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Measures of Dispersion

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The measure of dispersion is the measure of the spread of scores in a data set. It is the extent to which the values vary around the central, or average value.

Let’s use the same example that we used in the explanation on Measures of central tendency:

Imagine that you are a first-year university student, and a friend asks you about the ages of people in your psychology course. You’ll say: ‘Well, most people are 18, there are a few in their 20’s and two or three over 40.’

In this example, you actually gave the dispersion of the ages of people. That is, how much they varied from the average age of 18 (a few in their 20s, two or three over 40.)

In a low dispersion data set, the values do not have much variation, e.g., 20, 22, 23, 24, 25, 27, 28.

In a high dispersion data set, there is a lot of variation, e.g., 9, 10, 14, 26, 35, 37, 39.

The importance of measures of dispersion

Measures of dispersion are important because if we don't know the dispersion, a mean value can be misleading.

Suppose there are two companies and we took a look at the wages they pay employees. The average wage might be the same, but the variation or dispersion of the wages might be very different. In Company A, all workers get a similar amount of wages. However, in Company B there is a large variation between the lowest-paid and the highest-paid employees.

Ways to measure dispersion

For A-Level psychology, there are two measures of dispersion that we focus on: the range and standard deviation.

Range

The range is the easiest way to calculate dispersion. The range is calculated by subtracting the lowest number from the highest number in a data set.

The highest value is 50, the lowest value is 12, the range would be 50-12 = 38.

• We are able to include extreme values (outliers) when calculating the range.

• It is easy to calculate.

• As extreme scores are included, the range could be distorted.

• The range does not tell us much information about the dispersion of values between the highest and lowest scores. For example, it does not give information about whether the values are close to the mean or more spaced out.

Standard deviation

The standard deviation (SD) is normally used when the mean is the measure of central tendency. The SD is a measure that calculates the distance of the individual scores from the mean of the dataset.

• Large SD: the scores are widely spread out above and below the mean. It indicates the mean is not representative of the data set.

• Small SD: the mean is a good representation of the scores in the data set.

Normally, statistics programs can calculate the SD but it is good to see the maths and understand how the SD is calculated. This is the formula for calculating SD:

s = standard deviation

∑ = sum of

X = each value in the data set

x̅ = the mean

n = number of values in the sample

Procedure

This is what you should do to find the standard deviation:

1. Find the mean of the data set (x̅).

2. Subtract the mean from each value in the data set, this is the deviation from the mean (x - x̅).

3. Square each deviation.

4. Find the sum of the squared deviations (∑).

5. Divide this number by n-1 (the total number of values in the data set minus 1).

6. Find the square root of this number.

Let us try this with a data set. Suppose we have a data set of numbers 48, 71, 34, 62, 54, 43.

1. Find the mean:

x̅ = (48 + 71 + 34 + 62 + 54 + 43) ÷ 6 = 52

1. Subtract the mean from each value in the data set:

47-52 = -5

70-52 = 18

33-52 = -19

61-52 = 9

53-52 = 1

42-52 = -10

1. Square each deviation:

(-5) ² = 25, 18² = 324, (-19) ² = 361, 9² = 81, 1² = 1, (-10) ² = 100

1. Find the sum of the squared deviations:

25 + 324 + 361 + 81 + 1 + 100 = 892

1. Divide this number by n-1:

892 / 6-1 = 892/5 = 178.4

1. Find the square root of this number:

√178.4 = 13.36

Thus, SD = 13.36.

For A-Level psychology, you won't be asked to calculate the SD. However, you might be asked to interpret and explain the SD for a data set.

• The SD can be used in estimates of population parameters, this is a reflection about the general population from a data set.
• The SD is the most sensitive measure of dispersion as all values in the data set are taken into account.

• The SD is distorted by extreme values.
• It is rather complicated to calculate.

Measures of Dispersion - Key takeaways

• The measure of dispersion is the measure of the spread of scores in a data set. It is the extent to which the values vary around the central or average value.

• If the dispersion is not known a mean value can be misleading.

• The range is the difference between the top and bottom values of a data set.

• The standard deviation is a measure of the distance of scores in a data set from the mean.

• A large standard deviation shows the scores are widely spread out above and below the mean, indicating the mean is not representative of the data set. A small standard deviation shows the mean is a good representation of the scores in the data set.

Measures of central tendency and dispersion both tell us vital information about a data set. Central tendency is the average or central  value of a data set and dispersion how spread out the values in a data set vary around the average (central tendency) value.

We have looked at two measures of dispersion, the range and standard deviation. Another measure of dispersion not covered at this stage is the interquartile range.

The range is the difference between the top and bottom values of a data set. To calculate the standard deviation, a special formula is used.

The standard deviation is the most sensitive measure of dispersion as all values in the data set are taken into account.

The measure of dispersion is the measure of the spread of scores in a data set. It is the extent to which the values vary around the central, or average value.

Final Measures of Dispersion Quiz

Question

What are measures of dispersion?

The measure of dispersion is the measure of the spread of scores in a data set. It is the extent to which the values vary around the central or average value.

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Question

Why are measures of dispersion important?

If we don’t know the dispersion, a mean value can be misleading. E.g., two datasets have the same mean, but there is a large difference in the datasets' variation of values.

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Question

How do you calculate the range?

The range is the difference between the highest and lowest values of a data set. For example, if the highest value is 50, and the lowest value is 12, the range would be 50-12 = 38.

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Question

What are the advantages of using the range?

• We are able to include extreme values (outliers) when calculating the range.

• It is easy to calculate

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Question

What are the disadvantages of using the range?

• As extreme scores are included, the range could be distorted.

• The range does not tell us much information about the dispersion of values between the top and bottom scores.

It does not give information about whether the values are close to the mean or more spaced out.

Show question

Question

What is the standard deviation a measure of?

The standard deviation is a measure of the mean distance of scores in a data set from the mean.

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Question

What does a large standard deviation indicate?

The scores are widely spread out above and below the mean, therefore the mean is not representative of the data set.

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Question

What does a small standard deviation indicate?

The mean is a good representation of the scores in the data set.

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Question

What are the advantages of using the standard deviation?

• The SD can be used in estimates of population parameters.

• The SD is the most sensitive measure of dispersion as all values in the data set are taken into account.

Show question

Question

What are the disadvantages of standard deviation?

• The SD is distorted by extreme values.

• It is rather complicated to calculate.

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