How to Calculate T When Using Confidence Limits

Calculate the sample mean by taking the sum of the observations, divided by the total number of observations. You could measure, for example, the average time spent waiting in line for a sample of 121 of the 5,000 people who went to a movie premiere. If the total number of minutes the 121 people spent waiting in line was 23,792.5, the average would be 23,792.5/121, or 196.6. The sample average of 196.6 minutes is your estimate of how long each of the 5,000 people spent waiting in line.

Calculate the standard deviation of your sample mean. Take each value and subtract it from the sample mean. Square each of these values by multiplying it by itself. Add up these squared deviations from the mean and divide by the number of observations minus one. The square root of this number is the standard deviation. For example, if the first person in your sample waited 188.3 minutes and the average waiting time was 196.6 minutes, subtract 188.3 from 196.6 to equal (-8.3). This difference squared is 69. If the total of all the squared deviances from the mean is 47,692.5, and this number divided by 120 equals a standard deviation of 397.4. The square root of 397.5 is 19.94 minutes.

Select the confidence level, which is the confidence the statisticians wishes to have that the sample statistic accurately reflects the population parameter. A confidence level of 95 percent is a common selection.

Compute standard error, which is the standard deviation divided by the square root of the sample size. In this example 19.94/11 is 1.81.

Calculate alpha with the formula: 1 minus the confidence level divided by 100. In this example, 1-(95/100) = .05

Find the critical probability by using the formula: 1 minus alpha divided by 2. In this example, (1-.05)/2 = .975.

Calculate degrees of freedom by subtracting the 1 from the total sample. In this example, 121-1 = 120.

Look up the t-value by using a t-distribution table available in the back of most statistics books. Find the value at the intersection of the degrees of freedom on the left and the alpha level across the top. In this example, the value that intersects 120 degrees of freedom at an alpha of .05 is 1.98.

Compute the margin of error by using the formula: t-value multiplied by standard error. In this example, 1.98 x 1.81 = 3.58.

Report the average minutes spent waiting plus or minus the confidence interval. In this example, 196.6 +/- 3.58 minutes.