The population mean is 102.4. The population standard deviation is 106.5. What percentage of the 50 regions have stores within plus minus 1 standard deviations of the mean?

To calculate the percentage of the 50 regions that have stores within plus or minus 1 standard deviation of the mean, we can use the empirical rule (also known as the 68-95-99.7 rule). This rule states that for a normal distribution:

- Approximately 68% of the data lies within ±1 standard deviation from the mean.
- Approximately 95% of the data lies within ±2 standard deviations from the mean.
- Approximately 99.7% of the data lies within ±3 standard deviations from the mean.

Given that the population mean is \( 102.4 \) and the population standard deviation is \( 106.5 \):

- The range within ±1 standard deviation is: 
  \[
  102.4 \pm 106.5 = (102.4 - 106.5, 102.4 + 106.5) = (-4.1, 208.9)
  \]

Assuming the data is normally distributed, approximately 68% of the 50 regions would have stores within this range.

### Summary
- **Percentage of regions within ±1 standard deviation:** Approximately 68%.

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Would you like to explore more details, or do you have any further questions?

### Related Questions:
1. How do you calculate the percentage for ±2 or ±3 standard deviations?
2. What assumptions are made about the distribution of the data?
3. How would this calculation change if the distribution were skewed?
4. Can you calculate the exact percentage using z-scores?
5. What is the significance of the empirical rule in statistics?
6. How do you interpret standard deviations in real-world scenarios?
7. What are the implications if the data is not normally distributed?
8. How would you calculate the percentage if the standard deviation were smaller?

### Tip:
The empirical rule is most accurate for data that follows a normal distribution. If the data distribution is skewed or has outliers, the percentages may differ.

Calculate the percentage of regions within plus or minus 1 standard deviation of the mean in statistics using the empirical rule. Given a population mean of 102.4 and a standard deviation of 106.5, approximately 68% of the 50 regions would have stores within this range. Learn more about statistical distributions and the empirical rule.

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