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Population Variance: Definition, Formula & Example

Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

Population variance can sometimes seem tricky, but after learning what it is and how to figure it out it's a breeze. This lesson will define population variance, teach you the formula and give you some practice.

Population Variance

Karen is a new biologist studying adult lions in the wild. In this dangerous field of study, she definitely wants to study under someone who knows the ropes, which is why she found a native wildlife scientist, Anan, to study under. It took a few months, but the pair had finally taken the weight measurements of all adult female lions in their study area. As it happened, this was only ten females because there had been some hardships for the lion population. One day Anan asked Karen to compute the variance of this measurement among all the adult female lions in the study area. Let's see what she did.

Definition & Formula for Population Variance

Population variance is a fancy term for how much a specific measurement is expected to vary in a given population. If the measurement varies widely from individual to individual, it will have a high variance, whereas if the measurement only varies by a small amount, it will have a small variance.

You may have heard of standard deviation, which is closely related to the variance. In fact, the standard deviation is simply the square root of the variance. The equation used to calculate the variance is:

population varience

The calculations to figure out the variance are rather time consuming for humans, yet very quick for computer programs. Unfortunately for Karen, her laptop had gotten trampled on by a herd of wildebeests and she would have to wait three weeks to get a new one. In the meantime, she still needed to calculate the variance. She wanted to keep her data organized, so she made a table in her notebook with some room to add more columns on the right:

Female Lion Subject Weight (lbs.)
A34 307
A117 351
B17 285
B193 317
B194 325
C12 277
C26 287
C29 299
D99 333
D153 319

From the equation, Karen knew she had to compute the mean weight of the adult lionesses first.

Mean weight = (307 + 351 + 285 + 317 + 325 + 277 + 287 + 299 + 333 + 319) / 10

Mean weight = 3,100 / 10

Mean weight = 310

Next, she needed to determine the difference between each weight and the mean weight of the population, and then square that number. She added two more columns and one row to her table, and slowly started filling in the numbers. After quite a while, she was able to come up with the following:

Lion Subject Weight (lbs.) (Xi - X) (Xi - X)(Xi - X)
A34 307 -3 9
A117 351 41 1,681
B17 285 -25 625
B193 317 7 49
B194 325 10 100
C12 277 -33 1,089
C26 287 -23 529
C29 299 -11 121
D99 333 23 529
D153 319 9 81
Totals 3,100 - 4,938

Now she took the total of the squared differences, 4,938, and divided by the total number of female lions in the population she was studying.

4,938 / 10 = 493.8 lb^2

She had her answer!

Her elation didn't last long, however. She wasn't sure if her units were right: what was a square pound? And 493 of them sounded like quite a lot. She took her notebook to Anan to see if he could help her.

Anan smiled and nodded, 'You have done this correctly. It just looks strange because that's how variance works. If you want a less strange number, take the square root of that number - that will be the standard deviation. Also, you could have waited until you had your computer to do these computations.'

The square root of 493.8 = 22 lbs. So, roughly 65% of adult female lions in this area are expected to be within 22 lbs. of 310 lbs., or between 388 lbs. and 332 lbs.

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