A group of 50 observations has A.M. 61 and S.D. 8. Another group of 100 observations have A.M. 70 and S.D. 9. Find A.M. and S.D. for the combined group of 150 observations.

To find the **combined Arithmetic Mean (A.M.)** and **Standard Deviation (S.D.)** of two groups, we can use the following formulas:

### Combined Arithmetic Mean (A.M.):
\[
\text{Combined A.M.} = \frac{(n_1 \cdot \bar{x}_1) + (n_2 \cdot \bar{x}_2)}{n_1 + n_2}
\]
where:
- \(n_1 = 50\) (number of observations in the first group)
- \(n_2 = 100\) (number of observations in the second group)
- \(\bar{x}_1 = 61\) (A.M. of the first group)
- \(\bar{x}_2 = 70\) (A.M. of the second group)

### Combined Standard Deviation (S.D.):
\[
\text{Combined Variance} = \frac{(n_1 \cdot \sigma_1^2) + (n_2 \cdot \sigma_2^2) + n_1 \cdot (\bar{x}_1 - \text{Combined A.M.})^2 + n_2 \cdot (\bar{x}_2 - \text{Combined A.M.})^2}{n_1 + n_2}
\]
Then, the combined standard deviation is the square root of the combined variance:
\[
\text{Combined S.D.} = \sqrt{\text{Combined Variance}}
\]
where:
- \(\sigma_1 = 8\) (S.D. of the first group)
- \(\sigma_2 = 9\) (S.D. of the second group)

Let's compute these step by step.

### Step 1: Calculate the Combined A.M.
\[
\text{Combined A.M.} = \frac{(50 \cdot 61) + (100 \cdot 70)}{50 + 100}
\]
\[
\text{Combined A.M.} = \frac{3050 + 7000}{150} = \frac{10050}{150} = 67
\]

### Step 2: Calculate the Combined Variance
We first need to calculate the individual variances for the two groups and the adjustments due to the differences in means.

\[
\sigma_1^2 = 8^2 = 64, \quad \sigma_2^2 = 9^2 = 81
\]
\[
(\bar{x}_1 - \text{Combined A.M.})^2 = (61 - 67)^2 = (-6)^2 = 36
\]
\[
(\bar{x}_2 - \text{Combined A.M.})^2 = (70 - 67)^2 = 3^2 = 9
\]

Now, calculate the combined variance:
\[
\text{Combined Variance} = \frac{(50 \cdot 64) + (100 \cdot 81) + (50 \cdot 36) + (100 \cdot 9)}{150}
\]
\[
\text{Combined Variance} = \frac{3200 + 8100 + 1800 + 900}{150} = \frac{14000}{150} = 93.33
\]

### Step 3: Calculate the Combined S.D.
\[
\text{Combined S.D.} = \sqrt{93.33} \approx 9.66
\]

### Final Answer:
- **Combined A.M.** = 67
- **Combined S.D.** = 9.66

Would you like more details or have any questions?

Here are 5 related questions:
1. How is variance related to standard deviation?
2. What are the properties of the arithmetic mean in data sets?
3. How does sample size affect the combined mean and standard deviation?
4. Can we combine standard deviations without knowing the means?
5. What are the limitations of using standard deviation as a measure of spread?

**Tip:** Always check the consistency of your data before combining means and standard deviations to avoid errors.

Learn how to calculate the combined Arithmetic Mean (A.M.) and Standard Deviation (S.D.) for two groups with different A.M. and S.D. values. This problem involves key statistical concepts like variance and is suitable for Grades 10-12. It provides a step-by-step solution for combining 50 observations with A.M. 61 and S.D. 8, and 100 observations with A.M. 70 and S.D. 9.

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