Measures of Dispersion and Variability in Statistics

Note: While the range is easy to calculate, it is highly sensitive to outliers. A single extremely high or low value can significantly inflate the range, potentially giving a misleading impression of the typical spread of the data.

Variance and Standard Deviation

To get a more sophisticated understanding of variability, we use variance, which measures the average squared distance between each observation and the arithmetic mean. Because we square the differences, variance is always a positive number, and it heavily penalizes observations that are far from the mean.

Depending on whether you are analyzing an entire group or just a subset, you will use one of two formulas:

Population Variance

If you have data for every single member of the group you are studying (the population), you calculate the population variance ($\sigma^2$) as follows:

$\sigma^{2}=\frac{\sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2}}{N}$

Where:

• $\mu$ is the population mean.
• $N$ is the population size.
• $x_i$ is the $i^{th}$ value in the population.

Sample Variance

In most real-world scenarios, we only have access to a subset of the population (a sample). To account for the fact that a sample is only an estimate, we use the sample variance ($s^2$), which uses $n-1$ in the denominator instead of $n$:

$s^{2}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n-1}$

Where:

• $\bar{x}$ is the sample arithmetic mean.
• $n$ is the sample size.

Standard Deviation

While variance is mathematically useful, it can be difficult to interpret because its units are squared (e.g., if you are measuring height in meters, variance is in "meters squared"). To solve this, we use the standard deviation ($s$ or $\sigma$), which is simply the square root of the variance:

$\text{Standard Deviation} = \sqrt{\text{Variance}}$

The standard deviation is the most commonly used measure of variability because it returns the measurement to the same units as the original data, making it much easier to interpret.

The Coefficient of Variation (CoV)

A common question in statistics is: "Is this standard deviation high or low?" The answer depends on the scale of the data. A standard deviation of £100 is massive if you are measuring the price of a loaf of bread, but it is negligible if you are measuring the price of a luxury car.

To compare variability across different datasets or different units, we use the Coefficient of Variation (CoV). This is a measure of relative consistency that expresses the standard deviation as a proportion of the arithmetic mean:

$\text{Coefficient of Variation} = \frac{\text{Standard Deviation}}{\text{Arithmetic Mean}}$

• High CoV (e.g., > 1): Suggests the data is widely dispersed relative to the mean. In such cases, the arithmetic mean might not be a representative "typical" value for the dataset.
• Low CoV: Suggests the data is more consistent and clustered closely around the mean.

Exam Tip: Remember that a high or low CoV is not inherently "good" or "bad"; it is simply a descriptive feature of the data. However, if you are using the mean to create a forecast, a high CoV serves as a warning that your forecast may be unreliable.