Math · Introductory statistics · Concept

Mean, median and standard deviation

Three numbers describe most data sets. The mean is the balance point, the median is the middle value, and the standard deviation measures how far values typically sit from the mean. The median resists outliers; the mean and the standard deviation do not.

The mean

Add the values and divide by how many there are. The mean is the balance point of the data: the deviations above it cancel the deviations below it.

\bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}

The median

Sort the values and take the middle one, or the average of the two middle ones when n is even. Half the data lie at or below the median, and half at or above it.

The standard deviation

Subtract the mean from each value, square the deviations, add them and divide by n − 1: that is the sample variance, and its square root is the sample standard deviation s. Squaring stops positive and negative deviations from cancelling, and the root returns the data’s own units.

s = \sqrt{\frac{\sum {(x_{i} - \bar{x})}^{2}}{n - 1}}

Why n − 1

The deviations are measured from the sample mean, which sits closer to the data than the unknown population mean does. Dividing by n − 1 instead of n corrects for that, so the sample variance estimates the population variance without bias. For a whole population, divide by N and write σ.

σ = \sqrt{\frac{\sum {(x_{i} - μ)}^{2}}{N}}

Quartiles and the five-number summary

The first and third quartiles, Q1 and Q3, cut off the lowest and highest quarters of the data, and the interquartile range Q3 − Q1 measures the spread of the middle half. Methods differ slightly: Chalk Inverse, like OpenStax and many calculators, takes the medians of the lower and upper halves, while much statistical software interpolates between neighboring sorted values.

Outliers

One extreme value pulls the mean and inflates the standard deviation but barely moves the median or the interquartile range. A common screen flags values more than 1.5 IQR below Q1 or above Q3.

Choosing a summary

For roughly symmetric data without outliers, report the mean and the standard deviation. For skewed data or data with outliers, the median and the interquartile range describe the center and spread more faithfully.

Common mistakes

Finding the median without sorting the data first.
Dividing by n for a sample standard deviation: a sample divides by n − 1.
Averaging the deviations without squaring them: they always add to zero.
Reporting the variance as the spread: it is in squared units, while the standard deviation is back in the data’s units.

Key terms

Mean: The average: mean = (sum of the values) ÷ (number of values), written x̄ = ∑x ÷ n. One extreme value can pull it a long way.
Median: The middle value when the data are in order; with an even count, the average of the two middle values. Extreme values affect it less than the mean.
Standard deviation: How spread out the values are around the mean, in the data’s own units: the square root of the variance. It describes individual values, not how precise the mean is.
Sample variance: The average squared distance from the sample mean, with n − 1 in the denominator: s² = Σ(x − x̄)²/(n − 1). Dividing by n − 1 instead of n corrects the tendency to underestimate the population variance.
Variance: The average of the squared distances from the mean, a measure of spread in squared units. The population variance divides by N; the sample variance divides by n − 1.
Quartile: The values that split ordered data into four equal parts: Q1 (25th percentile), the median (50th) and Q3 (75th). Methods differ slightly for samples; Chalk Inverse uses the median of each half.
Interquartile range: Q3 − Q1, the spread of the middle half of the data. Extreme values affect it much less than the full range.
Outlier: A value far from the rest of the data, often flagged when it lies more than 1.5 × IQR beyond a quartile. It may be an error, a rare real case or a sign of something different, so don’t delete it without a reason.

Work through an example

Eight students report how many hours they slept: 4, 8, 6, 5, 3, 7, 9 and 6. Find the mean, the median and the sample standard deviation.

Find the mean, median and standard deviation →

How an outlier moves the mean, not the median →

Find quartiles and the interquartile range →

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Summarize the data in Statistics Check the arithmetic in Math Open worked example on a board Sample standard deviation in Math Reference

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