Standard deviation-formula and example

Standard deviation definition

Like Variance, it is a measure of variability of the scores of our sample, how much they differ from the mean. It is variance standardized by taking the square root of variance. We may develop the concept of standard deviation by examining the standard normal curve:

The surface area between two standard deviations can be calculated. Study the graphs below and always remember them. They are the basis of all other distributions (t-dstribution, F-distribution).


The standard normal curve. The shaded area represents 100% of the area.


The standard normal curve. The shaded area represents 99% of the area.


The standard normal curve. The shaded area represents 95% of the area.


The standard normal curve. The shaded area represents 34% of the area.


The standard normal curve. The shaded area represents 5% of the area.


Observe that in the standard normal curve there are 3 standard deviations σ on each side of the mean 0. The percentages shown are percentages of the area under the curve. The total area is 100%. Knowing the percentage of area between the mean and a point on the standard deviation line allows us to calculate the number of scores contained between these points.

Standard deviation formula

The formula for standard deviation for population is:
$$σ =\sqrt{\sum{({X}-{{μ})}}^2 \over {N}}$$

The formula for standard deviation for sample is $$s =\sqrt{\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$

We read this: Standard deviation is the square root of variance. The symbol for variance is \(s^2\). Variance is the sum of the difference of each score from the mean squared, over the sample size minus 1 (n-1). Another way of saying this is: To find variance we subtract each score from the mean and then square this difference. Then we add all of these squared differences and divide by the number of scores minus 1. (Please review the chapter on variance.)

Standard deviation practice example

SUMMARY OF THE ABOVE CALCULATIONS

X	$X-\bar{X}$	$(X-\bar{X})^2$
34 35 39 30 39 40	-2.17 -1.17 2.83 -6.17 2.83 3.83	4.7089 1.3689 8.0089 38.0689 8.0089 14.6689
ΣΧ=217 $\bar{X}=36.17$		$\sum{(X-\bar{X})^2}=74.8334 $.

Now we plug in the calculated values above to the variance formula.


$$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$
$s^2 ={74.8334 \over {6-1}}=12.47$ This is the variance of our data.


Finally the formula for standard deviation is: $$s =\sqrt{\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$ $$s =\sqrt{{s}^2 }$$ $$s =\sqrt{{s}^2 }=\sqrt{12.47}=3.53$$
This is the standard deviation of our data.