Standard Deviation of One Through Five

What is the standard deviation of the numbers $1, 2, 3, 4, 5$ ?

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#Mean and deviations

The mean is

\bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3. \tag{1}

The deviations from the mean are $-2, -1, 0, 1, 2$ , whose squares $4, 1, 0, 1, 4$ sum to $10$ .

The values spread evenly about the mean of 3. The shaded band reaches one population standard deviation, the square root of two, to either side, capturing the typical distance of a value from the centre.

#As a whole population

Taking the five numbers as the entire population, the variance is the mean squared deviation,

\sigma^2 = \frac{10}{5} = 2, \qquad \sigma = \sqrt{2} \approx 1.414. \tag{2}

#As a sample

If the five are instead a sample from something larger, divide by $n - 1 = 4$ ,

s^2 = \frac{10}{4} = 2.5, \qquad s = \sqrt{2.5} \approx 1.581. \tag{3}

#Result

The population standard deviation is $\sqrt{2} \approx 1.41$ , and the sample standard deviation is $\sqrt{2.5} \approx 1.58$ . State which you mean, since the only difference is dividing the summed squared deviations by $5$ or by $4$ .