Measures of Center

Introduction

Measures of center are statistical values that describe a typical or central value for a data set. The three most common measures of center are the mean, median, and mode. Choosing the appropriate measure depends on the type and distribution of the data.

Mean

Sample Mean

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

• Formula:

$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $

• Where: $x_i$ is the ith data value, and $n$ is the number of data values in the sample.

Example 1: Calculating the Sample Mean

Given the data: 5, 6, 8, 10, 4, 6, 9

Sample mean: $ \bar{x} = \frac{5+6+8+10+4+6+9}{7} = \frac{48}{7} = 6.9 $

Example 2: Using the Mean to Find a Data Value

Suppose the mean cost of five songs is $1.23, and four songs cost $1.29 each. To find the price of the fifth song ($x$):

$ \frac{1.29 + 1.29 + 1.29 + 1.29 + x}{5} = 1.23 $

Solve for $x$:

$ 5 \times 1.23 = 4 \times 1.29 + x $ $ 6.15 = 5.16 + x $ $ x = 6.15 - 5.16 = 0.99 $

Weighted Mean

The weighted mean is used when data values have different levels of importance (weights).

• Formula:

$ \bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} $

• Where: $x_i$ is the ith data value, $w_i$ is the weight of the ith value.

Example 3: Calculating a Weighted Mean

Suppose a final grade is determined by: Tests (40%), Homework (20%), Quizzes (10%), Final Exam (30%). Scores: 83, 98, 90, 87.

Weighted mean: $ \bar{x}_w = \frac{0.4 \times 83 + 0.2 \times 98 + 0.1 \times 90 + 0.3 \times 87}{1.0} = 87.9 $

Example 4: Finding a Required Value for a Weighted Mean

Given known grades and weights, solve for the unknown grade needed to achieve a desired final average using the weighted mean formula.

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 ($n$) is odd, the median is the middle value.

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

Example 5: Finding the Median

Data: 3, 4, 6, 7, 2, 8, 9 (ordered: 2, 3, 4, 6, 7, 8, 9). Median is 6 (middle value).

Data: 5, 7, 8, 1, 4, 9, 8, 9 (ordered: 1, 4, 5, 7, 8, 8, 9, 9). Median is $ \frac{7+8}{2} = 7.5 $.

Mode

Definition of Mode

The mode is the value that occurs most frequently in a data set.

• If all values occur once or equally often, there is no mode.

• If one value occurs most often, the data set is unimodal.

• If two values occur equally most often, the data set is bimodal.

• If more than two values occur equally most often, the data set is multimodal.

Example 6: Finding the Mode

Data: 6, 4, 6, 1, 7, 8, 7, 2, 5, 7. Mode is 7 (occurs three times), so the data set is unimodal.

Choosing an Appropriate Measure of Center

Guidelines for Choosing a Measure

• 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 7: Choosing the Most Appropriate Measure

• T-shirt sizes (S, M, L, XL): Mode (qualitative)

• Salaries for a professional baseball team: Median (skewed by high salaries)

• Prices of homes in a subdivision of similar homes: Mean (no outliers)

• Professor rankings (best, average, worst): Mode (qualitative)

Graphs and Measures of Center

Relationship of Mean, Median, and Mode in Distributions

• 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 or the tail of a skewed distribution.

Example 8: Interpreting a Distribution Graph

In a right-skewed distribution:

• Line A (at the peak) is the mode.

• Line B (divides area in half) is the median.

• Line C (farthest right) is the mean.

Table: Properties of Mean, Median, and Mode