Sample Variance from Population Variance

Variance describes the distance between each data point and the center. Statisticians realized quickly that if they just added each distance, then the total distance would be 0.

Data Value Score Distance from Mean
Person1 100 100-95 = 5
Person2 95 95 – 95 = 0
Person3 90 90 – 95 = -5

Mean of Scores: 95

Instead of adding 5 + 0 + -5 = 0, statisticians square each distance. This also makes small distances smaller and big distances bigger. So, this variance calculation better describes how far each datapoint is from the mean.

Data Value Score Squared distance from Mean
Person1 100 (100-95)2 = 25
Person2 95 (95 – 95)2 = 0
Person3 90 (90 – 95)2 = 25

Now the total distance is 25 + 0 + 25 = 50, and the average distance is 50 divided by 3, about 16.67.

Here is an example of finding the variance within a population of two people.

Population of 2 Distance between data point and mean Squared Distance Between data point and mean
John 15 15-25= -10 (15-25)2= 100
Pam 35 35-15= 10 (35-25)2= 100
Mean: 25 Total Squared Distance 100+100=200
Average Squared Distance 200 ÷ 2 = 100

The mean is 25 so the squared distance for both people are 100. The average distance is (100 + 100) ÷ 2 = 100.

This second example has a population of 4 people, and the mean is still 25.

Population of 4 Distance between data point and mean Squared Distance Between data point and mean
John 15 15-25= -10 (15-25)2= 100
Pam 35 35-15= 10 (35-25)2= 100
Ann 30 15-25= -10 (30-25)2= 25
Tommy 20 35-15= 10 (20-25)2= 25
Mean: 25 Total Squared Distance 100+100+25+25=250

So the population variance decreases to 62.5 because the two new scores are closer to the mean. Your question is based on the idea that the average of 100, 100, 25, 25 is equal to the average of 62.5, 62.5, 62.5, 62.5. We could represent this in the calculation of the total squared distances.

Population of 4 Squared Distance between data point and mean Hypothetical Squared Distance
100 62.5
100 62.5
25 62.5
25 62.5
Sum: 250 250
Mean: 250 ÷ 4 = 62.5 250 ÷ 4 = 62.5

So, if you know the population variance is 25. Then, you could estimate on average that each person has an individual variance of 25. In a sample of 20, the table would look like this:

Population of 20 Squared Distance between data point and mean Hypothetical Squared Distance
Person 1 Data Unknown 25
Person 2 25
Person 3 Actual distance unknown 25
Person 4 25
Person 5 25
Person 6 25
Person 7 25
Person 8 25
Person 9 25
Person 10 25
Person 11 25
Person 12 25
Person 13 25
Person 14 25
Person 15 25
Person 16 25
Person 17 25
Person 18 25
Person 19 25
Person 20 25
Sum: 500
Mean: 25

This means that the Sum of Squared Distances (SS) is 500, on average. If we wanted to calculate a sample deviation, then we would divide SS/df, 500 ÷ 19 = 26.32.

The question is hard conceptually because it isn’t realistic. You almost never know the population variance, and it isn’t used to estimate the variance in theoretical samples.

But, you can calculate SS by estimating that every person in the sample has the population variance. Then, calculating that sum.

If you have any further questions, just shoot me a text! Khan Academy walks through these calculations step by step.