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Chapter 3: Problem 72

The following dotplot shows the distribution of passing rates for the bar exam law schools in the United States in. The five number summary is $$ 0.60,0.84,0.90,0.94,1.00 $$ Draw the boxplot and explain how you determined where the whiskers go.

Short Answer

Expert verified
The boxplot has whiskers that extend from Q1 to the minimum value and from Q3 to the maximum value. They are drawn as lines showing the spread of the data outside the middle 50%, thus reaching from 0.60 to 0.84 on the lower end and from 0.94 to 1.00 on the upper end.

Step by step solution

01

Drawing the Box

To begin with, we draw a rectangular box that starts at Q1 (0.84) and ends at Q3 (0.94). The box represents the interquartile range where 50% of the values reside.
02

Drawing the Median Line

Next, we draw a line inside the box to represent the median (0.90). This line divides our box into two parts representing the values below (25%) and above (25%) the median.
03

Drawing the Whiskers

The whiskers of the boxplot are the two lines that extend from either end of the box indicating variability outside the upper and lower quartiles. The lower whisker begins at the minimum value (0.60) and stretches to Q1 (0.84). Similarly, the upper whisker starts at Q3 (0.94) and goes up to the maximum value (1.00). To visually represent this, we extend two lines from each end of the box, reaching to the minimum and maximum values respectively.

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Key Concepts

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

Five Number Summary
The five number summary is an essential tool in statistics to provide a quick overview of a data set. It comprises five key data points that give us a concise indication of the data distribution's spread and center.
  • Minimum: This is the smallest data point in the set. In the context of the bar exam passing rates mentioned, the minimum value is 0.60.
  • First Quartile (Q1): This value represents the 25th percentile, meaning 25% of the data falls below this point. Here, it is 0.84.
  • Median: The median indicates the middle point of the data set, where half the values are below it and half are above. In this case, it is 0.90.
  • Third Quartile (Q3): This denotes the 75th percentile. It shows that 75% of the data falls below this point. For the passing rates data, Q3 is 0.94.
  • Maximum: This is the largest data point, and here it is 1.00.
These numbers are powerful for understanding the initial spread and structure of the data set, giving us a summary before delving further with a boxplot.
Interquartile Range
The interquartile range (IQR) is a robust measure of statistical dispersion. It essentially measures the range within which the central 50% of the data points lie. Given by the formula IQR = Q3 - Q1, it effectively captures the spread of the middle half of the dataset.

  • In our example, the interquartile range is calculated as follows: IQR = 0.94 - 0.84 = 0.10.
  • This small interquartile range suggests that the central portion of the passing rates data is closely clustered, indicating relatively less variability among these rates.
  • The IQR can help identify outliers, which are data points that fall significantly lower than Q1 or higher than Q3.
By focusing on the IQR rather than the total range, we limit the impact of outliers and provide a more meaningful understanding of the data's spread.
Median
The median is a central value that effectively splits a data set into two halves. Often referred to as the "middle" of the data, it is particularly useful because it is not skewed by extremely high or low values. This makes it a simple yet powerful measure of central tendency.

  • In the given data of law school bar exam passing rates, the median is 0.90. This tells us that half of the schools have a passing rate below 90%, and the other half above it.
  • The median is depicted in the boxplot as a line inside the box, showing its position within the distribution of data.
  • This statistic is preferable over the mean when dealing with skewed distributions or outliers as it presents a clearer picture of the data's center.
Incredible in its simplicity, the median helps us understand more than just the center; it gives us insights into the symmetry and balance of the data.
Whiskers in Boxplot
The whiskers in a boxplot are lines that extend from either end of the box; they provide a visual representation of data variability outside the upper and lower quartiles.

  • In our example, the lower whisker extends from the lower end of the box (Q1 at 0.84) to the minimum value (0.60) in the data set.
  • The upper whisker extends from the upper end of the box (Q3 at 0.94) to the maximum value (1.00).
  • These whiskers help in understanding the spread of data beyond the middle 50%, highlighting the overall range of the dataset.
  • The whiskers often provide insight into potential outliers. If data points fall far outside these whiskers, they are typically considered outliers.
Accounting for data variability, the whiskers offer a straightforward visual cue regarding the entire distribution, bridging the key measures offered by the five number summary and the interquartile range.

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Most popular questions from this chapter

Wechsler IQ tests have a mean of 100 and a standard deviation of 15 . Which is more unusual: an IQ above 110 or an IQ below 80 ?

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