Below are the jersey numbers of 11 players randomly selected from a football team. Find the​ range, variance, and standard deviation for the given sample data. What do the results tell​ us?
26 98 89 57 76 24 1 83 91 45 43
Question content area bottom
Part 1
Rangeequals

97 ​(Round to one decimal place as​ needed.)
Part 2
Sample standard deviationequals

enter your response here ​(Round to one decimal place as​ needed.)

Let's solve this step by step.

### 1. Range
The range is calculated as the difference between the maximum and minimum values in the dataset.

- **Maximum value**: 98
- **Minimum value**: 1

\[
\text{Range} = 98 - 1 = 97
\]

This matches the range you provided.

### 2. Variance and Standard Deviation
The standard deviation and variance for the sample can be calculated as follows:

#### Step 1: Calculate the mean (average) of the data.
\[
\text{Mean} = \frac{26 + 98 + 89 + 57 + 76 + 24 + 1 + 83 + 91 + 45 + 43}{11}
\]
\[
\text{Mean} = \frac{633}{11} = 57.5455 \quad \text{(rounded to four decimal places)}
\]

#### Step 2: Calculate each deviation from the mean and square it.
For each data point \( x_i \), the squared deviation is \( (x_i - \text{Mean})^2 \).

#### Step 3: Calculate the sample variance.
\[
\text{Sample Variance} = \frac{\sum (x_i - \text{Mean})^2}{n-1}
\]
Where \( n \) is the number of observations.

#### Step 4: Calculate the sample standard deviation.
\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}}
\]

Let's calculate these values.

I'll calculate these now.### Results:

- **Mean**: 57.5 (rounded to one decimal place)
- **Sample Variance**: 1042.1 (rounded to one decimal place)
- **Sample Standard Deviation**: 32.3 (rounded to one decimal place)

### Interpretation:
- **Range**: The range of 97 indicates a wide spread in the jersey numbers, with values ranging from 1 to 98.
- **Variance and Standard Deviation**: The high standard deviation of 32.3 shows that the jersey numbers are quite dispersed around the mean of 57.5, indicating variability in the numbers chosen by the players.

Would you like more details or have any questions?

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Here are five related questions to deepen your understanding:
1. How does the standard deviation change if the dataset is more concentrated around the mean?
2. What is the effect of an outlier on the range and standard deviation?
3. Can you explain why the sample variance uses \(n-1\) instead of \(n\) in the denominator?
4. How would you interpret the standard deviation in the context of a normal distribution?
5. How does the range compare to other measures of dispersion like interquartile range?

**Tip:** When calculating variance and standard deviation, always check if the data represents the entire population or just a sample. Use \(n-1\) for sample data (Bessel's correction) to get an unbiased estimate.

Learn how to calculate the range, variance, and standard deviation for a set of football team jersey numbers (26, 98, 89, 57, 76, 24, 1, 83, 91, 45, 43). Understand the implications of these statistical measures on data dispersion and central tendency. Detailed step-by-step solution provided.

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