The measure of dispersion or variability is a measure to determine how observed values differ from each other. Variability is a charateristic of a distribution of data indicating how spread out or homogeneous the data is.

1. Know the charateristics of and be able to compute the range, interquartile range , and semi-interquartile range *

The range is the difference between the the highest and lowest values in a distribution of data.

The interquartile range is the difference between the 1st and 3rd quartiles (25th and 75th percentile) of a distribution of data. It ignores the most extreme observations and contain 50% of the middle observation values.

The semi-interquartile range is half the interquartile range.

2. Know the meaning of sum of squares.

In measures of average deviations the following intermediate statistics are computed:

Difference: X - mean (sample or population), where X is the set of sample observations.

When one value or number is substracted from another this is call a deivation.

When the deviations are squared and added together, we have a sum of square. Sometimes called a sum of square deviation. Example if you have a series of errors, e and you find the average, , then the sum of square of the errors (SSE) is .

When you take the sum of square and divide it by the sample size you get the mean of the squared deviations.

3. Know the meaning of the variance in terms of the mean sum of squared deviations.

The variance is the average sum of squared deviations from the mean or the mean of the squared deviations.

The population variance is denoted by the Greek symbol, sigma squared or  .

The sample variance is denoted by the symbol, s² .

4. Know how to determine the standard deviation.

The standard deviation is the square root of the variance. 

5. Know how to compute the sample and population variances and standard deviations.

These are the formulas for the population and sample variances and standard deviations:
 

Population variance, : , where X are the data values, mu,  is the mean and N is the size of the population.  Population standard deviation, :	Sample variance, s2: , where X are the data values, x-bar,  is the mean and n is the size of the sample.  Sample standard deviation, s: ,
Group Variance and  , where X are the data values, x-bar,  is the group mean and n is the size of the sample and f is the class frequency.. Group Standard Deviations  ,	Alternate Formulas for Variance and standard deviation , , and

Example: The age of officers on and a local committee are: 55, 62, 73, 72, 62, 48, 50

If this are the total ages of committe members, find the committee average age and the variation in their ages according to the formulas above.
 

Data arrange from smallest to largest: 48, 50, 55, 62, 62, 72, 73, N=7 and  Population Mean,  , , So standard deviation,

Table of Values , N = 7, N2 =49     X X2 48 -12.29 150.94 2,304 50 -10.29 105.8 2,500 55 -5.29 27.94 3,025 62 1.71 2.94 3,844 62 1.71 2.94 3,844 72 11.71 137.22 5,184 73 12.71 161.65 5,329 422   Sum = 589.43   178084     26,030

Example: A group of boys and men gathered each night to play basketball at the neighborhood court, one night their age were recorded as: 12, 14, 13, 11, 23, 15, 19

Use this sample the get and ideal about the variance of the age of the boys playing basket ball on any given evening.
 

12, 14, 13, 11, 23, 15, 19, n=7 and  Sample Mean,  , , So standard deviation,

Table of Values , n-1 = 6     X X2 12 -3.29 10.8 144 14 -1.29 1.65 196 13 -2.29 5.22 169 11 -4.29 18.37 121 23 7.71 59.51 529 15 -0.29 0.08 225 19 3.71 13.8 361   Sum =109.42   11449     1745

Workshop Problem: Find the variance and standard deviation for the following catehorized data:
 

Data     Class Midpoint Frequency 25 5 75 16 125 117 175 236 225 331 275 78 325 27 375 8

Solution:(click to see solution)

Use worksheet below:

Class Index, i	Class Midpoint X	Frequency f			X2
1
2
3
4
5
6
7
8
9
10
n=

Group mean =

Group Variance: = Group standard deviation: