Question

During his 20 seasons in the NHL, Wayne Gretzky scored \(50 \%\) more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: \(\begin{aligned} &79,80,80,80,74,80,80,79,64,78,73,78,74,45,81,48,80, \\ &82,82,70 \end{aligned}\) a) Create a stem-and-leaf display for these data, using split stems. b) Describe the shape of the distribution. c) Describe the center and spread of this distribution. d) What unusual feature do you see? What might explain this?

Short answer

Gretzky's games are right-skewed. 78.5 is the median with a range of 37. Low counts may reflect injuries or shortened seasons.

Step-by-step solution

Sort Data for Stem-and-Leaf Display

First, arrange the number of games played in ascending order: 45, 48, 64, 70, 73, 74, 74, 78, 78, 79, 80, 80, 80, 80, 80, 81, 82, 82, 82.

Construct Stem-and-Leaf Display

Split the stems into two groups each (0-4 and 5-9) for each decade. Then distribute the leaves appropriately: 4 | 5 8 6 | 4 7 | 0 3 4 4 8 8 9 8 | 0 0 0 0 1 2 2 2.

Analyze the Shape

The distribution appears to be right-skewed, with a higher frequency of larger numbers towards the right.

Determine Center and Spread

Calculate the median, which lies between the 10th and 11th terms of the sorted list: 78 and 79. Thus, the median is 78.5. The spread is from 45 to 82, so the range is 37 games.

Identify Unusual Features

There is a notable dip in the number of games (45, 48) significantly less than other values. This could be due to unusual circumstances like injuries or a shortened season.

Key concepts

Stem-and-Leaf Plot

A stem-and-leaf plot is a great way to display data. It helps us see the distribution clearly and quickly spot patterns. In a stem-and-leaf plot, each number is split into a stem and a leaf. The stem represents the leading digits, while the leaf represents the last digit.
For example, in Wayne Gretzky's season game numbers, if we look at "79," the "7" becomes the stem and the "9" the leaf. When we have a wide range of numbers or many repeating ones, slight adjustments can be made.
One such method is splitting stems. By dividing stems into two groups (e.g., 0-4 and 5-9), we achieve a clearer display of the data, capturing more detail.

• The numbers 45 and 48 become 4 | 5, 8.
• The numbers group between 70 and 79 become 7 | 0, 3, 4, 4, 8, 8, 9.

This simple, visual arrangement makes it easy to see the frequency and distribution of the data.

Distribution Shape

When analyzing data, the shape of the distribution is crucial. It tells us about the data's overall pattern and trend.
In our example, the distribution of the number of games Gretzky played each season is right-skewed. This term means that most values are amassed on the left with tails going out further to the right.
Right-skewed distributions often arise when there are fewer entries in the higher range of data. For Gretzky's games, this skew reflects how he played mostly in the 70 to 82 range, but had a few lower seasons in the 40s and 60s. Overall, understanding the shape of a distribution helps in identifying where most values lie and potentially spotting any outliers or anomalies.

Median and Range

The median and range are two important measures in statistics that describe a data set's center point and spread.
The median is the middle value when the data is ordered. If there's an even number of observations, the median is the average of the two middle numbers.

• For Gretzky's game data, a sorted order identifies the median between the 10th and 11th numbers: 78 and 79.
• This results in a median of 78.5 games per season.

The range measures how spread out the data is, calculated by subtracting the smallest value from the largest.
Here, the lowest number is 45, and the highest is 82, yielding a range of 37 games. These measures give insight into Gretzky's typical and potential variation in gameplay each season.

Unusual Features in Data

Unusual features in a data set can reveal unique insights or identify potential errors or exceptional cases. In the context of Gretzky's game data, the notable observation is the seasons with dramatically fewer games, namely the seasons with only 45 and 48 games.
Such deviations from the norm often warrant investigation.

• There could be logical explanations like injuries preventing him from playing a full season.
• Alternatively, these low numbers may result from shortened seasons due to external reasons, such as lockouts.

Identifying unusual features is crucial as they can provide context specific to historical events or player circumstances, enriching our understanding of the data.