We are given the smallest score $X_{\text{min}} = 5$ and the largest score $X_{\text{max}} = 10$. We need to analyze what we can conclude about the standard deviation.

Understanding the Standard Deviation

The standard deviation ($\sigma$) measures the dispersion of data points around the mean. It is calculated as:

$\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}$

where:

• $X$ represents individual data points,
• $\mu$ is the mean,
• $N$ is the number of data points.

Evaluating the Answer Choices

1. "Mean is greater than 10"

   • This is incorrect because the mean cannot exceed the highest value in the population (which is 10).

2. "Standard deviation is smaller than 5"

   • This is true in most reasonable distributions since the range is 5 (i.e., $10 - 5 = 5$). The standard deviation cannot exceed the range, so it must be less than 5.

3. "Standard deviation is less than 6"

   • Since the maximum possible range is 5, the standard deviation must be less than 6. This is also correct.

4. "Standard deviation is greater than 6"

   • This is incorrect because the standard deviation is always at most the range divided by a scaling factor. In this case, it's impossible for the standard deviation to be greater than 6 if the range is only 5.

Best Answer:

✅ "Standard deviation is less than 6" is the most general and correct conclusion.

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Do you need further clarification or details?

Related Questions:

1. How is the standard deviation affected by extreme values in a dataset?
2. What is the difference between population and sample standard deviation?
3. Can the mean be equal to the smallest or largest value in a dataset?
4. What role does the range play in estimating standard deviation?
5. How do you calculate variance, and how is it related to standard deviation?

Tip: The standard deviation is always less than or equal to half of the range in a uniform distribution! 🚀