What is Dispersion?

Dispersion, also known as variability or spread, measures the extent to which individual data points deviate from the central value. It provides information about the spread or distribution of data points in a dataset.

Common measures of dispersion include

• Range
• Variance
• Standard Deviation
• Interquartile Range (IQR)

Range

In statistics, the range refers to the difference between the highest and lowest values in a dataset. It provides a simple measure of variability, indicating the spread of data points. The range is calculated by subtracting the lowest value from the highest value.

For example, in a dataset {4, 6, 9, 3, 7}, the range is 9 – 3 = 6.

Variance

Variance is a statistical measure that quantifies the amount of variation or dispersion of a set of numbers from their mean value. Specifically, variance is defined as the expected value of the squared deviation from the mean. It is calculated by:

1. Finding the mean (average) of the data set.
2. Subtracting the mean from each data point to get the deviations from the mean.
3. Squaring each of the deviations.
4. Calculating the average of the squared deviations. This is the variance.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion of a set of values from the mean value. It is calculated as the square root of the variance, which is the average squared deviation from the mean.

Interquartile Range (IQR)

Interquartile Range (IQR) is a measure of statistical dispersion that represents the middle 50% of a data set. It is calculated as the difference between the 75^th percentile (Q₃) and the 25^th percentile (Q₁) of the data i.e., IQR = Q₃ − Q₁.

Examples for Dispersion

Let’s consider the same dataset of daily temperatures recorded over a week: 22°C, 23°C, 21°C, 25°C, 22°C, 24°C, and 20°C.

Range: Maximum temperature – Minimum temperature = 25°C – 20°C = 5°C

Variance: Variance = (Sum of squared differences from the mean) / (Number of data points)

Mean = 21.86 °C

Sum of squared differences from the mean = (22 – 21.86)² + (23 – 21.86)² + (21 – 21.86)² + (25 – 21.86)² + (22 – 21.86)² + (24 – 21.86)² + (20 – 21.86)²

= (0.14)² + (1.14)² + (-0.86)² + (3.14)² + (0.14)² + (2.14)² + (-1.86)²

= 0.0196 + 1.2996 + 0.7396 + 9.8596 + 0.0196 + 4.5796 + 3.4596

= 19.0772

Thus, Variance = 19.0772 / 7 ≈ 2.725 °C

Standard Deviation: Take the square root of the variance to get the standard deviation.

Thus, Standard Deviation ≈ √2.725 ≈ 1.65 °C

Interquartile Range (IQR): First Quartile (Q₁) = 21°C Third Quartile (Q3) = 24°C

Thus, IQR = Q₃ − Q₁ = 24°C – 21°C = 3°C

Read More about Measure of Dispersion.

Measures of central tendency and dispersion are statistical measures used to describe the characteristics of a dataset. Central tendency helps us identify a single representative value around which data tends to cluster, whereas measures of dispersion tell us how deviated data is from the central value. Measures of central tendency include mean, median, and mode, while measures of dispersion include range, variance, standard deviation, and interquartile range. In this article, we will be discussing all these measures in detail.