## More Information

### Worked Example

If you would like to estimate the mean systolic blood pressure of British adults with 95% confidence and a margin of error no larger than 2mmHg, how many samples are required? Assuming a population variance of 100 then a sample size of 96 is sufficient. Increasing the margin of error to 3mmHg, reduces the sample size to 43, whilst increasing the variance to 225 increases the sample size to 216.

### Formula

This calculator uses the following formula for the sample size n:

n = N*X / (1 + X-1),

where,

X = Z_{α/2}^{2} * σ^{2} / MOE^{2},

and Z_{α/2} is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, σ^{2} is the population variance, and N is the population size. Note that a Finite Population Correction has been applied to the sample size formula.

### Discussion

The above sample size calculator provides you with the recommended number of samples required to estimate the true population mean with the required margin of error and confidence level.

You can use the Alternative Scenarios to see how changing the four inputs (the margin of error, confidence level, population size and population variance) affect the sample size. By watching what happens to the alternative scenarios you can see how each input is related to the sample size and what would happen if you didn’t use the recommended sample size. The larger the sample size, the more certain you can be that the estimates reflect the population, so the narrower the confidence interval. However, the relationship is not linear, e.g., doubling the sample size does not halve the confidence interval.

For some further information, see our blog post on The Importance and Effect of Sample Size.

### Definitions

#### Margin of error

The margin of error is the the level of precision you require. This is the plus or minus number that is often reported with an estimated mean and is also called the confidence interval. It is the range in which the true population mean is estimated to be. Note that the actual precision achieved after you collect your data will be more or less than this target amount, because it will be based on the population variance estimated from the data and not your expected variance.

#### Confidence level

The confidence level is the probability that the margin of error contains the true mean. If the study was repeated and the range calculated each time, you would expect the true value to lie within these ranges on 95% of occasions. The higher the confidence level the more certain you can be that the interval contains the true mean.

#### Population size

This is the total number of distinct individuals in your population. In this formula we use a finite population correction to account for sampling from populations that are small. If your population is large, but you don’t know how large you can conservatively use 100,000. The sample size doesn’t change much for populations larger than 100,000.

#### Population variance

This is calculated as:

σ^{2} = (1/N)* ∑^{N}_{i=1}(x_{i}-μ)^{2},

where,

μ = (1/N)* ∑^{N}_{i=1}x_{i}

and gives you an indication of how variable the population is. When performing significance tests, the sample variance provides an estimate of the population variance for inclusion in the formula.

#### Sample size

This is the minimum sample size you need to estimate the true population mean with the required margin of error and confidence level. Note that if some people choose not to respond they cannot be included in your sample and so if non-response is a possibility your sample size will have to be increased accordingly. In general, the higher the response rate the better the estimate, as non-response will often lead to biases in your estimate.