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 values from different data sets and identify significant or unusual observations.

z Scores

Definition and Calculation

• z Score: The number of standard deviations a data value (x) is above or below the mean.

• For a sample:

• For a population:

• Round z scores to two decimal places.

Properties of z Scores

• z scores are unitless.

• A z score less than or equal to −2 indicates a significantly low value.

• A z score greater than or equal to +2 indicates a significantly high value.

• Negative z scores indicate values below the mean.

Using z Scores to Identify Significant Values

• Significantly low values:

• Significantly high values:

• Values not significant:

Example: Comparing Data Values

• To compare values from different data sets, convert each to a z score.

• Example: A body temperature of 99°F has a z score of 1.29; a quarter weighing 5.7790 g has a z score of 2.26. The quarter's weight is more extreme relative to its data set.

Example: Identifying Significant Earthquake Magnitude

• Given: Mean = 2.572, Standard deviation = 0.651, Value = 4.01

• z score:

• Interpretation: Since 2.21 ≥ 2, the magnitude is significantly high.

Percentiles

Definition and Interpretation

• Percentiles divide a data set into 100 groups, each containing about 1% of the values.

• Notation: is the kth percentile (e.g., is the 25th percentile).

Finding the Percentile of a Data Value

• Count the number of values less than the given value, divide by the total number of values, and multiply by 100. Round to the nearest whole number.

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

Converting a Percentile to a Data Value

• Calculate the locator (where is the percentile and is the number of values).

• If is not a whole number, round up to the next whole number. The value at this position in the sorted data is the percentile value.

Quartiles

Definition and Description

• Quartiles divide data into four groups, each containing about 25% of the values.

• (first quartile): Same as ; separates the lowest 25% from the rest.

• (second quartile): Same as and the median; separates the lowest 50% from the highest 50%.

• (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, , Median (), , Maximum.

• Example (Space Mountain wait times): 10, 25, 35, 50, 110 (all in minutes).

Boxplots (Box-and-Whisker Diagrams)

Definition and Construction

• A boxplot is a graphical representation of 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.

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.

Skewness

Identifying Skewness with Boxplots

• A distribution is skewed if it is not symmetric and extends more to one side.

• Boxplots can help visually identify skewness in data distributions.

Identifying Outliers and Modified Boxplots

Procedure for Identifying Outliers

1. Find , , and .

2. Calculate .

3. Compute .

4. A value is an outlier if it is below or above .

Modified Boxplots

• Modified boxplots use special symbols (e.g., asterisks) to mark outliers.

• The whiskers extend only to the most extreme non-outlier values.