Population distribution

The population distribution is the probability distribution using all elements of a population.

Population versus sample

Suppose there are 5 students in a math class and their scores on the final exam are the followings:

80          85          85         90           92

Notice that we are only using 5 scores! Nonetheless, the 5 scores above are the scores for the population since we are using all 5 scores.

A population does not need to have 10,000,000 items in order for it to be called a population! It could have just 5 items as long we use all the elements.

Calculating the population distribution for the 5 scores above

Let x be the score of a student. We can make a frequency distribution table of scores as shown in the table below.


x f Relative Frequency
 80 1     1/5  = 0.20
 85  2     2/5  = 0.40
 90  1     1/5 = 0.20
 92  1     1/5 = 0.20
  5       Sum = 1

Here is the population probability distribution


x P(x)
80 0.20
85 0.40
90 0.20
92 0.20
  ΣP(x) =1

Calculating the mean using the population probability distribution

μ =  ΣxP(x) = 80 × 0.20 + 85 × 0.40 + 90 × 0.20 + 92 × 0.20 =  86.4

μ is a population parameter and it gives the average grade for the population distribution.

The value of μ or 86.4 is fixed or constant. In other words, there is only 1 value of the population mean.

The value of the standard deviation, which we do not compute here is also fixed or constant.

Sampling distribution



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