Standard Deviation & Standard Error
Values which estimate SEM
(Standard Error of the Mean)


The purpose of this handout is to summarize the meaning of information provided by a standard deviation regardless of the name given to the standard deviation and regardless of how it might be calculated.
Given:
  • You have a set of numbers. You calculate the standard deviation (SD). What can that standard deviation tell you?
    • How scattered the scores are. The SD is a descriptive statistic.
    • How scattered the scores in a new sample will be. The SD is a descriptive statistic.
  • Divide the standard deviation by the square-root of sample size.
    • The new number will tell you the SD you can expect for sample means.
    • The standard deviation for sample means is called the Standard Error.
What you do
The Calculation
What the value estimates
Draw one sample of size n from a population Calculate the standard deviation of the observations.

The sample standard deviation.

The standard deviation of the population

"How much single observations jump around from one observation to the next."

1. Draw one sample of size n from a population.

2. Calculation the standard deviation of the observations.

Divide the standard deviation by

The standard error of the mean (SEM).

 

The standard error of the mean

"How much sample means jump around from one sample to the next."

1. Draw r samples of size n from a population.

2. Calculate the sample mean for each sample.

Calculate the standard deviation of the r sample means.

The standard deviation of the sample means.

The standard error of the mean (SEM).

There is no notation for this other than standard deviation.

The standard error of the mean

"How much sample means jump around from one sample to the next."

In the real world we would not know the population standard deviation and population mean--all we would know is estimates calculated from samples. But since the samples on this page were generated via computer-programmed Monte Carlo techniques we do. The population mean was 100 and the standard deviation was 10.

Three Problems
  1. Pretend that the last 10 times I've taught this course I've had exactly 16 students each time. I've kept track of the average score on the final exam. The average of the averages was 160. The standard devation of these 10 numbers was 12. What does this number, the standard deviation tell me?
  1. Based on the information in the first question, would you be surprised if the mean performance on the final exam in this course was 190? Why?
  1. Based on the information in the first question, what standard deviation would you expect in the scores of the final exam during this semester?

© 2002 by BurrtonWoodruff. All rights reserved. Modified Sunday, March 25, 2007