Effect of sample size on sampling error

The larger the sample size, the smaller the sampling error. However when the population is large, sampling a small proportion of the population may still give accurate estimates.

Sampling error depends much more strongly on the sample size than on the proportion of the population that is sampled.

For example, a sample of 10 from a population of 10,000 people will estimate the proportion of males almost as accurately as a sample of size 10 from a population of 100.

The cost savings from using a sample instead of a full census can be huge.


Cabbage varieties

A market gardener plants a field of 900 cabbages and notices that there are two different varieties of cabbage growing — seeds of the two varieties must have been mixed before planting. Variety A is known to be less resistant to dry conditions and will need to be harvested earlier, so the farmer wants to discover the proportion of this variety.

The diagram below illustrates how sampling can be used to estimate this proportion. Cabbages in the population of Variety A are shown as blue crosses.

Click the button Take sample to randomly select 180 of the 900 cabbages in the field. We would use the sample proportion of Variety A as an estimate of the population proportion (an unknown in a real field).

Click Take sample a few more times. The sample proportion is usually within 0.04 of the population proportion, and this would be accurate enough for the farmer to assess the work involved in harvesting this variety early.


The seed merchant who supplied the cabbage seeds above decides to test a container of 1,000,000 cabbage seeds.

Use the button Take sample a few times. The difference between the sample proportion of seeds of Variety A and the population proportion (0.200) is the sampling error.

Use the pop-up menu to investigate how the sample size affects the accuracy of the estimate. You should observe that the sampling error is usually smaller when the sample size is large.

In practice, a sample size of 1000 seeds would give the seed merchant a sufficiently accurate estimate of the proportion that are of Variety A. It is certainly hard to imagine a situation where more than 1% of this population would need to be sampled!