In statistics, a confidence interval is also known as a margin of error. Given a defined sample size, or the number of test results that were produced from identical repetitions, a confidence interval will report a particular range within which a certain percentage of certainty in the results can be established. For example, a scientist may only be able to say with 90% certainty that the results fall within 48 and 52 in his experiment. The 48-52 range would be a confidence interval, and the 90% would be a confidence level. In order to determine a confidence interval, the original test data must be analyzed.

## Confidence Interval of a Sample

Calculate the mean of your data set. The mean is also known as the average. Add up all the numbers within your data set and divide by the quantity of values within your data set, also known as sample size, to determine the average. For example, if your data set has the numbers 2, 5 and 7, you would need to add these together (a total of 14) then divide by 3 for a mean of 4.67.

Calculate the standard deviation of your data set, which is outlined in Section 2.

Take the square root of your sample size. Divide the standard deviation calculated in Step 2 by the square root of the sample size. The resulting number is known as the standard error of the mean.

Subtract one from your sample size to determine your sample's degrees of freedom. Decide next on the percentage confidence level you would like your sample to have. Examples of common percentage confidence levels include 95%, 90%, 80 and 70%.

Refer to the t-table chart (See Resource) to determine the sample's critical value, or t. Find the row which has your number of degrees of freedom. Follow that row across until you stop at the column which matches your decided upon value for the confidence level percentage, which is listed at the bottom of the table.

Multiply the standard error calculated in Step 3 with the critical value just found on the t-table. Subtract this number from the original mean of the sample to determine the confidence interval's lower limit. Add the value to the mean to determine the confidence interval's upper limit.

## Standard Deviation of a Sample

- Statistics or experiment results
- Calculator

Locate the first value in your data set. Subtract from it the mean of your entire sample size. Square this value, and record it. Locate the second value in your data set. Subtract from it the mean of your entire sample size. Square this value and record it. Continue this process for all numbers in your data.

Add all values determined in Step 1 together. Divide this value by the degrees of freedom of your data set, which is the number of values in your data set minus one.

Take the square root of the value calculated in Step 2 to arrive at the standard deviation of the sample.

#### Things You'll Need

References

About the Author

Bailey Richert is a 2010 graduate of Rensselaer Polytechnic Institute with a dual bachelor's degree in environmental engineering and hydrogeology, as well as a master's degree in systems engineering. After several years in the environmental consulting industry, she is now attending MIT for graduate school. An accomplished traveler, she has visited 23 countries and published her first book about international travel in 2014.

Photo Credits

% image by Marko Kovacevic from Fotolia.com