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.

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

= population standard deviation

= sum of...

= 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

**Designing and Conducting Health Systems Research Projects:**Module 27 (Page 4) of this WHO guide provides instruction on the use of measures of dispersion.-
Open Learning Initiative: Probability and Statistics Course: Unit 1 of this free, online course gives a detailed introduction into examining distributions and measures of dispersions. Through its practical exercises, it offers instructions on how to generate these measures using a range of statistical software.

### Source

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

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