Measures of Dispersion

What is Dispersion in Statistics?

In statistics, dispersion describes how far the values of a variable are spread out – whether they cluster tightly around the center or scatter widely. Dispersion is also called variability, scatter or spread. The most important measures of dispersion are the standard deviation, the variance, the range and the interquartile range; in some textbooks they are also called dispersion parameters or measures of variability.

Measures of location describe the center of your data; measures of dispersion describe how widely the values are spread around that center.

Note that dispersion itself is not a single number you calculate – it is the general concept. Variance, standard deviation and range are the concrete measures used to quantify the dispersion of data.

Standard Deviation & Variance

The most common measures of dispersion for metric variables are the standard deviation and variance. Both are based on the deviations of the individual values from the mean and indicate how widely the values scatter around it.

What is the Standard Deviation?

The standard deviation indicates how widely values are spread around their mean. More precisely, it is the square root of the mean squared deviation from the mean. It is always zero or positive: a value of zero means that all observations are identical, and a larger value indicates greater spread when comparing data measured on the same scale.

If the individual values scatter widely around the mean, the standard deviation is large. There are two related formulas, depending on whether the data represent the entire population or a sample. If all values in the population are available, use the population formula:

Often, the entire population is not available. A sample is then used to estimate the population variance and standard deviation. The sample formula is:

In the population variance, the sum of squared deviations is divided by N. In the usual sample variance, it is divided by n - 1; this correction makes the sample variance an unbiased estimator of the population variance under random sampling. It is customary to use s for the standard deviation of a sample and σ for the standard deviation of the population.

What is the Variance?

Like the standard deviation, the variance measures dispersion around the mean. For a population, it is the sum of the squared deviations from the population mean divided by N. For a sample used to estimate population variance, the sum of squared deviations from the sample mean is usually divided by n - 1.

Variance is therefore the average squared deviation when describing a population, with the adjustment above when estimating it from a sample. Because deviations are squared, variance is expressed in squared units rather than the original unit, which makes it less intuitive to interpret directly.

Variance vs. Standard Deviation

Variance is based on squared deviations from the mean. Standard deviation is the square root of variance, so it is expressed in the same unit as the original data. Standard deviation is not the ordinary mean of the absolute distances from the mean; that is a different measure called the mean absolute deviation.

However, this squaring results in a key figure that is difficult to interpret, since the unit does not correspond to the original data. Standard deviation is therefore often reported when describing spread. Variance remains important in statistical methods such as analysis of variance and regression, so the appropriate measure depends on the purpose.

Range

The range, also called span, is the measure of dispersion that is calculated by subtracting the smallest value from the largest value. It is therefore the distance between the minimum and the maximum of a distribution. For example, if the height of 7 people is measured and the largest value is 1.90 m and the smallest is 1.50 m, the range is 1.90 m - 1.50 m = 0.4 m.

The range, often abbreviated with R, is therefore calculated by

Range vs. standard deviation: Both describe the dispersion of data, but in different ways. The range only uses the two most extreme values, which makes it very easy to calculate but also very sensitive to outliers. The standard deviation takes every value into account and summarizes the squared deviations from the mean. That is why the standard deviation is usually the preferred measure of dispersion for metric data, while the range is best suited for a quick first impression.

Quartiles

Quartiles are cut points that divide ordered data into four parts, as equal in size as possible. To calculate them, first sort the data from smallest to largest. Statistical software can use slightly different calculation conventions, especially for small data sets, so quartile values may occasionally differ between programs.

• First quartile (Q1): The 25th percentile; about 25% of observations are at or below this value.
• Second quartile (Q2): The median, or 50th percentile.
• Third quartile (Q3): The 75th percentile; about 75% of observations are at or below this value.

Because finite data sets and tied values do not always split exactly, these percentages should be understood as the positions the quartiles represent rather than a guarantee that exactly 25% or 75% of observed values are strictly below them.

Interquartile Range

The interquartile range (IQR) describes the spread of the middle 50% of the data. It is the distance from the first quartile to the third quartile and is calculated as IQR = Q3 - Q1. Unlike the full range, it is not determined by the minimum and maximum, so it is robust to outliers.

Example: Calculate Range, Variance and Standard Deviation

The calculation of range, variance and standard deviation is now illustrated with an example. For this purpose, the results of students in a statistics exam (scores) are used.

How it works in numiqo: The calculator for descriptive statistics reports the range, sample variance and sample standard deviation. Copy the above data into the Online Statistics Calculator, click on Descriptive Statistics and select the Score variable. The result will look like this:

Calculate Variance:

Because these exam scores are treated as a sample, the sample variance is the sum of the squared deviations from the sample mean divided by n - 1. Here, the mean is 11 and the sample variance is 31.778.

Calculate standard deviation:

The standard deviation is the square root of the variance.

Calculate range:

The range is obtained by subtracting the smallest value from the largest value.