CK-12-Pre-Calculus Concepts

15.5. Variance http://www.ck12.org

This method turns out to be extraordinarily powerful in statistics. One downside is that most of the time you
cannot get data from the entire population, you usually only get it from a sample. Over time people realized that
samples were typically less variable than their populations and dividing by the number of data points was consistently
underestimating the true variance of the population. In other words, ifnis the size of the sample then multiplying
the sum of the square differences by^1 nmakes the variance too small. Research and theory progressed until it was
realized that multiplying the sum of the square differences byn−^11 made the fraction slightly larger and properly
estimated the variance of the population. Thus, there are two ways to calculate variance, one for populations and
one for samples.

Hey wait, by squaring the differences, doesn’t that mean that the units are squared? What if I want to describe the
spread in the regular units? Should I just take the square root of the variance?
This is why the Greek letter lowercase sigma,σ, is used for standard deviation of a population (which is the square
root of the variance) andσ^2 is the symbol for variance of a population. The letterss,s^2 are used for sample standard
deviation and sample variance. The Greek letter mu,μ, is the symbol used for mean of a population, while xis the
symbol used for mean of a sample.
Mean and variance for the population:x 1 ,x 2 ,x 3 ,...,xn

μ=^1 n·

n

i∑= 1 xi

σ^2 =^1 n·

n

i∑= 1 (μ−xi)^2

Mean and variance for a sample from a population:x 1 ,x 2 ,x 3 ,...,xm

x=m^1 ·

m

i∑= 1 xi

s^2 =m−^11 ·

m

i∑= 1 (x−xi)^2

Remember that variance is a measure of the spread of data. The bigger the variance, the more spread out the data
points.
Example A
Calculate the variance and mean for rolling a fair six sided die.
Solution:μ=^16 ( 1 + 2 + 3 + 4 + 5 + 6 ) =^16 · 21 = 3. 5
Since the population for a six sided die is entirely known, you would use the population variance.

σ^2 =^16 [( 3. 5 − 1 )^2 +( 3. 5 − 2 )^2 +( 3. 5 − 3 )^2 +( 3. 5 − 4 )^2 +( 3. 5 − 5 )^2 +( 3. 5 − 6 )^2 ]

=^16 [ 6. 25 + 2. 25 + 0. 25 + 0. 25 + 2. 25 + 6. 25 ]

≈ 2. 9167

Example B
Calculate the mean and variance of the following data sample of lap times.