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Chapter 2: Problem 65

Female strength The High School Female Athletes data file on the text CD has data for 57 high school female athletes on the maximum number of pounds they were able to bench press, which is a measure of strength. For these $$ \text { data, } \bar{x}=79.9, \mathrm{Q} 1=70, \text { median }=80, \mathrm{Q} 3=90 . $$ a. Interpret the quartiles. b. Would you guess that the distribution is skewed or roughly symmetric? Why?

Short Answer

Expert verified
a. Quartiles show data dispersion: 25% below 70, 50% below 80, 75% below 90. b. Distribution is roughly symmetric since mean 鈮� median.

Step by step solution

01

Understanding Quartiles

Quartiles divide a data set into four equal parts. \( \text{Q1} \) represents the first quartile, meaning 25% of the data is below this value (70 pounds in this case). The median divides the data in half, so 50% of the values are below 80 pounds. \( \text{Q3} \) is the third quartile, indicating that 75% of the data is below this value (90 pounds).
02

Describe the Data Distribution with Quartiles

Given \( \text{Q1} = 70 \), median = 80, and \( \text{Q3} = 90 \), each quartile represents a 10-pound increase. This can suggest a balanced distribution between the first quartile and the median, and between the median and the third quartile.
03

Consider the Skewness based on Quartiles and Mean

The mean \( \bar{x} = 79.9 \) is almost equal to the median (80), indicating that the data is likely to be symmetric. In a symmetric distribution, the mean and median are similar. Skewness would typically move the mean relative to the median.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quartiles
Quartiles are essential in summarizing a data set by dividing it into four equal parts. These points help in understanding the spread and center of the data:
  • First Quartile (Q1): This is the point below which 25% of the data falls. For our data, Q1 is 70 pounds. It means that a quarter of the athletes can bench press less than 70 pounds.
  • Median: This is the midpoint of the data, dividing it into two equal halves. For this exercise, the median is 80 pounds, showing that half of the athletes can bench press less than this amount, and the other half more.
  • Third Quartile (Q3): It represents the 75% mark of the distribution. Here, Q3 is 90 pounds, which tells us that 75% of the athletes bench press below this weight, meaning only the top 25% can surpass 90 pounds.
Understanding these quartiles allows us to analyze and interpret the athletes' strength clearly and concisely.
Data Distribution
Data distribution explains how data values are spread or arranged. Understanding the structure of data distribution helps in identifying patterns and variations, which is crucial for statistical analysis.
  • In our case, the range that quartiles cover is 20 pounds (from Q1 to Q3).
  • This range shows the central 50% data spread, from 70 to 90 pounds.
  • A consistent increase between each quartile (10 pounds) suggests a balanced distribution.
With quartiles spaced relatively evenly, as seen in this data, we get an idea that the distribution does not have extreme variations or outliers that dominate. The spread between quartiles suggests how spread out the data is around the mean and median, offering insights into the uniformity or spread of bench press abilities among the athletes.
Skewness
Skewness indicates whether data is symmetrical or tilted towards one side. It's a measure of asymmetry in data distribution.
  • If the mean and median are close together, like in this example (79.9 is almost equal to 80), the distribution is likely symmetric.
  • Skewness moves the mean away from the median. If, for example, the mean were significantly higher or lower than the median, it would indicate skewness.
  • An equal distance between Q1, median, and Q3, further supports the idea that the data is not skewed.
Hence in the case of these athletes, since the mean (79.9) and the median (80) are nearly equal, this suggests that there isn't significant skewness in the data, indicating a fairly balanced distribution on either side of the center.
Symmetric Distribution
A symmetric distribution is one where the left and right sides of the histogram are approximately mirror images.
  • For the data to be symmetric, the mean, median, and mode should all align closely.
  • In our exercise, the mean and median are very close, implying symmetry.
  • This type of distribution means that data points are evenly distributed around the central point, in this case, around the median value of 80.
Symmetric distributions often imply that there are no unusual surprises like long tails, that is, values far removed from the center. In terms of interpreting data on athletes' bench pressing abilities, this symmetry gives us confidence that their performance levels are generally consistent with fewer extreme variations.

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Most popular questions from this chapter

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