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Mean, median and standard deviation

The mean is the sum divided by the count, while the median is the middle value in sorted order, and is less sensitive to extreme values. The sample variance $s^2$ uses $n-1$ in the denominator, and the standard deviation $s$ is the square root of the variance — it has the same unit as the data.

\bar x = \frac{1}{n}\sum_i x_i

mean

s^2 = \frac{1}{n-1}\sum_i (x_i - \bar x)^2

sample variance

s = \sqrt{s^2}

sample standard deviation

Symbols

$\bar x$	mean of the sample
$n$	number of observations
$s$	sample standard deviation

Example

Data: 2, 4, 4, 4, 5, 5, 7, 9. Sum $= 40$ , $\bar x = 40/8 = 5$ .

The median of the same data (sorted, 8 values) is the average of the two middle values: $4.5$ .

Use

n-1

(not

n

) in the sample variance — it gives an unbiased estimate of the population variance.

Practise distributions for free →

← Poisson and exponential distribution · Hypothesis testing and p-value →

Part of Statistics and Risk Analysis: Distributions.