate the standard deviation (s_d) of paired data sets A and B. Follow a step-by-step guide with examples and formulas. Suitable for high school students studying statistics.

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