The results of inferential statistics carry a margin of error. One example of this is in an opinion poll in which support for an issue is gauged at a certain percent, plus or minus a given percent. Another example is when we state that at a certain level of confidence, the mean is x̄ +/- *E*, where *E* is the margin of error. This range of values is due to the nature of the statistical procedures that are done, but the calculation of the margin of error relies upon a fairly simple formula.

Although we can calculate the margin of error just by knowing the sample size, population standard deviation and our desired level of confidence, we can flip the question around. What should our sample size be in order to guarantee a specified margin of error?

### Design of Experiment

This sort of basic question falls under the idea of experimental design. For a particular confidence level, we can have a sample size as large or as small as we want. Assuming that our standard deviation remains fixed, the margin of error is directly proportional to our critical value (which relies upon our level of confidence), and inversely proportional to the square root of the sample size.

The margin of error formula has numerous implications for how we design our statistical experiment:

- The smaller the sample size is, the larger the margin of error.
- To keep the same margin of error at a higher level of confidence, we would need to increase our sample size.
- Leaving everything else equal, in order to cut the margin of error in half we would have to quadruple our sample size. Doubling the sample size will only decrease the original margin of error by about 30%.

### Desired Sample Size

To calculate what our sample size needs to be, we can simply start with the formula for margin of error, and solve it for *n* the sample size. This gives us the formula *n* = (*z*_{α/2}σ/*E*)^{2}.

### Example

The following is an example of how we can use the formula to calculate a desired sample size.

The standard deviation for a population of 11th graders for a standardized test is 10 points. How large of a sample of students do we need to ensure at a 95% confidence level that our sample mean is within 1 point of the population mean?

The critical value for this level of confidence is *z*_{α/2} = 1.64. Multiply this number by the standard deviation 10 to obtain 16.4. Now square this number to result in a sample size of 269.

### Other Considerations

There are some practical matters to consider. Lowering the level of confidence will give us a smaller margin of error. However, doing this will mean that our results are less certain. Increasing the sample size will always decrease the margin of error. There may be other constraints, such as costs or feasibility, that do not allow us to increase the sample size.