Measures of dispersion

Measures of dispersion provide information about how much variation there is in the data, including the range, inter-quartile range and the standard deviation.

Range

The range is simply the difference between the maximum and minimum values in a data set. 

Range = max - min

So in a data set of 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 11, 13, 15, 15, 17, 19, 20, the range is the difference between 2 and 20.

18 = 20 - 2

While it is useful in seeing how large the difference in observations is in a sample, it says nothing about the spread of the data.

Inter-quartile range

The inter-quartile range (IQR) gives more information about how the observation values of a data set are dispersed. It shows the range of the middle 50% of observations.

The IQR is found by first finding the median (see Measures of Central Tendency) of a data set, then by finding the medians of the bottom 50% of the data and the top 50% of the data. 

  1. Find the median: 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 11, 13, 15, 15, 17, 19, 20
  2. Find the first quartile (Q1), which is the median of the bottom 50%: 2, 2, 3, 4, 5, 5, 6, 7, 8
  3. Find the second quartile (Q2), which is the median of the top 50%: 8, 9, 11, 13, 15, 15, 17, 19, 20

The IQR is the difference between Q1 and Q2.

IQR = Q2 - Q1

​10 = 15 - 5

The IQR is a necessary measure of spread when using the median as a measure of central tendency.

Standard Deviation

The standard deviation indicates the average distance between an observation value, and the mean of a data set. In this way, it shows how well the mean represents the values in a data set. Like the mean, it is appropriate to use when the data set is not skewed or containing outliers.

The formula for calculating the standard deviation of a sample is:

mathematical equation for standard deviation

 

standard deviation sign

 = population standard deviation

sum sign

 = sum of...

mathematical symbol for population mean, greek mu

 = population mean
n = number of scores in sample.

(taken from Laerd Statistics, 2013)

However the standard deviation can be easily calculated with statistical programs and online calculators.

Resources

Sources

Laerd Statistics (2013). 'Standard deviation' [Webpage]. Retrieved from https://statistics.laerd.com/statistical-guides/measures-of-spread-standard-deviation.php

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