   SEARCH HOME Math Central Quandaries & Queries  Question from Anita, a student: The average height of a sample of basketball players is 6 feet, 2 inches or 74 inches. The standard deviation of this sample of players is 4 inches. If each person's height were to be multiplied by 2.54, what would be the value of the resulting sample variance? Hi Anita,

There are 2.54 centimeters in an inch so you are converting the units from inches to centimeters.

To find the average of a set of n heights you add them and them and then divide by n. If you multiply each height by 2.54 before you add them then the new sum will be 2.54 times the old sum and hence the new average will be 2.54 times the old average or 2.54 × 74 centimeters.

What about the variance? The standard deviation of the heights in inches is 4 inches so the variance is 42 = 16 inches squared. To calculate the variance you take each player's height, subtract the average, add them up and divide by n - 1. Thus for the heights in inches you found (height - 74)2, added them and divided by n - 1. When you convert to centimeters you have to find (2.54 × height - 2.54 × 74)2, add these and divide by n - 1. But

(2.54 × height - 2.54 × 74)2 = [2.54 × (height - 74)]2 = 2.542 × (height - 74)2

Now when you add these and divide by n - 1 you see that the new variance is 2.542 times the old variance.

I hope this helps,
Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.