Describing, Exploring, and Comparing Data

Measures of Relative Standing and Boxplots

This section covers statistical methods for describing the position of data values within a data set. Key concepts include z scores, percentiles, quartiles, the 5-number summary, and boxplots. These tools help compare data values, identify outliers, and visualize data distribution.

z Scores

Definition and Calculation

A z score (also called standard score or standardized value) indicates how many standard deviations a data value x is above or below the mean. It allows for comparison across different data sets.

• Sample z score:

• Population z score:

Round-off Rule: Round z scores to two decimal places (e.g., 2.31).

Properties of z Scores

• A z score is the number of standard deviations a value is above or below the mean.

• z scores have no units of measurement.

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

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

Example: Comparing Data Values

Given two data values from different sets:

• 99°F body temperature (mean = 98.20°F, s = 0.62°F):

• 5.7790 g quarter (mean = 5.63930 g, s = 0.06194 g):

The quarter's weight is more extreme (farther above the mean) than the body temperature.

Using z Scores to Identify Significant Values

• Significantly low:

• Significantly high:

• Not significant:

Example: Earthquake Magnitude

Given mean = 2.572, s = 0.651, magnitude = 4.01:

Since , the magnitude is significantly high.

Percentiles

Definition

Percentiles are measures of location, denoted , dividing data into 100 groups with about 1% of values in each group.

Finding the Percentile of a Data Value

To find the percentile for a value x:

Example: Percentile Calculation

For a wait time of 45 minutes among 50 sorted values, with 36 values less than 45:

Interpretation: 45 minutes is the 72nd percentile ().

Notation

• n: total number of values

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

• L: locator for position in sorted list ( means 12th value)

• : kth percentile

Converting a Percentile to a Data Value

• Compute

• If is not a whole number, round up to next whole number

• The th value in the sorted list is

Example: 25th Percentile

• ,

• (round up to 13)

• 13th value is 25 minutes ()

Quartiles

Definition

Quartiles are measures of location, denoted , , , dividing data into four groups with about 25% of values in each group.

Descriptions of Quartiles

• (First quartile): Same as ; separates bottom 25% from top 75%.

• (Second quartile): Same as and the median; separates bottom 50% from top 50%.

• (Third quartile): Same as ; separates bottom 75% from top 25%.

Caution: Procedures for finding percentiles and quartiles may vary between technologies.

Statistics Defined Using Quartiles and Percentiles

• Interquartile range (IQR):

• Semi-interquartile range:

• Midquartile:

• 10–90 percentile range:

5-Number Summary

Definition

The 5-number summary for a data set consists of:

1. Minimum

2. First quartile ()

3. Second quartile (, median)

4. Third quartile ()

5. Maximum

Example: Finding a 5-Number Summary

All values are in minutes (from the "Space Mountain" wait times example).

Boxplot (Box-and-Whisker Diagram)

Definition

A boxplot (or box-and-whisker diagram) is a graphical representation of a data set, showing the minimum, , median (), , and maximum. It helps visualize the spread and skewness of data.

Procedure for Constructing a Boxplot

1. Find the 5-number summary.

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

3. Draw a box from to , with a line at the median ().

Additional info:

• Boxplots are useful for identifying skewness and outliers.

• Skewness is present if the boxplot is not symmetric and extends more to one side.

• Outliers can be identified using modified boxplots, where values beyond from or are marked as outliers.