Measures of Variability

Measures of Variability

The measures of variability tell you about the spread or variation of values around the mean. They are also called measures of dispersion, and they indicate the degree of diversity in a data set. These measures also show the difference within the group itself.

Two different data sets can have the same mean, median, or mode but completely different levels of dispersion — or the opposite can happen, where the spread is similar but the central values differ. The common measures of dispersion or variability are Range and Standard Deviation.

Range

Range is the difference between the maximum and minimum values of the data — the largest value minus the smallest value. It can be calculated only for numerical data. Range tells you about the coverage or spread of data values, for example the difference in salaries of employees, marks of a student, or price of toys.

Since range is calculated based on only the two extreme values, any outlier in the data badly influences the result. If M is the largest or maximum value and S is the smallest or minimum value in the data, then:

Range = M – S, or Maximum – Minimum

Example: For a set of heights where the minimum height is 85 cm and the maximum height is 115 cm, the range is 115 – 85 = 30 cm.

Standard Deviation

Standard deviation refers to differences within the group or set of data of a variable. Like range, it also measures the spread of data. However, unlike range which only uses two extreme values, the calculation of standard deviation considers all the given data.

It is calculated as the positive square root of the average of the squared difference of each value from the mean value of the data. A smaller value of standard deviation means the data are less spread, while a larger value means the data are more spread.

Given n values x₁, x₂, x₃, ... xₙ, and their mean x̄, the standard deviation (represented by the Greek letter sigma, σ) is computed as:

σ = √[ Σ(xᵢ – x̄)² / n ]

Example: Let us compute the standard deviation of the height of nine students. The mean (x̄) was calculated to be 101.33 cm. Subtract each value from the mean and take the square of that value. Dividing the sum of square values by the total number of values and taking its square root gives the standard deviation.

n = 9, x̄ = 101.33, Σ(x – x̄) = 0.03, Σ(x – x̄)² = 938.00

σ = √(938.00 / 9) = √104.22 ≈ 10.2 cm

Choose the Suitable Statistical Technique

Let us look at the following problems and select the suitable statistical technique to be applied (Mean / Median / Mode / Range / Standard Deviation). This is a self-check exercise from the textbook — try to work out your own answer for each before moving on; the book does not supply the answers.