In order to prove that a process has been improved, you must measure the process capability before and after improvements are implemented. This allows you to quantify the process improvement (e.g., defect reduction or productivity increase) and translate the effects into an estimated financial result â€“ something business leaders can understand and appreciate. If data is not readily available for the process, how many members of the population should be selected to ensure that the population is properly represented? If data has been collected, how do you determine if you have enough data?

Determining sample size is a very important issue because samples that are too large may waste time, resources and money, while samples that are too small may lead to inaccurate results. In many cases, we can easily determine the minimum sample size needed to estimate a process parameter, such as the population mean .

When sample data is collected and the sample mean is calculated, that sample mean is typically different from the population mean . This difference between the sample and population means can be thought of as an error. The margin of error is the maximum difference between the observed sample mean and the true value of the population mean :

is known as the critical value, the positive value that is at the vertical boundary for the area of in the right tail of the standard normal distribution.

is the population standard deviation.

is the sample size.

Letâ€™s put all this statistical mumbo-jumbo to work. Take for example that we would like to start an Internet service provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model.

### Sample Size Calculation Example

**Problem**

We would like to start an ISP and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 95 percentÂ sure that the sample mean is within 1 minute of the population mean . Assume that a previous survey of household usage has shown = 6.95 minutes.

**Solution**

We are solving for the sample size .

A 95% degree confidence corresponds to = 0.05. Each of the shaded tails in the following figure has an area of = 0.025. The region to the left of and to the right of = 0 is 0.5 – 0.025, or 0.475. In the table of the standard normal () distribution, an area of 0.475 corresponds to a value of 1.96. The critical value is therefore = 1.96.