What are the main measures of central tendency and when should each be used?

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AcadiFi · Accepted Answer

The three main measures of central tendency are the arithmetic mean, the median, and the mode. Each answers a different question.

The arithmetic mean sums all values and divides by n. For annual returns of {5%, 8%, 12%, -4%, 45%}, the mean is 13.2%. It's the go-to estimator when data are roughly symmetric, but it is extremely sensitive to outliers — the 45% observation pulls the average up.

The median is the middle value after ordering the data. For the same series, the median is 8%. It's robust to outliers and is preferred when the distribution is skewed, like hedge-fund returns or household wealth. For example, median net worth tells a more representative story than the mean because billionaire outliers distort the average.

The mode is the most frequent value. It matters for categorical data (most common credit rating in a bond portfolio, say BBB) and for discrete distributions. It's rarely used alone for continuous financial data.

When to choose what:
- Mean: symmetric, well-behaved distributions (e.g., index returns over long periods after log-transformation).
- Median: skewed distributions, presence of outliers (returns during crisis periods, salary data).
- Mode: categorical classifications, identifying peaks in multimodal distributions.

CFA candidates should also know the geometric mean for compounded returns and the harmonic mean for average ratios. Choosing the wrong measure can make a strategy look acceptable when tail risk is actually severe.

What are the main measures of central tendency and when should each be used?

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