Mean and Median
The mean and the median are summary
measures used to describe the most "typical" value in a set of
values. Statisticians sometimes refer to the mean and median as measures of central
tendency.
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The Mean and the Median
The difference between the mean and median can be illustrated with an example.
Suppose we draw a sample of five teenage boys and measure their
weights. They weigh 100 pounds, 100 pounds, 130 pounds, 140 pounds, and 150
pounds.
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
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.
(A) I only
(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.