In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.


Super Bowl Jersey Numbers Listed below are the jersey numbers of the 11 offensive players on the starting roster of the New England Patriots when they won Super Bowl LIII. What do the results tell us?


12 26 46 15 11 87 77 62 60 69 61

Verified step by step guidance

1. Step 1: Organize the data set. The given jersey numbers are: 12, 26, 46, 15, 11, 87, 77, 62, 60, 69, 61. Ensure the data is sorted if necessary for easier calculations.
2. Step 2: Calculate the range. The range is the difference between the maximum and minimum values in the data set. Identify the maximum value (87) and the minimum value (11), then compute the range using the formula: $\mathrm{Range}=\mathrm{Max}-\mathrm{Min}$.
3. Step 3: Compute the variance. First, find the mean of the data set using the formula: $\mathrm{Mean}=\frac{\sum x}{n}$, where $x$ represents each data point and $n$ is the number of data points. Then, calculate the squared differences between each data point and the mean, sum them up, and divide by $n-1$ (since this is a sample). The formula for sample variance is: $\mathrm{Variance}=\frac{\sum (x-\mathrm{Mean}{)}^2}{n-1}$.
4. Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance. Use the formula: $\mathrm{Standard}\mathrm{Deviation}=\sqrt{\mathrm{Variance}}$.
5. Step 5: Interpret the results. The range, variance, and standard deviation provide insights into the spread and variability of the jersey numbers. Discuss what these measures tell us about the distribution of the data, such as whether the numbers are tightly clustered or widely spread.

Key Concepts

Range

The range is a measure of variation that represents the difference between the highest and lowest values in a data set. It provides a simple way to understand the spread of the data. For example, in the jersey numbers provided, the range would be calculated by subtracting the smallest number from the largest number, giving insight into how diverse the jersey numbers are.

Variance

Variance quantifies the degree to which data points in a set differ from the mean. It is calculated by averaging the squared differences between each data point and the mean. A higher variance indicates that the data points are more spread out, while a lower variance suggests they are closer to the mean. Understanding variance is crucial for assessing the consistency of the jersey numbers.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of the average distance of each data point from the mean. It is expressed in the same units as the data, making it more interpretable. A smaller standard deviation indicates that the data points tend to be close to the mean, while a larger one suggests greater variability. This concept helps in understanding how much the jersey numbers deviate from the average number.