Find the standard deviation of 4, 8, 6, 5, 3.

Step-by-step solution

1. 1

   Compute the mean first — every later step depends on it.

   $\bar{x} = \frac{4 + 8 + 6 + 5 + 3}{5} = \frac{26}{5} = 5.2$

   With five whole numbers the mean is not an integer, so keep it as $5.2$ exactly rather than rounding to $5$ — rounding here would distort every squared deviation.

2. 2

   Square each deviation from the mean. Deviations can be negative, and squaring is what removes the sign so that they do not cancel:

   $(4-5.2)^2 = 1.44, \quad (8-5.2)^2 = 7.84, \quad (6-5.2)^2 = 0.64$

   $(5-5.2)^2 = 0.04, \quad (3-5.2)^2 = 4.84$

3. 3

   Add them to get the sum of squares.

   $\sum (x - \bar{x})^2 = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8$

   A quick sanity check: the raw deviations $-1.2, 2.8, 0.8, -0.2, -2.2$ add to $0$, which they must if the mean is right.

4. 4

   Decide on the divisor: n or n − 1. Treating the five numbers as the entire population divides by $n = 5$; treating them as a sample drawn from something larger divides by $n - 1 = 4$ (Bessel’s correction, which compensates for the mean being estimated from the same data):

   $\sigma^2 = \frac{14.8}{5} = 2.96, \qquad s^2 = \frac{14.8}{4} = 3.7$

5. 5

   Take square roots.

   $\sigma = \sqrt{2.96} \approx 1.72, \qquad s = \sqrt{3.7} \approx 1.92$

   With a short list like this the two answers differ noticeably, so state which convention you used. Unqualified textbook exercises on a complete list of five numbers usually want the population value $1.72$.