Measures of Center

Introduction to Measures of Center

Measures of center are statistical values that describe a typical or central value for a data set. The three most common measures are the mean, median, and mode. Each measure provides a different perspective on the data and is appropriate in different contexts.

Mean

Sample Mean

The sample mean is the arithmetic average of a set of data values. It is calculated by summing all the data values and dividing by the number of values in the sample.

• Formula: where is the ith data value and is the number of data values.

• Example: For the data set 5, 6, 8, 10, 4, 6, 9, the mean is

• Application: The mean is sensitive to outliers and skewed data.

Using the Mean to Find a Data Value

• Example: If the mean cost of five songs is

Weighted Mean

The weighted mean is used when different data values contribute unequally to the overall mean. Each value is multiplied by a weight representing its relative importance.

• Formula: where is the ith data value and is its weight.

• Example: If a final grade is based on tests (40%), homework (20%), quizzes (10%), and a final exam (30%), and the scores are 83, 98, 90, and 87, respectively:

Median

Finding the Median of a Data Set

The median is the middle value when the data are ordered. It divides the data set into two equal halves.

• Order the data from smallest to largest.

• If the number of values () is odd, the median is the middle value.

• If is even, the median is the mean of the two middle values.

• Example: For the data set 3, 4, 6, 7, 2, 8, 9 (ordered: 2, 3, 4, 6, 7, 8, 9), the median is 6.

Mode

Definition of Mode

The mode is the value that appears most frequently in a data set. A data set may have one mode (unimodal), two modes (bimodal), or more (multimodal). If all values occur with the same frequency, there is no mode.

• Example: For the data set 6, 4, 6, 1, 7, 8, 7, 2, 5, 7, the mode is 7 (appears most often).

Choosing an Appropriate Measure of Center

Guidelines for Choosing a Measure of Center

• For qualitative data, use the mode.

• For quantitative data without outliers or skewness, use the mean.

• For quantitative data with outliers or skewness, use the median.

• Example:

  • T-shirt sizes (S, M, L, XL): Use mode.

  • Salaries for a professional baseball team: Use median (due to outliers).

  • Prices of homes in a subdivision of similar homes: Use mean.

  • Professor rankings (best, average, worst): Use mode.

Graphs and Measures of Center

Relationship Between Graphs and Measures of Center

• The mode is at the highest peak of the distribution.

• The median divides the area under the distribution in half.

• The mean is pulled toward outliers and the tail in a skewed distribution.

• In a right-skewed distribution (as shown above):

  • Line A: Mode (at the peak)

  • Line B: Median (divides area in half)

  • Line C: Mean (pulled toward the right tail)

Table: Properties of Mean, Median, and Mode