Statistics Resources

Computing SEM and Constructing Confidence Levels

You can use this resource to learn more about computing the standard error of the mean (SEM) and using that value to construct a confidence interval. This handout will not delve into the conceptual pieces of this process, so if you’d like to learn more about the concepts, please use ASC Chat or Ask a Coach for additional assistance.

Computing SEM

To compute the standard error of the mean (SEM), you’ll need the standard deviation (σ) and the sample size (n).  Here is a basic formula that you can use:

SEM = σ/√n

Example

You are studying IQ’s, which have a known mean and standard deviation of 100 and 15, respectively.  For a randomly selected group of 49 students, what is the SEM?

SEM = σ/√n = 15/√49 = 15/7 = 2.14

Constructing Confidence Intervals

A confidence interval allows you to estimate the range of values that contains the true population mean for the selected sample. To construct this interval, you’ll need a point estimate of the population parameter (i.e. sample mean (x̄) to estimate population mean), a critical value (Z), and the SEM (σ_E). Here is a basic formula that you can use:

CI = x̄ ± Z(σ_E)

The value of Z will depend on the level of confidence given.  Since Z is from a standardized distribution, this value does not change based on sample size.  Here is a table of common values that you can use:

Confidence Level                                            Critical Z

            90%                                                        1.645

            95%                                                        1.960

            99%                                                        2.576

Example

Let’s say your sample of 49 students had a mean IQ of 92. Construct the 95% confidence interval for the population mean.

95% CI = x̄ ± Z(σ_E) = 92 ± 1.960(2.14) = 92 ± 4.19

= 92 – 4.19 = 87.81

= 92 + 4.19 = 96.19

95% CI = (87.81, 96.19)