Question

How many points do football teams score in the Super Bowl? Here are the total numbers of points scored by both teams in each of the first 42 Super Bowl games: \(45,47,23,30,29,27,21,31,22,38,46,37,66,50,37,47,44\), \(47,54,56,59,52,36,65,39,61,69,43,75,44,56,55,53,39\), \(41,37,69,61,45,31,46,31\) a) Find the median. b) Find the quartiles. c) Write a description based on the 5 -number summary.

Short answer

The median is 45, Q1 is 37, and Q3 is 55.

Step-by-step solution

Organize the Data

First, we need to arrange the given data in ascending order. Doing so gives us: 21, 22, 23, 27, 29, 30, 31, 31, 31, 37, 37, 37, 38, 39, 39, 39, 41, 43, 44, 44, 45, 45, 46, 46, 47, 47, 47, 50, 52, 53, 54, 55, 56, 59, 61, 61, 65, 66, 69, 69, 75.

Find the Median

The median is the middle number in a sorted list. With 42 numbers (an even set), the median is the average of the 21st and 22nd values.21st number: 45 22nd number: 45So the median is: \[\text{Median} = \frac{45 + 45}{2} = 45\]

Calculate the Quartiles

For quartiles, we divide the data set into quarters. For 42 numbers: - **Q1 (Lower Quartile)**: Median of the first 21 numbers. With numbers: 21, 22, 23, ..., 44 - Q1 is the 11th value: 37 - **Q3 (Upper Quartile)**: Median of the last 21 numbers. With numbers: 45, 45, 46, ..., 75 - Q3 is the 32nd value: 55

5-Number Summary

The 5-number summary consists of the minimum, Q1, median, Q3, and maximum values: - **Minimum**: 21 - **Q1**: 37 - **Median**: 45 - **Q3**: 55 - **Maximum**: 75 This provides insights into the spread and center of the data.

Key concepts

Median

The median is a measure of central tendency that represents the middle value in a data set when arranged in ascending order. It provides a useful summary of the data center, especially when the data includes outliers or extreme values, which could skew other statistical measures like the mean.
In the Super Bowl data of football scores, with an even number of observations (42 total scores), the median is calculated by averaging the 21st and 22nd scores in the ordered list. Both are 45, hence the median score is 45. This indicates that half of the Super Bowl games had scores below this value, and the other half had scores above it.

Quartiles

Quartiles divide a data set into four equal parts, making it easier to understand the distribution of data within segments. They help in identifying the spread and concentration of data values across the dataset.
- **First Quartile (Q1)**: This marks the 25th percentile and is the median of the first half of the data. In the Super Bowl scores, Q1 is the 11th value in the sorted data, which is 37. This means that 25% of the scores are below 37. - **Third Quartile (Q3)**: This marks the 75th percentile and is the median of the second half of the data. For the Super Bowl scores, Q3 is the 32nd value, 55, indicating that 75% of the scores fall below 55.
These quartiles help us understand where the bulk of the data lies and its variability.

5-number summary

A 5-number summary provides a quick and informative way to describe a set of data. It includes five key numerical descriptions: Minimum, Lower Quartile (Q1), Median, Upper Quartile (Q3), and Maximum.

• **Minimum (21)**: The lowest score, providing a baseline for the sample's range.
• **Q1 (37)**: The score below which 25% of the scores fall, indicating the lower quartile.
• **Median (45)**: The central value that divides the data into two equal halves.
• **Q3 (55)**: The upper quartile, showing the score below which 75% of the data lies.
• **Maximum (75)**: The highest score, reflecting the upper end of the data set range.

These elements together give a comprehensive picture of the dataset’s spread and center, helping to visualize its distribution and identify any potential outliers.

Data Organization

Organizing data is a crucial step in descriptive statistics as it lays the foundation for further analysis. By arranging data in ascending order, we can easily calculate measures like the median, quartiles, and the 5-number summary.
For the Super Bowl scores, sorting the scores from lowest (21) to highest (75) allowed for accurate determination of the median and quartiles. This systematic arrangement aids not only in simplifying calculations but also in revealing patterns or trends in the data that may not be immediately apparent when the data is unordered.

Statistical Analysis

Statistical analysis involves interpreting data to uncover patterns, trends, and insights. Using measures like the median and quartiles, alongside a 5-number summary, provides succinct information about the dataset, highlighting its central tendency, spread, and any potential outliers.
In the context of the Super Bowl score data, statistical analysis reveals that while most games' total scores cluster around 45, there is considerable range and variability, with some games having significantly higher scores. Such analysis can be pivotal in understanding past trends and potentially forecasting future events, thereby offering practical insights into data-driven decision making.