Probability & Statistics

Standard Deviation of One Through Five

The numbers 1 through 5 have mean 3 and summed squared deviations 10. Dividing by n = 5 gives the population value sqrt(2), dividing by n - 1 = 4 the sample value sqrt(2.5), and an honest answer names its convention.

solvedeasy1 min

What is the standard deviation of the numbers 1,2,3,4,51, 2, 3, 4, 5?

Reveal solutionHide solution

#Mean and deviations

The mean is

xˉ=1+2+3+4+55=3.(1)\bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3. \tag{1}

The deviations from the mean are −2,−1,0,1,2-2, -1, 0, 1, 2, whose squares 4,1,0,1,44, 1, 0, 1, 4 sum to 1010.

mean 312345
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,

σ2=105=2,σ=2≈1.414.(2)\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=4n - 1 = 4,

s2=104=2.5,s=2.5≈1.581.(3)s^2 = \frac{10}{4} = 2.5, \qquad s = \sqrt{2.5} \approx 1.581. \tag{3}

#Population against sample

The population standard deviation is 2≈1.41\sqrt{2} \approx 1.41, and the sample standard deviation is 2.5≈1.58\sqrt{2.5} \approx 1.58. State which you mean, since the only difference is dividing the summed squared deviations by 55 or by 44.