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Location and spread answer different questions

The mean and median describe where data are centered. Variance and standard deviation describe how dispersed the values are around a center. The range uses only the minimum and maximum, while quantile-based summaries such as the interquartile range focus on the middle portion of the data.

Worked example: a compact dataset

Use the ordered dataset 2, 4, 4, 4, 5, 5, 7, 9.

The mean and median are close here, but they are not identical. That distinction becomes much more visible when the distribution contains an extreme value.

Population variance and standard deviation

For a population of N observed values with mean μ, population variance averages the squared deviations from that mean:

For the dataset above, the deviations from 5 are −3, −1, −1, −1, 0, 0, 2, and 4. Their squared deviations sum to 32. Dividing by 8 gives a population variance of 4, and the population standard deviation is 2.

The squaring step prevents positive and negative deviations from canceling. It also gives more influence to values farther from the mean. The tradeoff is that variance is measured in squared units, which is why standard deviation is often easier to interpret in the context of the original measurement.

Sample variance uses a different divisor

When the observed data are treated as a sample and the goal is to estimate population variability, the common sample-variance formula uses n − 1 in the denominator rather than n.

Using the same eight values, the squared deviations still sum to 32, but the divisor is 7. The sample variance is approximately 4.571, and the sample standard deviation is approximately 2.138. A statistics calculator should therefore make clear whether it is reporting population or sample values.

What an outlier does

Replace the final value 9 with 25. The new dataset is 2, 4, 4, 4, 5, 5, 7, 25.

The median stays at 4.5 because the two middle ordered values are unchanged. The mean increases from 5 to 7, while variance and standard deviation increase sharply because the extreme value contributes a large squared deviation. This is why the choice of summary should reflect the shape and purpose of the data rather than habit.

Median and interquartile summaries can be more robust

NIST’s Engineering Statistics Handbook distinguishes measures of location and measures of scale and discusses robust alternatives for data affected by heavy tails or outliers. In practical analysis, the median and interquartile range can provide a useful complement to the mean and standard deviation when extreme observations would otherwise dominate the summary.

That does not make the median automatically better. If every observation is meaningful and the arithmetic average is the quantity of interest, the mean may be exactly the right measure. Robustness is a property to consider, not a universal rule for choosing a statistic.

Common mistakes

• Using sample and population formulas interchangeably. The denominator changes, so the result changes.
• Reporting variance as if it used the original units. Variance is in squared units; standard deviation returns to the original units.
• Using only the range to describe spread. Range ignores every value except the minimum and maximum.
• Assuming mean and median must be similar. Skewed data and outliers can separate them substantially.
• Interpreting descriptive statistics as causal evidence. A mean or standard deviation summarizes the values supplied; it does not explain why they occurred.
• Rounding too early. Intermediate rounding can change the final variance or standard deviation.

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Authoritative references

• NIST Engineering Statistics Handbook: Measures of Location
• NIST Engineering Statistics Handbook: Measures of Scale