Question

The number of passes6 completed by Aaron Rodgers, quarterback for the Minnesota Vikings, was recorded for each of the 15 regular season games that he played in the fall of 2010 (www. ESPN,com): \(19 \quad 19\) \(\begin{array}{llllll}34 & 12 & 27 & 18 & 21 & 15\end{array}\) \(27 \quad 22\) \(\begin{array}{lllll}26 & 21 & 7 & 25 & 19\end{array}\) a. Draw a stem and leaf plot to describe the data. b. Calculate the mean and standard deviation for Aaron Rodgers' per game pass completions. c. What proportion of the measurements lie within two standard deviations of the mean?

Short answer

Explain why the mean and standard deviation are useful in this situation. The mean and standard deviation are useful in this situation because they provide a summary of the central tendency and variability of the data. The mean (21.6 passes per game) provides an average performance, which can be used as a benchmark for comparing individual game performances. The standard deviation (6.8) shows the variability in the number of passes completed per game. A smaller standard deviation would indicate that the data points are closely grouped around the mean, while a larger standard deviation indicates a greater spread of data points. In this case, knowing the standard deviation can help understand the consistency of Aaron Rodgers' performance. Furthermore, the proportion of measurements within two standard deviations (100%) indicates how well the data aligns with the normal distribution – a higher percentage of data points within two standard deviations suggests that the performance is more predictable and consistent. This information is valuable for coaches and analysts when evaluating Aaron Rodgers' performance and making decisions related to game strategies.

Step-by-step solution

Organize the data

First, sort the data in ascending order: 7, 12, 15, 18, 19, 19, 21, 21, 22, 25, 26, 27, 27, 34

Create the stem and leaf plot

Now, create the stems based on the tens place and the leaves based on the ones place. Stem | Leaf -----|----- 0 | 7 1 | 2 5 8 9 9 2 | 1 1 2 5 6 7 7 3 | 4 #b. Calculating the mean and standard deviation#

Calculate the mean

Use the formula for the mean: \(\mu = \frac{\sum x}{N}\), with N being 15. \(\mu = \frac{7 + 12 + 15 + 18 + 19 + 19 + 21 + 21 + 22 + 25 + 26 + 27 + 27 + 34}{15} = 21.6\) The mean of passes completed per game is 21.6.

Calculate the standard deviation

Use the formula for the standard deviation: \(\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}\) \(\sigma = \sqrt{\frac{(7 - 21.6)^2 + (12 - 21.6)^2 + \cdots + (34 - 21.6)^2 }{15}} \approx 6.8\) The standard deviation of passes completed per game is approximately 6.8. #c. Finding the proportion within two standard deviations#

Determine the range of values within two standard deviations

Calculate the lower and upper bounds where the data points lie within two standard deviations of the mean. Lower bound: \(21.6 - 2 \times 6.8 = 8\) Upper bound: \(21.6 + 2 \times 6.8 = 35.2\)

Count data points within the range

Count the number of data points within the range of 8 to 35.2. There are 15 data points in this range, which is the total amount of data points.

Calculate the proportion

Divide the number of data points within the range by the total number of data points. Proportion of measurements within two standard deviations: \(\frac{15}{15} = 1\) This means that 100% or all of the measurements lie within two standard deviations of the mean.

Key concepts

Stem and Leaf Plot

A Stem and Leaf Plot is a great way to display data in an organized manner. This method provides a clear picture of data distribution and maintains the original data values for reference.

To create a Stem and Leaf Plot, follow these steps:

• First, sort the data from smallest to largest to make it easier to work with. For Aaron Rodgers' pass completions, the sorted sequence is: 7, 12, 15, 18, 19, 19, 21, 21, 22, 25, 26, 27, 27, 34.
• Next, divide each number into a 'stem' (all but the last digit) and a 'leaf' (the last digit). For example, 34 has a stem of '3' and a leaf of '4'.
• Arrange these numbers into a table where all values with the same stem are listed with their leaves organized horizontally.

This not only helps in quick data visualization but also in identifying the mode and detecting any outliers.

Mean and Standard Deviation

Calculating the mean and standard deviation of a data set provides vital insights into the typical value and variability of the dataset.

The **mean** is calculated by summing all data values and dividing by the count of the data points. For Aaron Rogers, it is: \(\mu = \frac{7 + 12 + 15 + 18 + 19 + 19 + 21 + 21 + 22 + 25 + 26 + 27 + 27 + 34}{15} = 21.6\).This means on average, there are 21.6 pass completions per game.

The **standard deviation** measures data spread around the mean, calculated by:\[ \sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}} \],where \(N\) is the number of values. For this data, the standard deviation is approximately 6.8, which indicates how much the pass completions vary each game.

Data Analysis

Data Analysis uses statistical tools to interpret and make sense of data. Understanding tools like the mean and standard deviation helps in summarizing and drawing conclusions from data. Keep the following in mind:

• The mean provides a central value around which data points are centered.
• The standard deviation helps in understanding the spread and variation within data. A higher standard deviation means more variation in pass completions, while a lower one means more consistency.
• Data analysis helps in quickly spotting trends, such as whether Aaron Rogers typically completes around 21-22 passes per game.

With these methods, we can make informed decisions and better predictions.

Proportion of Measurements

The proportion of measurements within a specific range is an important aspect of data analysis, offering insights about data distribution.

It can show how much of the data is within a comfortable or expected range. In this case, we're interested in how many pass completions lie within two standard deviations from the mean.

• First, calculate the boundaries: from the mean, subtract and add twice the standard deviation: \( 21.6 - 2 \times 6.8 = 8\) and \(21.6 + 2 \times 6.8 = 35.2\).
• Count all the data points that fall within these limits. Here, all original data points (7 through 34) fall within this range.
• Finally, express this count as a proportion of the total number of data points, resulting in a proportion of 1 (or 100%).

Knowing that all data lies within this range provides assurance of data's consistency and predictability.