AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

Measures of Variation

Measures of Variation

Statistical measures of variation are numerical values that indicate the variability inherent in a set of data measurements. The most common measures of variation are the range, variance and standard distribution.

Range

The range of a set of observations is the absolute value of the difference between the largest and smallest values in the set. It measures the size of the smallest contiguous interval of real numbers that encompasses all the data values.

EX. Given the following sorted data:

1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8

The range of this set of data is 3.8 - 1.2 = 2.6.

 

Variance and Standard Deviation

The variance of a set of data is a cumulative measure of the squares of the difference of all the data values from the mean.
The population and sample variance are calculated as follows:

Given the set of data values x1, x2, .... xN from a finite population of size N, the population variance is calculated as

 3smdmov1

Given the set of data values x1, x2, .... xn from a sample of size n, the sample variance s2 is calculated as

3smdmov2

 

Note that the population variance is simply the arithmetic mean of the squares of the difference between each data value in the population and the mean. On the other hand, the formula for the sample variance is similar to the formula for the population variance, except that the denominator in the fraction is (n-1) instead of n. Using the above formula, the sample variance is statistically proven to be a most effective estimator for the variance of the population to which the sample belongs.

The standard deviation of a set of data is the positive square root of the variance.

EX. Given the following sorted data:

1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8

x = 2.48 as computed earlier

s2 = ( 1 / (10-1)) × ( (1.2 - 2.48)2 + (1.5 - 2.48)2 + (1.9 - 2.48)2 + (2.4 - 2.48)2 + (2.4 - 2.48)2 + (2.5 - 2.48)2 + (2.6 - 2.48)2 + (3.0 - 2.48)2 + (3.5 - 2.48)2 + (3.8 - 2.48)2 )

= (1 / 9) × (1.6384 + 0.9604 + 0.3364 + 0.0064 + 0.0064 + 0.0004 + 0.0144 + 0.2704 + 1.0404 + 1.7424)
= 0.6684
s = ( 0.6684 )1/2 = 0.8176

The sample variance can also be calculated as follows:

3smdmov3

EX. Given the above data, we can calculate s2 using the above formula:

3smdmov4

= 1.44 + 2.25 + 3.61 + 5.76 + 5.76 + 6.25 + 6.76 + 9.00 + 12.25 + 14.44

= 67.52

3smdmov5

s2 = ( 1 / (10 × 9) ) × ( 10 × 67.52 - (24.8)2 )

= 0.6684

 

 

 

Subject: 
Statistics
Subject X2: 
Statistics

Need Help?

We hope your visit has been a productive one. If you're having any problems, or would like to give some feedback, we'd love to hear from you.

For general help, questions, and suggestions, try our dedicated support forums.

If you need to contact the Course-Notes.Org web experience team, please use our contact form.

Need Notes?

While we strive to provide the most comprehensive notes for as many high school textbooks as possible, there are certainly going to be some that we miss. Drop us a note and let us know which textbooks you need. Be sure to include which edition of the textbook you are using! If we see enough demand, we'll do whatever we can to get those notes up on the site for you!