Module 1 · Descriptive Statistics · Topic 3

Measures of Center

Central Tendency. Measures of center describe a typical or representative value in a data set. The three main measures are the mean, median, and mode.

Median and Mode. The median is the middle value when data is sorted. For an even number of observations, average the two middle values. The mode is the most frequently occurring value - a data set can have no mode, one mode, or multiple modes.

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Key formulas

Mean (Arithmetic Average)

x̄ = Σxᵢ / n

From the Descriptive Statistics formula sheet

Mean: xbar = sum(x_i) / n — The arithmetic average of a data set.
Sample Variance: s^2 = sum((x_i - xbar)^2) / (n - 1) — Average squared distance from the sample mean with Bessel correction.

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Worked example

Worked Example

Data: {3, 7, 7, 12, 21}. Mean = (3 + 7 + 7 + 12 + 21) / 5 = 50 / 5 = 10. Median = 7 (middle value). Mode = 7 (appears twice).

When to use it

When to Use Which

Use the mean for symmetric data without outliers. Use the median for skewed data or when outliers are present. Use the mode for categorical data or to find the most common value.

Related glossary terms

Outlier: A data point that is unusually far from the rest of the data. Example: In 2, 4, 5, 6, 50, the value 50 is a likely outlier.

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