Everything Maths Grade 11

CHAPTER 18. STATISTICS 18.2

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5

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45

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60

Heads Tails

Frequency (%)

Face of Coin

Tails-Mean

Heads-Mean

Figure 18.1: The graphshows the results of 100 tosses of a fair coin, with 45 heads and 55 tails. The
mean value of the tosses is shown as a verticaldotted line. The difference between the meanvalue
and each data value is shown.

Population Variance

Let the population consist of n elements{x 1 ; x 2 ;.. .; xn}, with mean ̄x (read as ”x bar”). The variance
of the population, denoted by σ^2 , is the average of the square of the distance of each data value from
the mean value.

σ^2 =

(

�

(x− ̄x))^2

n

. (18.1)

Since the population variance is squared, it isnot directly comparablewith the mean and thedata
themselves.

Sample Variance

Let the sample consist of the n elements{x 1 ,x 2 ,... ,xn}, taken from the population, with mean ̄x. The
variance of the sample,denoted by s^2 , is the average of the squared deviations from the sample mean:

s^2 =

�

(x− ̄x)^2

n− 1

. (18.2)

Since the sample variance is squared, it is alsonot directly comparablewith the mean and thedata
themselves.

A common question atthis point is ”Why is thenumerator squared?” One answer is: to get rid of the
negative signs. Numbers are going to fall aboveand below the mean and, since the variance is looking
for distance, it would becounterproductive if those distances factored each other out.

Difference between Population Variance and Sample Variance

As seen a distinction is made between the variance, σ^2 , of a whole populationand the variance, s^2 of
a sample extracted fromthe population.

When dealing with the complete population the(population) variance isa constant, a parameter which
helps to describe the population. When dealing with a sample fromthe population the (sample)
variance varies from sample to sample. Its valueis only of interest as anestimate for the population
variance.