When analyzing the sample data from an experiment or research study, perhaps one of the most important statistical parameters is the mean: the numerical average of all the data points. However, statistical analysis is ultimately a theoretical model imposed on a set of concrete, physical data. To account for the inherent imprecision of statistical modeling, use confidence intervals to evaluate the reliability of the mean (and other parameters). A confidence interval is a range of values within which a parameter is likely to be found. The larger the interval, the higher the probability of it including the actual parameter.

### Calculate The Standard Deviation

Add together the value of every data point in the sample.

Divide this sum by the total number of data points. This is the mean value for the sample.

Subtract the mean from the lowest value of all the data points. For example, in the set of five data points with values of 3, 6, 11, 2 and 4, the mean would be 5.2, or (3+6+11+2+4)/5 = (26)/5 = 5.2. Since "2" is the lowest value, subtract 5.2 from 2 to get -3.2.

Square this value and write down the result.

Repeat Steps 3 and 4 for every data point in the entire sample.

Add together all of the values you wrote down in Step 4.

Divide the total from Step 6 by the total number of data points.

Find the square root of the result from Step 7. The result will be the standard deviation for the sample.

Divide the standard deviation by the square root of the total number of data points. The result is called the standard error of the mean.

### Calculating the Confidence Interval

Dtermine the critical value or "z" for the specific percentage you want the interval to be. Do this by accessing an online table (see Resources).

Scroll down the second calculator on the page and check the box next to "Between."

In the text field next to "Area", enter the percentage you want (in decimal form). For example, if you want a 95 percent confidence interval, type 0.95. If you want a 99 percent confidence interval, type 0.99.

Write down the number that appears next to "Between." This is the critical value for the interval.

Multiply the critical value by the standard error of the mean (calculated in Section 1, Step 9).

Subtract the result from the parameter you want to set the confidence interval around (the mean). This is the "lower boundary" of the confidence interval.

Add the result from Section 2, Step 5 to the parameter. This is the upper boundary of the confidence interval.