Sampling one value from a finite population

Random sampling from populations is described using probability. If one value is sampled from a finite population of N distinct values, we say that

The definition can be extended to populations where some values occur more than once. In particular, when one value is randomly selected from a categorical population, the probability of obtaining a particular value is the proportion of population values equal to it.

The probability that a single sampled value is x is the proportion of times this value occurs in the population.

Categorical example

In the population of 44 categorical values below, there are 27 'success' and 17 'failure' values. The probability that a single value sampled from this population is a success is therefore 27/44.

Household size in Mauritius

The bar chart below shows the sizes of all households in Mauritius in its 2011 census. Dual axes are shown to display both the number of households and proportion of each size.

If a single household is randomly selected in Mauritius, the probability that it will be of any particular size equals the population proportion of households of that size in the census.

Click on the bars to read off the probabilities.

Probability of getting one of several values

When one value is sampled from a population, the probability of getting a particular value, x, is the proportion of population values that equal x. A similar definition is used for the probability that the sampled value is either x, y, ...

The probability that a single sampled value is either x, y, ... is the proportion of population values that are either x, y, ... .

If the values are numerical, this definition gives the probability of getting a value within some range. For example, if 12 values in a population of 100 values are under 3.5, we say that the probability that a single sampled value will be under 3.5 is 12/100 = 0.12. To express this in an equation, we use the symbol X for the value that is sampled and write

Prob( X < 3.5 )   =   0.12

More generally,

Prob( a < X < b )   =   propn of values between a and b.

Wheat yields

The diagram below shows a jittered dot plot of wheat yields (tonnes per hectare) from all 60 farms in one region of the United States in 1998.

The farms with yields between 2.5 and 2.8 tonnes per hectare are highlighted. The probability that a single farm selected at random from those in the region will have a yield between 2.5 and 2.8 is the proportion of highlighted values.

Drag the left and right edges of the highlighted area to display the probabilities of getting a value in other ranges.