Variance - formula and example

What is variance

The definition of variance: Variance is a measure of variability of the scores of our sample, how much they differ from the mean. The concept of variance is most important as it is the basis of statistical tests of significance such as the t-test and analysis of variance, ANOVA.

The formula for variance

The formula for variance is:
$$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$ We read this: variance is the sum 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.

The mean - formula

What is the Mean? The mean or average gives an average of a series of numbers. For example, the average body weight of my friends. The formula for calculating the mean is $$\bar{X} ={\sum{X} \over {n}}$$ X with the line on top we read: X bar (eks bar). It sands for the mean, the average of the scores in our sample. X stands for score. The capital Greek letter we read: Sigma. It stands for the Sum of the scores. The n we read: n. It stands for the number of scores in our sample.

Variance -practice example

An experimenter wanted to know what is the variance in the quantity of food young rats eat. He chose 6 rats randomly from the colony of rats in his lab. He weighed the food each ate in 24 hours in grams. Here are the data, The symbol for each score is X.

X 34 35 39 30 39 40


Looking at the variance formula above, we begin by finding the mean, the average. We add all of scores and divide by the number of scores. The symbol for the number of scores is n. In this case the n is 6. The sum of the scores is 217 . Now we divide this by 6, the n, and we find the mean 36.17.

To compute the variance we have to subtract each score from the mean $$X-\bar{X}$$ and then square this $$(X-\bar{X})^2$$ We then sum these squared deviations $$\sum{(X-\bar{X})^2}$$ Important! This is the sum of squared deviations of each sore from the mean.
It is the SS term on the ANOVA summary table!

and then divide by n, the number of scores, $${\sum{(X-\bar{X})^2}}\over{n-1}$$ This is variance $$s^2 ={\sum{({X}-{\bar{X})}}^2 \over {n-1}}$$
Here are the calculations
$$X-\bar{X}$$ -2.17 -1.17 2.83 -6.17 2.83 3.83
$$(X-\bar{X})^2$$ 4.7089 1.3689 8.0089 38.0689 8.0089 14.6689

$${\sum{({X}-{\bar{X})}}^2}$$ 74.8334

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.