Chapter III, “Summarizing Distributions” from Online Statistics Education: An Interactive Multimedia Course of Study comprises public domain material by David M. Lane, project leader, Rice University.

Measures of Central Tendencyby David M. Lane

Prerequisites• Chapter 1: Percentiles • Chapter 1: Distributions• Chapter 3: Central Tendency

Learning Objectives1. Compute mean2. Compute median3. Compute modeIn 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.

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 MeanThe 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:

Measures of central tendency

=���

� =���

=��� =

63431 = 20.4516

(4 + 7)2 = 5.5

Additional measures of central tendency

����� =(�25 + 2�50 + �75)

4

(15 + 2 × 20 + 23)4 =

784 = 19.5

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:

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Measures of central tendency

=���

� =���

=��� =

63431 = 20.4516

(4 + 7)2 = 5.5

Additional measures of central tendency

����� =(�25 + 2�50 + �75)

4

(15 + 2 × 20 + 23)4 =

784 = 19.5

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.

Measures of central tendency

=���

� =���

=��� =

63431 = 20.4516

(4 + 7)2 = 5.5

Additional measures of central tendency

����� =(�25 + 2�50 + �75)

4

(15 + 2 × 20 + 23)4 =

784 = 19.5

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.

MedianThe 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

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scores above the 16th score. The median can also be thought of as the 50th percentile.

Computation of the MedianWhen 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:

Measures of central tendency

=���

� =���

=��� =

63431 = 20.4516

(4 + 7)2 = 5.5

Additional measures of central tendency

����� =(�25 + 2�50 + �75)

4

(15 + 2 × 20 + 23)4 =

784 = 19.5

When there are numbers with the same values, then the formula for the third definition of the 50th percentile should be used.

ModeThe 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

500-600600-700700-800800-900

900-10001000-1100

365501

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Measures of Variabilityby David M. Lane

Prerequisites• Chapter 1: Percentiles• Chapter 1: Distributions• Chapter 3: Measures of Central Tendency

Learning Objectives1. Determine the relative variability of two distributions2. Compute the range3. Compute the inter-quartile range4. Compute the variance in the population5. Estimate the variance from a sample6. Compute the standard deviation from the variance

What is Variability?Variability refers to how “spread out” a group of scores is. To see what we mean by spread out, consider graphs in Figure 1. These graphs represent the scores on two quizzes. The mean score for each quiz is 7.0. Despite the equality of means, you can see that the distributions are quite different. Specifically, the scores on Quiz 1 are more densely packed and those on Quiz 2 are more spread out. The differences among students were much greater on Quiz 2 than on Quiz 1.

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Quiz 1

0

1

2

3

4

5

6

7

4 5 6 7 8 9 10

Quiz 2

0

1

2

3

4

5

4 5 6 7 8 9 10

Figure 1. Bar charts of two quizzes.

The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is. Just as in the section on central tendency where we discussed measures of the center of a distribution of scores, in this chapter we will

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discuss measures of the variability of a distribution. There are four frequently used measures of variability: range, interquartile range, variance, and standard deviation. In the next few paragraphs, we will look at each of these four measures of variability in more detail.

RangeThe range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life. The range is simply the highest score minus the lowest score. Let’s take a few examples. What is the range of the following group of numbers: 10, 2, 5, 6, 7, 3, 4? Well, the highest number is 10, and the lowest number is 2, so 10 - 2 = 8. The range is 8. Let’s take another example. Here’s a dataset with 10 numbers: 99, 45, 23, 67, 45, 91, 82, 78, 62, 51. What is the range? The highest number is 99 and the lowest number is 23, so 99 - 23 equals 76; the range is 76. Now consider the two quizzes shown in Figure 1. On Quiz 1, the lowest score is 5 and the highest score is 9. Therefore, the range is 4. The range on Quiz 2 was larger: the lowest score was 4 and the highest score was 10. Therefore the range is 6.

Interquartile RangeThe interquartile range (IQR) is the range of the middle 50% of the scores in a distribution. It is computed as follows:

IQR = 75th percentile - 25th percentile

For Quiz 1, the 75th percentile is 8 and the 25th percentile is 6. The interquartile range is therefore 2. For Quiz 2, which has greater spread, the 75th percentile is 9, the 25th percentile is 5, and the interquartile range is 4. Recall that in the discussion of box plots, the 75th percentile was called the upper hinge and the 25th percentile was called the lower hinge. Using this terminology, the interquartile range is referred to as the H-spread.

A related measure of variability is called the semi-interquartile range. The semi-interquartile range is defined simply as the interquartile range divided by 2. If a distribution is symmetric, the median plus or minus the semi-interquartile range contains half the scores in the distribution.

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VarianceVariability can also be defined in terms of how close the scores in the distribution are to the middle of the distribution. Using the mean as the measure of the middle of the distribution, the variance is defined as the average squared difference of the scores from the mean. The data from Quiz 1 are shown in Table 1. The mean score is 7.0. Therefore, the column “Deviation from Mean” contains the score minus 7. The column “Squared Deviation” is simply the previous column squared.

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Table 1. Calculation of Variance for Quiz 1 scores.

Scores Deviation from Mean Squared Deviation

9 2 4

9 2 4

9 2 4

8 1 1

8 1 1

8 1 1

8 1 1

7 0 0

7 0 0

7 0 0

7 0 0

7 0 0

6 -1 1

6 -1 1

6 -1 1

6 -1 1

6 -1 1

6 -1 1

5 -2 4

5 -2 4

MeansMeansMeans

7 0 1.5

One thing that is important to notice is that the mean deviation from the mean is 0. This will always be the case. The mean of the squared deviations is 1.5. Therefore, the variance is 1.5. Analogous calculations with Quiz 2 show that its variance is 6.7. The formula for the variance is:

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����

Measures of variability

� =�(� � �)

�

� =�(� ��)

� � 1

� =1 + 2 + 3 + 4 + 5

4 =124 = 3

� =(1� 3) + (2� 3) + (4� 3) + (5� 3)

4� 1 =4 + 1 + 1 + 4

3 =103 = 3.333

� =�� � (��)

��

� =�� � (��)

�� � 1

where σ2 is the variance, μ is the mean, and N is the number of numbers. For Quiz 1, μ = 7 and N = 20.

If the variance in a sample is used to estimate the variance in a population, then the previous formula underestimates the variance and the following formula should be used:

����

Measures of variability

� =�(� � �)

�

� =�(� ��)

� � 1

� =1 + 2 + 3 + 4 + 5

4 =124 = 3

� =(1� 3) + (2� 3) + (4� 3) + (5� 3)

4� 1 =4 + 1 + 1 + 4

3 =103 = 3.333

� =�� � (��)

��

� =�� � (��)

�� � 1

where s2 is the estimate of the variance and M is the sample mean. Note that M is the mean of a sample taken from a population with a mean of μ. Since, in practice, the variance is usually computed in a sample, this formula is most often used.

Let's take a concrete example. Assume the scores 1, 2, 4, and 5 were sampled from a larger population. To estimate the variance in the population you would compute s2 as follows:

����

Measures of variability

� =�(� � �)

�

� =�(� ��)

� � 1

� =1 + 2 + 3 + 4 + 5

4 =124 = 3

� =(1� 3) + (2� 3) + (4� 3) + (5� 3)

4� 1 =4 + 1 + 1 + 4

3 =103 = 3.333

� =�� � (��)

��

� =�� � (��)

�� � 1

There are alternate formulas that can be easier to use if you are doing your calculations with a hand calculator:

����

Measures of variability

� =�(� � �)

�

� =�(� ��)

� � 1

� =1 + 2 + 3 + 4 + 5

4 =124 = 3

� =(1� 3) + (2� 3) + (4� 3) + (5� 3)

4� 1 =4 + 1 + 1 + 4

3 =103 = 3.333

� =�� � (��)

��

� =�� � (��)

�� � 1

and

����

Measures of variability

� =�(� � �)

�

� =�(� ��)

� � 1

� =1 + 2 + 3 + 4 + 5

4 =124 = 3

� =(1� 3) + (2� 3) + (4� 3) + (5� 3)

4� 1 =4 + 1 + 1 + 4

3 =103 = 3.333

� =�� � (��)

��

� =�� � (��)

�� � 1

For this example,

150

�����=(1 + 2 + 4 + 5)�

4 =1444 = 36

� =(46 � 36)

4 = 2.5

�� =(46� 36)

3 = 3.333

Shapes of Distributions

3(���� � ������)

�(� � )�

�

�(� � )

� 3

Variance Sum law I

���� = �� + ��

�����=(1 + 2 + 4 + 5)�

4 =1444 = 36

� =(46 � 36)

4 = 2.5

�� =(46� 36)

3 = 3.333

Shapes of Distributions

3(���� � ������)

�(� � )�

�

�(� � )

� 3

Variance Sum law I

���� = �� + ��

�����=(1 + 2 + 4 + 5)�

4 =1444 = 36

� =(46 � 36)

4 = 2.5

�� =(46� 36)

3 = 3.333

Shapes of Distributions

3(���� � ������)

�(� � )�

�

�(� � )

� 3

Variance Sum law I

���� = �� + ��

as with the other formula.

Standard DeviationThe standard deviation is simply the square root of the variance. This makes the standard deviations of the two quiz distributions 1.225 and 2.588. The standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal (see Chapter 7) because the proportion of the distribution within a given number of standard deviations from the mean can be calculated. For example, 68% of the distribution is within one standard deviation of the mean and approximately 95% of the distribution is within two standard deviations of the mean. Therefore, if you had a normal distribution with a mean of 50 and a standard deviation of 10, then 68% of the distribution would be between 50 - 10 = 40 and 50 +10 =60. Similarly, about 95% of the distribution would be between 50 - 2 x 10 = 30 and 50 + 2 x 10 = 70. The symbol for the population standard deviation is σ; the symbol for an estimate computed in a sample is s. Figure 2 shows two normal distributions. The red distribution has a mean of 40 and a standard deviation of 5; the blue distribution has a mean of 60 and a standard deviation of 10. For the red distribution, 68% of the distribution is between 35 and 45; for the blue distribution, 68% is between 50 and 70.

151

Figure 2. Normal distributions with standard deviations of 5 and 10.

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