To solve the problem of finding the mean of the dataset $\{x_1, x_2, \ldots, x_{25}\}$ given that $\sum_{i=1}^{25} x_i^2 = 2568.25$ and the standard deviation is 5.2, let's use the formulas related to the mean and standard deviation.

Step 1: Recall the formula for the standard deviation

The standard deviation $\sigma$ for a dataset is given by: $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}$ where:

• $n$ is the number of data points.
• $\mu$ is the mean of the dataset.

Rearranging this formula to relate it to $\sum_{i=1}^{n} x_i^2$: $\sum_{i=1}^{n} x_i^2 = n\sigma^2 + n\mu^2$

Step 2: Plug in the known values

Given:

• $n = 25$ (the number of data points).
• $\sum_{i=1}^{25} x_i^2 = 2568.25$.
• The standard deviation $\sigma = 5.2$.

We substitute these values into the equation: $2568.25 = 25 \times (5.2)^2 + 25\mu^2$

Calculating $25 \times (5.2)^2$: $25 \times 27.04 = 676$

This simplifies the equation to: $2568.25 = 676 + 25\mu^2$

Step 3: Solve for the mean $\mu$

Rearrange the equation to solve for $\mu^2$: $25\mu^2 = 2568.25 - 676$ $25\mu^2 = 1892.25$ $\mu^2 = \frac{1892.25}{25} = 75.69$ $\mu = \sqrt{75.69} \approx 8.7$

Conclusion

The mean of the dataset is approximately $\mu = 8.7$.

Would you like more details or have any questions about this solution?

Here are five related questions you might find interesting:

1. How would the calculation change if the number of data points was different?
2. What is the relationship between the mean and standard deviation in normally distributed data?
3. How can we find the variance of a dataset from its standard deviation?
4. How would you find the mean if you were given the actual data points?
5. What are some practical applications of mean and standard deviation in data analysis?

Tip: Remember that the standard deviation measures how spread out the numbers are from the mean, while the mean is the average of the numbers.