When a general probability of an event is known about a process, it is possible to determine the precise number of observations to be taken. The required number of observations can be calculated based on the general probability of the event, the desired accuracy of that probability, and the desired confidence level.

## Calculation

Convert the general odds of the event to be observed to a percentage. The accuracy will be based on how close to this probability the answer should be. For example, if an estimated one in 10 products is manufactured incorrectly, the probability is 10 percent.

Determine the confidence level required. This will be a level of statistical accuracy in any results found in the observations. This value is between zero and 100 percent. According to “Modern Construction: Lean Project Delivery and Integrated Practices,” by Lincoln H. Forbes and Syed M. Ahmed, “a confidence level of 95 percent and a limit of error or accuracy of 5 percent is generally adequate.”

Determine the desired accuracy level. This value is typically between 1 percent and 10 percent. Accuracy level will be based on how close to the 10 percent probability set in Step 1 the data observations will be.

Look up the Z value, also called the standard normal deviate, for the desired confidence level on the Standard Normal (Z) Table. For a 95 percent confidence level, the Z value is 1.96.

Change the confidence level from a percent to a decimal. A 95 percent confidence level becomes 0.95.

Change the accuracy level from a percentage to a decimal. A 5 percent accuracy level becomes 0.05.

Subtract the probability of occurrence from 1. For a probability of occurrence estimated to be 10 percent, 1-0.10 = 0.90.

Multiply the result of Step 7 by the odds of occurrence. For a 10 percent probability of occurrence, this will be 0.90 multiplied by 0.10 to yield 0.09.

Square the Z value found at Step 4 by referencing the Standard Normal (Z) Table. Multiply the result with the value from Step 8. The Z value of 1.96 squared equals 3.8416, which multiplied by 0.09 equals 0.3457.

Square the desired accuracy level. For a desired accuracy level of 5 percent, this will be 0.05 squared, or 0.0025.

Divide the answer from Step 9 with the value from Step 10 to get the minimum required number of observations for work sampling. In this case, 0.3457 would be divided by 0.0025 for a result of 138.28.

Round up any fractional result to the next whole number. For the value of 138.28, round up to 139. This means that the process must be observed at least 138 times to record enough observations to have a 95 percent confidence level of any information recorded about the event that only occurs 10 percent of the time, plus or minus 5 percent.

#### Tips

According to “Work Measurement and Methods Improvement,” by Lawrence S. Aft, “The number of observations that an analyst must make of a particular job also depends on how much time is devoted to a particular task. The less time an operator spends doing a particular task, the more observations that will be required to ensure that the task is measured properly relative to its contribution or use of the operator’s time.” “Corrosion Tests and Standards” by Robert Baboian says, “Other things being equal, a larger number of observations is needed to detect a small change or to obtain a higher level of confidence in the result.”

#### Warnings

This calculation does assume that the events being observed are independent of each other. If the events are dependent upon each other, such as one failure causing another failure right after it, the actual number of observations required to get enough data will be fewer than the value found by this equation.

#### References

- “Modern Construction: Lean Project Delivery and Integrated Practices”; Lincoln Harding Forbes, Syed Ahmed; 2010
- “Work Measurement and Methods Improvement”; Lawrence S. Aft; 2000
- “Schaum's Outline of Theory and Problems of Operations Management”; Joseph Monks; 1996
- “Corrosion Tests and Standards”; Robert Baboian; 2004

#### Resources

#### Photo Credits

- Thomas Northcut/Photodisc/Getty Images