Let's calculate the range, sample variance, and sample standard deviation for the given sample data:

The data set provided is:
92, 49, 2, 38, 10, 74, 51, 89, 70, 63, 3

Part 1: Range

The range is simply the difference between the highest and lowest values in the data set.

• Maximum value: 92
• Minimum value: 2

$\text{Range} = \text{Maximum value} - \text{Minimum value} = 92 - 2 = 90$

So, the range is 90.

Part 2: Sample Variance and Sample Standard Deviation

Step 1: Find the mean (average)

To calculate the sample variance and sample standard deviation, we first need to find the mean:

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{92 + 49 + 2 + 38 + 10 + 74 + 51 + 89 + 70 + 63 + 3}{11}$

$\text{Mean} = \frac{541}{11} \approx 49.1$

Step 2: Calculate the squared differences from the mean

For each value in the data set, subtract the mean and square the result.

• $(92 - 49.1)^2 = 42.9^2 = 1842.41$
• $(49 - 49.1)^2 = (-0.1)^2 = 0.01$
• $(2 - 49.1)^2 = (-47.1)^2 = 2211.61$
• $(38 - 49.1)^2 = (-11.1)^2 = 123.21$
• $(10 - 49.1)^2 = (-39.1)^2 = 1527.21$
• $(74 - 49.1)^2 = 24.9^2 = 620.01$
• $(51 - 49.1)^2 = 1.9^2 = 3.61$
• $(89 - 49.1)^2 = 39.9^2 = 1592.01$
• $(70 - 49.1)^2 = 20.9^2 = 436.81$
• $(63 - 49.1)^2 = 13.9^2 = 193.21$
• $(3 - 49.1)^2 = (-46.1)^2 = 2127.21$

Step 3: Sum of squared differences

$\text{Sum of squared differences} = 1842.41 + 0.01 + 2211.61 + 123.21 + 1527.21 + 620.01 + 3.61 + 1592.01 + 436.81 + 193.21 + 2127.21 = 9655.21$

Step 4: Calculate the sample variance

For sample variance, divide the sum of squared differences by $n - 1$ (since this is a sample, not a population). Here $n = 11$, so we divide by $10$.

$\text{Sample variance} = \frac{9655.21}{10} = 965.5$

So, the sample variance is 965.5 (rounded to one decimal place).

Step 5: Calculate the sample standard deviation

The sample standard deviation is the square root of the sample variance.

$\text{Sample standard deviation} = \sqrt{965.5} \approx 31.1$

So, the sample standard deviation is approximately 31.1 (rounded to one decimal place).

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Final Answers:

• Range: 90
• Sample Variance: 965.5
• Sample Standard Deviation: 31.1

What do these results tell us?

• The range of 90 shows that the values vary greatly from the lowest (2) to the highest (92) in the dataset.
• The sample variance of 965.5 indicates how spread out the values are around the mean. A larger variance means the numbers are more spread out.
• The sample standard deviation of 31.1 confirms this spread, indicating that on average, the data points deviate from the mean by about 31.1 units.

Would you like further details or have any questions? Here are a few related questions:

1. How does variance differ from standard deviation, and why is it important to calculate both?
2. If this were a population instead of a sample, how would the variance and standard deviation change?
3. How can the range be affected by extreme outliers in the data?
4. What impact would adding more data points have on the variance and standard deviation?
5. How do you interpret the spread of data in a context like a sports team or business performance?

Tip: Always double-check your calculations, especially when squaring values. Small errors can lead to large changes in variance and standard deviation!