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  1. Introduction
  2. Graphing Distributions

  3. Summarizing Distributions
    1. Contents
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    2. Central Tendency
      Standard         Video
    3. What is Central Tendency
      Standard         Video
    4. Measures of Central Tendency
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    5. Balance Scale Simulation
      Standard         Video
    6. Absolute Differences Simulation
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    7. Squared Differences Simulation
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    8. Median and Mean
      Standard         Video
    9. Mean and Median Demo
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    10. Additional Measures
      Standard         Video
    11. Comparing Measures
      Standard         Video
    12. Variability
      Standard         Video
    13. Measures of Variability
      Standard         Video
    14. Variability Demo
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    15. Estimating Variance Simulation
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    16. Shapes of Distributions
      Standard         Video
    17. Comparing Distributions Demo
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    18. Effects of Linear Transformations
      Standard         Video
    19. Variance Sum Law I
      Standard         Video
    20. Statistical Literacy
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    21. Exercises
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  4. Describing Bivariate Data
  5. Probability
  6. Research Design
  7. Normal Distribution
  8. Advanced Graphs
  9. Sampling Distributions
  10. Estimation
  11. Logic of Hypothesis Testing
  12. Tests of Means
  13. Power
  14. Regression
  15. Analysis of Variance
  16. Transformations
  17. Chi Square
  18. Distribution Free Tests
  19. Effect Size
  20. Case Studies
  21. Calculators
  22. Glossary
 

Measures of Central Tendency

Author(s)

David M. Lane

Prerequisites

Percentiles, Distributions, Central Tendency

Learning Objectives
  1. Compute mean
  2. Compute median
  3. Compute mode

In the previous section we saw that there are several ways to define central tendency. This section defines the three most common measures of central tendency: the mean, the median, and the mode. The relationships among these measures of central tendency and the definitions given in the previous section will probably not be obvious to you. Rather than just tell you these relationships, we will allow you to discover them in the simulations in the sections that follow.

This section gives only the basic definitions of the mean, median and mode. A further discussion of the relative merits and proper applications of these statistics is presented in a later section.

Arithmetic Mean

The arithmetic mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers. The symbol "μ" is used for the mean of a population. The symbol "M" is used for the mean of a sample. The formula for μ is shown below:

μ = ΣX/N
where ΣX is the sum of all the numbers in the population and
N is the number of numbers in the population.


The formula for M is essentially identical:

M = ΣX/N
where ΣX is the sum of all the numbers in the sample and
N is the number of numbers in the sample.

As an example, the mean of the numbers 1, 2, 3, 6, 8 is 20/5 = 4 regardless of whether the numbers constitute the entire population or just a sample from the population.

Table 1 shows the number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season. The mean number of touchdown passes thrown is 20.4516 as shown below.

μ = ΣX/N
  = 634/31
  = 20.4516

Table 1. Number of touchdown passes.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

Although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by far the most commonly used. Therefore, if the term "mean" is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.

Median

The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the same number of scores is above the median as below it. For the data in Table 1, there are 31 scores. The 16th highest score (which equals 20) is the median because there are 15 scores below the 16th score and 15 scores above the 16th score. The median can also be thought of as the 50th percentile.

Computation of the Median

When there is an odd number of numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 4. When there is an even number of numbers, the median is the mean of the two middle numbers. Thus, the median of the numbers 2, 4, 7, 12 is (4+7)/2 = 5.5. When there are numbers with the same values, then the formula for the third definition of the 50th percentile should be used.

Mode

The mode is the most frequently occurring value. For the data in Table 1, the mode is 18 since more teams (4) had 18 touchdown passes than any other number of touchdown passes. With continuous data such as response time measured to many decimals, the frequency of each value is one since no two scores will be exactly the same (see discussion of continuous variables). Therefore the mode of continuous data is normally computed from a grouped frequency distribution. Table 2 shows a grouped frequency distribution for the target response time data. Since the interval with the highest frequency is 600-700, the mode is the middle of that interval (650).

Table 2. Grouped frequency distribution.

Range Frequency
500-600
600-700
700-800
800-900
900-1000
1000-1100		 3
6
5
5
0
1
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