To find the range, variance, and standard deviation of the given sample data, we follow these steps:

1. Range:

The range is the difference between the maximum and minimum values.

Given dataTo find the range, variance, and standard deviation of the given sample data, we follow these steps:

1. Range:

The range is the difference between the maximum and minimum values.

Given $45, 77, 35, 65, 11, 9, 73, 79, 78, 36, 96$

• Maximum value = 96
• Minimum value = 9

$\text{Range} = 96 - 9 = 87$

2. Variance:

Variance measures the spread of the data around the mean.

Step-by-step:

a. Find the mean (average):

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{45 + 77 + 35 + 65 + 11 + 9 + 73 + 79 + 78 + 36 + 96}{11}$

Let me calculate the mean.The mean (average) of the data is approximately $54.91$.

b. Calculate the squared differences from the mean for each value:

Each data point's deviation from the mean will be squared, and then we'll find the average of those squared deviations.

Let me calculate the variance and standard deviation for you.### 3. Variance: The variance of the data is approximately $860.69$.

4. Standard Deviation:

The standard deviation, which is the square root of the variance, is approximately $29.34$.

Interpretation:

• Range: The range of 87 indicates a wide spread between the highest and lowest jersey numbers.
• Variance and Standard Deviation: The high values for both variance and standard deviation suggest that the jersey numbers are spread out over a large range rather than being tightly clustered around the mean.

Would you like further details or have any questions?

Related questions:

1. How do we interpret high vs. low variance in a dataset?
2. What other measures of spread (besides standard deviation) can we use?
3. How does the standard deviation change if we add an outlier to the data?
4. Why do we use $n-1$ for variance in sample data?
5. How can we use variance to predict the spread in future samples?

Tip:

When working with small sample sizes, the standard deviation can be more sensitive to outliers, so it's important to consider the context of the data.