# The Mean and the Median

The two most common measures of central tendency are the median and the mean, which can be illustrated with an example. Suppose we draw a sample of five women and measure their weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150 pounds.

- To find the
**median**, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Thus, in the sample of five women, the median value would be 130 pounds; since 130 pounds is the middle weight.

- The
**mean**of a sample or a population is computed by adding all of the observations and dividing by the number of observations. Returning to the example of the five women, the mean weight would equal (100 + 100 + 130 + 140 + 150)/5 = 620/5 = 124 pounds. In the general case, the mean can be calculated, using one of the following equations:Population mean = μ = ΣX / N OR Sample mean = x = Σx / nwhere ΣX is the sum of all the population observations, N is the number of population observations, Σx is the sum of all the sample observations, and n is the number of sample observations.

When statisticians talk about the mean of a population, they use the Greek letter μ to refer to the mean score. When they talk about the mean of a sample, statisticians use the symbol x to refer to the mean score.

## The Mean vs. the Median

As measures of central tendency, the mean and the median each have advantages and disadvantages. Some pros and cons of each measure are summarized below.

- The median may be a better indicator of the most typical value if a set of scores has an
**outlier**. An outlier is an extreme value that differs greatly from other values.

- However, when the sample size is large and does not include outliers, the mean score usually provides a better measure of central tendency.

To illustrate these points, consider the following example. Suppose we examine a sample of 10 households to estimate the typical family income. Nine of the households have incomes between $20,000 and $100,000; but the tenth household has an annual income of $1,000,000,000. That tenth household is an outlier. If we choose a measure to estimate the income of a typical household, the mean will greatly over-estimate the income of a typical family (because of the outlier); while the median will not.

## Effect of Changing Units

Sometimes, researchers change units (minutes to hours, feet to meters, etc.). Here is how measures of central tendency are affected when we change units.

- If you add a constant to every value, the mean and median increase by the same constant. For example, suppose you have a set of scores with a mean equal to 5 and a median equal to 6. If you add 10 to every score, the new mean will be 5 + 10 = 15; and the new median will be 6 + 10 = 16.

- Suppose you multiply every value by a constant. Then, the mean and the median will also be multiplied by that constant. For example, assume that a set of scores has a mean of 5 and a median of 6. If you multiply each of these scores by 10, the new mean will be 5 * 10 = 50; and the new median will be 6 * 10 = 60.

## Test Your Understanding of This Lesson

**Problem 1**

Four friends take an IQ test. Their scores are 96, 100, 106, 114. Which of the following statements is true?

I. The mean is 103.

II. The mean is 104.

III. The median is 100.

IV. The median is 106.

II. The mean is 104.

III. The median is 100.

IV. The median is 106.

(A) I only

(B) II only

(C) III only

(D) IV only

(E) None is true

(B) II only

(C) III only

(D) IV only

(E) None is true

**Solution**

The correct answer is (B). The mean score is computed from the equation:

Mean score = Σx / n = (96 + 100 + 106 + 114) / 4 = 104

Since there are an even number of scores (4 scores), the median is the average of the two middle scores. Thus, the median is (100 + 106) / 2 = 103.