Math 225 Course Notes

Section 2.5: Descriptive Statistics - Measures of Dispersion

Key Concepts

The most important measure of spread is the standard deviation. There are differences in the formulas for the standard deviation from samples and populations.

Closely related to the standard deviation is the variance, which is simply the standard deviation squared.

A standard deviation can frequently be interpreted as a typical deviation from the mean.

You should know in principle how to compute the standard deviation by hand with small examples, but should learn to use you calculator for most calculations.

The Standard Deviation

In words, the standard deviation is the root-mean-square deviation from the mean.

To calculate it, do the operations root-mean-square in opposite order.

  1. Find the deviations from the mean by subtracting the mean from each data point.
  2. Square each deviation.
  3. Take the mean of the squared deviations.
  4. Take the square root of this number.
This procedure results in the population standard deviation. The notation for this is the Greek letter sigma.

To computer the standard deviation of a sample, the procedure is exactly the same, except that in step 3, the division is by n-1 instead of n.

The notation for the sample standard deviation is s.

The notation for the population and sample variances are respectively sigma^2 ans s^2. They are computed by simply not taking the square root in step 4.

Example: Find the standard deviation of the numbers 

  4  6  2  9  2
The mean of these numbers is 4.6.

A table can be used to assist in the calculations. 

 xi   (xi - x-bar)  (xi - x-bar)^2
----------------------------------
 4        -0.6           0.36
 6         1.4           1.96
 2        -2.6           6.76
 9         4.4          19.36
 2        -2.6           6.76
----------------------------------
           0.0          35.20
The population variance, sigma^2, is 35.20/5 = 7.04.

The population standard deviation, sigma, is square root(7.04) = 2.65.

The sample variance, s^2, is 35.20/4 = 8.80.

The sample standard deviation, s, is square root(8.8) = 2.97.

Leave time to do this example using a calculator.

Interpreting the Standard Deviation

A standard deviation may be thought of as a typical deviation from the mean, in many instances. We will see that for most cases, it is very unusual for an observation to be more than 3 standard deviations from its mean, in either direction.

In the above example, some of the observation are closer than a single standard deviation (about 2.5 or 3) to the the mean, while others are farther away.

No observation is as far as 3 standard deviations (about 9) from the mean.

Last modified: Jan 15, 1996

Bret Larget, larget@mathcs.duq.edu