How to Calculate Confidence Interval of Mean

The confidence interval of the mean is a statistical term used to describe the range of values the true mean, or average, could fall in based on your data and confidence level. The most commonly used confidence level is 95 percent, which means there is a 95 percent probability that the true mean lies within the confidence interval. To calculate the confidence interval, you need to know the mean, standard deviation, sample size and confidence level you are working with.

Things You'll Need

• Calculator
• T-table

Instructions

Confidence Interval of the Mean

1. Calculate the mean by adding all of the values in your data set and dividing by the number of values. For example, if your data set was 86, 88, 89, 91, 91, 93, 95 and 99, you would get 91.5 for the average.

2. Calculate the standard deviation for the data set (see section 2 for detailed instructions on how to calculate the standard deviation).

3. Determine the standard error of the mean by dividing the standard deviation by the square root of the sample size. In this example, you would divide 4.14 (the standard deviation) by the square root of 8 (the sample size) to get about 1.414 for the standard error.

4. Determine the critical value for t by using a t-table (see resources). The number of degrees of freedom is equal to one less than the number of data points in your set and the p-value is the confidence level. In the example, if you wanted a 95 percent confidence interval and you had seven degrees of freedom, your critical value for t would be 2.365.

5. Multiply the critical value from step 4 by the standard error from step 3. Continuing the example, you would multiply 2.365 by 1.414 and get 3.34411.

6. Subtract the result from step 5 from the mean from step 1 to find the lower limit of the confidence interval of the mean. For example, you would subtract 3.34411 from the mean of 91.5 to find the lower limit to be 88.15589.

7. Add the result from step 5 to the mean from step 1 to find the upper limit of the confidence interval of the mean. For example, you would add 3.34411 to the mean of 91.5 to find the upper limit to be 94.84411.

Standard Deviation

1. Subtract the mean from the first value in your data set and square the result. For example, using the data from section 1, you would subtract 91.5 from 86 and get -5.5 and then square -5.5 to get 30.25.

2. Repeat step 1 for each of the numbers in the data set.

3. Add the results from the previous two steps together. For example, you would add 30.25 plus 12.25 plus 6.25 plus 0.25 plus 0.25 plus 2.25 plus 12.25 plus 56.25 to get 120.

4. Subtract 1 from the number of items in the data set. For example, you would subtract 1 from 8 to get 7.

5. Divide the result from step 3 by the result from step 4. Continuing the example, you would divide 120 by 7 and get about 17.143.

6. Take the square root of the result from step 5 to calculate the standard deviation. The square root of 17.143 is about 4.14 so that is the standard deviation.

References

• Online Stat Book: Confidence Interval for the Mean
• Math Motivation: The Standard Deviation
• Online Stat Book: Standard Error of the Mean

Resources

• Read this Article in Spanish
• San Jose State University: T-Table