Question

In the Super Bowl, by how many points does the winning team outscore the losers? Here are the winning margins for the first 42 Super Bowl games: \(25,19,9,16,3,21,7,17,10,4,18,17,4,12,17,5,10,29,22\), \(36,19,32,4,45,1,13,35,17,23,10,14,7,15,7,27,3,27,3\), \(3,11,12,3\) a) Find the median. b) Find the quartiles. c) Write a description based on the 5 -number summary.

Short answer

The median is 17; quartiles are 7 and 23.5.

Step-by-step solution

Order the Data

First, list all the winning margins in ascending order. This helps to accurately find the median and quartiles.Ordered data: \[1, 3, 3, 3, 3, 4, 4, 5, 7, 7, 7, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 17, 17, 18, 19, 19, 21, 22, 23, 25, 27, 27, 29, 32, 35, 36, 45 \]

Find the Median

The median is the middle value in an ordered set. Since there are 42 data points, the median is the average of the 21st and 22nd values. In our ordered list, these are 17 and 17.Thus, the median is:\[\frac{17 + 17}{2} = 17\]

Calculate the Quartiles

To find the quartiles, divide the data set into four equal parts.- The first quartile \(Q1\) is the median of the first half of the data (values 1 through 21). The middle values here are the 10th and 11th values, both 7: \[Q1 = \frac{7 + 7}{2} = 7\]- The third quartile \(Q3\) is the median of the second half of the data (values 22 through 42). The middle values here are the 32nd and 33rd values, 25 and 27: \[Q3 = \frac{23 + 25}{2} = 23.5\]

Determine the Five-Number Summary

Compile the minimum, first quartile \(Q1\), median, third quartile \(Q3\), and maximum values to get the five-number summary:- Minimum value: 1- First quartile \(Q1\): 7- Median: 17- Third quartile \(Q3\): 23.5- Maximum value: 45The five-number summary is: 1, 7, 17, 23.5, 45.

Key concepts

Median

The median is a measure of central tendency that represents the middle value of a data set when it's ordered from smallest to largest. It splits the data into two equal halves. If you have an odd number of data points, the median is the center value directly. However, with an even number of data points, like in our Super Bowl scores example (with 42 data points), the median is calculated by taking the average of the two middle numbers.

To find the median, firstly, arrange all your data points in ascending order. In this situation, the 21st and 22nd numbers are both 17, hence, our median is:

• Median = \( \frac{17 + 17}{2} = 17 \)

The median gives a great indication of the dataset's center, as it's not skewed by extremely high or low values like the mean might be.

Quartiles

Quartiles are numbers that divide your data into four equal parts. Each part represents a quarter of the dataset. Understanding quartiles gives insights into the spread and distribution beyond what the median tells you.

Here's how you find them:

• The first quartile (\(Q1\)) is the median of the first half of your dataset.
• The third quartile (\(Q3\)) is the median of the second half of your dataset.

For our data on Super Bowl scores, the first quartile \(Q1\):

• Values: \(7 + 7\)
• \( Q1 = \frac{7 + 7}{2} = 7 \)

And the third quartile \(Q3\):

• Values: \(23 + 25\)
• \( Q3 = \frac{23 + 25}{2} = 23.5 \)

These quartiles help in understanding the range where the middle 50% of the data lies. Quartiles especially come in handy to detect outliers or errors in the data.

Five-number summary

The five-number summary is a comprehensive descriptor of a data set, offering a quick overview of its distribution. It includes the minimum, first quartile \(Q1\), median, third quartile \(Q3\), and the maximum value. With these five values, you can assess the entire distribution at a glance.

For the given Super Bowl margins:

• Minimum value: 1
• First quartile \(Q1\): 7
• Median: 17
• Third quartile \(Q3\): 23.5
• Maximum value: 45

This summary helps in identifying the spread and symmetry of the data. By using the five-number summary, you can visualize the data layout, quickly understand the range, and the distribution pattern of your dataset. It's particularly useful in comparing different datasets and identifying outliers easily.