I have data from two groups and need to calculate the composite standard deviation. Here are the details:Group 1:Arithmetic Mean ((\bar{X}_1)): 7Standard Deviation ((s_1)): 1.25Number of Observations ((n_1)): 25Group 2:Arithmetic Mean ((\bar{X}_2)): 10Standard Deviation ((s_2)): 2.2Number of Observations ((n_2)): 45

To calculate the **composite standard deviation** (also called the **pooled standard deviation**) for two groups, the formula is:

\[
s_{\text{pooled}} = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}
\]

Where:
- \( s_1 \) and \( s_2 \) are the standard deviations of Group 1 and Group 2, respectively.
- \( n_1 \) and \( n_2 \) are the number of observations in Group 1 and Group 2, respectively.

### Given dataTo calculate the **composite standard deviation** (also called the **pooled standard deviation**) for two groups, the formula is:

\[
s_{\text{pooled}} = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}
\]

Where:
- \( s_1 \) and \( s_2 \) are the standard deviations of Group 1 and Group 2, respectively.
- \( n_1 \) and \( n_2 \) are the number of observations in Group 1 and Group 2, respectively.

### Given 
- Group 1: \( s_1 = 1.25 \), \( n_1 = 25 \)
- Group 2: \( s_2 = 2.2 \), \( n_2 = 45 \)

Now, let’s compute the pooled standard deviation.

\[
s_{\text{pooled}} = \sqrt{\frac{(25 - 1)(1.25)^2 + (45 - 1)(2.2)^2}{25 + 45 - 2}}
\]

Let’s calculate this step-by-step.The composite (pooled) standard deviation for the two groups is approximately **1.92**.

Would you like further details on the calculation or have any other questions? Here are five related questions to deepen your understanding:

1. How does the sample size affect the pooled standard deviation?
2. What is the significance of using pooled standard deviation in hypothesis testing?
3. How does pooled standard deviation differ from weighted standard deviation?
4. Can we use the pooled standard deviation if the variances of the two groups differ significantly?
5. How would you interpret the pooled standard deviation in the context of your data?

**Tip:** Pooled standard deviation is most useful when the sample sizes and variances of the two groups are not drastically different.

I have data from two groups and need to calculate the composite standard deviation. Here are the details: Group 1: Arithmetic Mean (\bar{X}_1): 7, Standard Deviation (s_1): 1.25, Number of Observations (n_1): 25; Group 2: Arithmetic Mean (\bar{X}_2): 10, Standard Deviation (s_2): 2.2, Number of Observations (n_2): 45.

Learn how to calculate the composite (pooled) standard deviation for two groups using statistical concepts. This step-by-step guide walks through an example with Group 1 (Mean: 7, SD: 1.25, n=25) and Group 2 (Mean: 10, SD: 2.2, n=45). Ideal for college-level statistics courses.

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