To solve this, we need to calculate the mean and standard deviation of the combined set of 100 numbers, using the given data.

1. Mean of the Combined Set:

The mean of a set of numbers is calculated as the sum of the numbers divided by the number of elements.

We are given the following information:

• The sum of the first set of 25 numbers is 1000.
• The sum of the second set of 75 numbers is 5000.

The total sum of all 100 numbers is: $\text{Total Sum} = 1000 + 5000 = 6000$

The total number of elements is: $\text{Total Count} = 25 + 75 = 100$

Thus, the mean of the combined set is: $\text{Mean} = \frac{\text{Total Sum}}{\text{Total Count}} = \frac{6000}{100} = 60$

2. Standard Deviation of the Combined Set:

The standard deviation for a combined set of numbers can be calculated using the formula for pooled variance: $\sigma_{\text{combined}}^2 = \frac{(n_1 - 1) \sigma_1^2 + (n_2 - 1) \sigma_2^2 + \frac{n_1 n_2}{n_1 + n_2}(M_1 - M_2)^2}{n_1 + n_2 - 1}$ Where:

• $n_1 = 25$, the number of elements in the first set,
• $n_2 = 75$, the number of elements in the second set,
• $\sigma_1 = 16$, the standard deviation of the first set,
• $\sigma_2 = 14$, the standard deviation of the second set,
• $M_1 = 40$, the mean of the first set (since 1000/25 = 40),
• $M_2 = 66.67$, the mean of the second set (since 5000/75 = 66.67).

Now, calculate the variance and standard deviation of the combined set:

Step 1: Calculate the variances:

$\sigma_1^2 = 16^2 = 256$ $\sigma_2^2 = 14^2 = 196$

Step 2: Plug values into the pooled formula:

$\sigma_{\text{combined}}^2 = \frac{(25 - 1) \cdot 256 + (75 - 1) \cdot 196 + \frac{25 \cdot 75}{25 + 75} \cdot (40 - 66.67)^2}{25 + 75 - 1}$ $\sigma_{\text{combined}}^2 = \frac{24 \cdot 256 + 74 \cdot 196 + 25 \cdot 75 \cdot (40 - 66.67)^2 / 100}{99}$

Step 3: Calculate each term:

$24 \cdot 256 = 6144$ $74 \cdot 196 = 14464$ $25 \cdot 75 = 1875, \quad (40 - 66.67)^2 = (-26.67)^2 = 711.11$ $1875 \cdot 711.11 / 100 = 13333.33$

Step 4: Sum up the results:

$\sigma_{\text{combined}}^2 = \frac{6144 + 14464 + 13333.33}{99} = \frac{34041.33}{99} \approx 344.97$

Step 5: Take the square root to find the standard deviation:

$\sigma_{\text{combined}} = \sqrt{344.97} \approx 18.57$

Final Results:

• Mean of the combined set: $60$
• Standard deviation of the combined set: $18.57$

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Would you like more details on any of the steps or further clarification?

Here are some related questions you might find interesting:

1. How does the formula for the combined standard deviation change if the two sets have the same mean?
2. What are the key differences between variance and standard deviation in this context?
3. How would the combined standard deviation change if the standard deviations of both sets were higher?
4. What other methods can be used to combine data sets with different sample sizes?
5. How does the spread of numbers in each set affect the overall standard deviation of the combined set?

Tip: When combining data sets, it's crucial to consider both the individual variances and the difference between the means, as both affect the overall spread of the data.