Measures of Central Tendency and 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.

What is Central Tendency?

Central tendency is a statistical concept that refers to the tendency of data to cluster around a central value or a typical value.

It identifies a single value as representative of an entire data distribution. In other words, the central tendency is a way of describing the centre or midpoint of a dataset. The three most common measures of central tendency are:

• Mean
• Median
• Mode

Let’s discuss these as follows:

Mean

The mean, also known as the average, is calculated by summing up all the values in a dataset and dividing the sum by the total number of values. It is sensitive to extreme values, making it susceptible to outliers.

Median

The median is the middle value in a dataset when the values are arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. Unlike the mean, the median is less affected by outliers.

Mode

The mode is the value that occurs most frequently in a dataset. It is particularly useful for categorical data but can also be applied to numerical data. A dataset may have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

Note: The choice of which central tendency measure to use depends on the properties of the data. For example, the mean is best for symmetric distributions, while the median is better for skewed distributions with outliers. The mode is useful for categorical data.

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

• Mean: (22 + 23 + 21 + 25 + 22 + 24 + 20) / 7 = 21.86°C

• Median: Arranging the temperatures in ascending order: 20°C, 21°C, 22°C, 22°C, 23°C, 24°C, 25°C. The median is 22°C.

• Mode: The mode is 22°C as it occurs most frequently in the dataset.

Read More about Measure of Central Tendency.

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.

Differences between Central Tendency and Dispersion

The key differences between central tendency and dispersion are listed in the following table:

Aspect	Central Tendency	Dispersion
Definition	Indicates the typical or central value around which data tends to cluster	Indicates the spread or variability of data points in a dataset
Purpose	Provides a single representative value summarizing the dataset	Describes how spread out the values are from each other and from the central value
Examples	Mean, median, mode	Range, variance, standard deviation, interquartile range
Calculation	Calculated using the values of the dataset	Calculated using deviations from the central value (usually the mean)
Interpretation	Helps in understanding the center of the data distribution	Helps in understanding the spread of the data distribution
Measure of Location	Indicates where the data is centered	Indicates how widely the data is spread out

Read More,

• Qualitative Data
• Mean, Median, and Mode of Grouped Data
• Difference Between Mean, Median, and Mode with Examples
• Variance and Standard Deviation
• Quartiles and Quartile Deviation

FAQs on Central Tendency and Dispersion

Define central tendency.

Central tendency refers to the tendency of data to cluster around a central or typical value in a dataset. It provides a summary of the center of the data distribution.

What are the common measures of central tendency?

The common measures of central tendency are mean, median, and mode.

When is the median preferred over the mean?

The median is preferred when the data is skewed or has outliers, as it is less affected by extreme values.

What is dispersion in statistics?

Dispersion refers to the spread or variability of data points in a dataset. It provides information about how spread out the values are from each other and from the measures of central tendency.

What are the common measures of dispersion?

The common measures of dispersion include range, variance, standard deviation, and interquartile range (IQR).

How does range measure dispersion?

Range measures dispersion by calculating the difference between the maximum and minimum values in a dataset.

What does variance indicate about a dataset?

Variance indicates the average squared deviation of each data point from the mean of the dataset. It measures how far the data points are spread out from the mean.

When is the interquartile range (IQR) used?

The interquartile range (IQR) is used to measure the spread of the middle 50% of the data, ignoring outliers. It is less sensitive to extreme values compared to range and standard deviation.