Measures of Relative Standing and Boxplots

Introduction

This section explores statistical tools used to describe the position of individual data values within a data set. Key concepts include z scores, percentiles, quartiles, and the construction and interpretation of boxplots. These measures help compare data values, identify outliers, and summarize data distributions.

z Scores

Definition and Calculation

A z score (also called a standard score or standardized value) indicates how many standard deviations a data value (x) is above or below the mean. It is calculated as follows:

• Sample z score:

• Population z score:

Round z scores to two decimal places.

Properties of z Scores

• A z score shows the number of standard deviations a value is from the mean.

• z scores are unitless.

• A value is significantly low if and significantly high if .

• If a value is less than the mean, its z score is negative.

Example: Comparing Data Values

Suppose we compare a body temperature of 99°F and a quarter weighing 5.7790 g. By converting both to z scores, we find:

• 99°F: (1.29 standard deviations above the mean)

• 5.7790 g: (2.26 standard deviations above the mean)

The quarter's weight is more extreme relative to its data set.

Using z Scores to Identify Significant Values

• Significantly low:

• Significantly high:

• Not significant:

For example, an earthquake magnitude of 4.01 with a mean of 2.572 and standard deviation of 0.651 yields , which is significantly high.

Percentiles

Definition and Interpretation

Percentiles divide a data set into 100 groups, each containing about 1% of the values. The kth percentile, , is the value below which k% of the data fall.

Finding the Percentile of a Data Value

To find the percentile for a value x:

• Count the number of values less than x.

• Divide by the total number of values, multiply by 100, and round to the nearest whole number.

Example: If 36 out of 50 wait times are less than 45 minutes, then 45 minutes is at the 72nd percentile ().

Notation

• n: Total number of values

• k: Percentile (e.g., for the 25th percentile, k = 25)

• L: Locator (position in the sorted list)

• Pk: kth percentile value

Converting a Percentile to a Data Value

To find the value at the kth percentile:

• Compute

• If L is not a whole number, round up to the next whole number. The value at this position is .

Example: For in a list of 50 values, ; round up to 13. The 13th value is $P_{25}$.

Quartiles

Definition and Description

Quartiles divide a data set into four groups, each containing about 25% of the values:

• Q1: First quartile (same as ), separates the lowest 25% from the rest.

• Q2: Second quartile (same as or the median), separates the lowest 50% from the highest 50%.

• Q3: Third quartile (same as ), separates the lowest 75% from the highest 25%.

Note: There is not universal agreement on the exact procedure for calculating quartiles; results may vary by method or technology.

Statistics Defined Using Quartiles and Percentiles

• Interquartile Range (IQR):

• Semi-interquartile Range:

• Midquartile:

• 10–90 Percentile Range:

5-Number Summary

Definition and Example

The 5-number summary consists of:

• Minimum

• First quartile ()

• Median ()

• Third quartile ()

• Maximum

Example: For "Space Mountain" wait times, the 5-number summary is 10, 25, 35, 50, and 110 minutes.

Boxplots (Box-and-Whisker Diagrams)

Definition and Construction

A boxplot is a graphical summary of data based on the 5-number summary. It consists of a box from to , a line at the median (), and "whiskers" extending to the minimum and maximum values.

• Find the 5-number summary.

• Draw a line from the minimum to the maximum value.

• Draw a box from to with a line at .

Skewness

Identifying Skewness with Boxplots

A distribution is skewed if it is not symmetric and extends more to one side. Boxplots can help identify skewness:

• Skewed right: Longer whisker on the right

• Skewed left: Longer whisker on the left

• Symmetric: Whiskers of similar length

Identifying Outliers and Modified Boxplots

Procedure for Identifying Outliers

• Find , ,

• Compute

• Calculate

• A value is an outlier if it is below or above

Example: For "Space Mountain" data, , . Outliers are values ≤ -12.5 or ≥ 87.5. Thus, 105 and 110 are outliers.

Modified Boxplots

A modified boxplot marks outliers with a special symbol (e.g., asterisk) and extends the whiskers only to the most extreme non-outlier values.