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  10095 = 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  (10095)^{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  1525= 10  (1525)^{2}= 100 
Pam  35  3515= 10  (3525)^{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  1525= 10  (1525)^{2}= 100 
Pam  35  3515= 10  (3525)^{2}= 100 
Ann  30  1525= 10  (3025)^{2}= 25 
Tommy  20  3515= 10  (2025)^{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.
