Dispersion

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Dispersion

Measures of dispersion offer us an idea of how spread out the scores are, or how wide is the distribution of scores.

There are 5 primary measures of dispersion; 3 of which will be used repeatedly during the rest of this course.
- Range
- Sums of Squares
- Variance
- Standard Deviation
- Coefficient of Variation
All measures of dispersion must not be zero.
- If a measure of dispersion is zero, then you do not have a variable, you have a constant.
- If our scores are: (5, 5, 5, 5, 5) then dispersion is zero and this is a constant.

The Range

The range is simply the maximum score, minus the minimum score. Examples from our oil data:

Barrels: $228 - 159 = 69$
Costs: $580 - 510 = 70$

Disadvantages:

It is calculated from only 2 scores.
Those two values are the most extreme in the distribution (obviously sensitive to outliers).
The range can change dramatically from sample to sample (of the same variable).
The range is not terribly informative.

Sums of Squares: symbol = $SoS$

The Sums of Squares are the sum of the squared deviations from the mean for a distribution of scores.

Though not informative or used as a measure of dispersion, it is very frequently used in the calculation of other statistics.

The general formula for calculating a variable's SoS is:

$SoS = \sum{\left(X - \overline{X}\right)}^{2}$

Variance: sample symbol = $S ^{2}$ , population symbol = $\sigma\ ^{2}$

The variance is the average of each score's squared difference from the mean.

The general formula for calculating a variable's sample variance is:
$S ^{2} = \frac{\sum{\left(X - \overline{X}\right)}^{2}}{n - 1}$ or $S ^{2} = \frac{SoS}{n - 1}$
The formula for calculating a variable's population variance is:
$\sigma\ ^{2} = \frac{\sum{\left(X - \mu\right)}^{2}}{N}$
Note: with a sample, we divide by $n - 1$ ; if we divided by $n$ , our variance statistic would be less representative of the variance parameter (i.e., the sample value would be systematically smaller than the population value).
Also note: when referring to total number of scores
in a population we use $N$ , in a sample we use $n$ .

Standard Deviation: sample symbol = $S$ , population symbol = $\sigma\$

The Standard Deviation is the square root of the variance and allows us to compare the dispersion of one distribution to another.

It is the most commonly reported measure of dispersion³.
It is very easy to calculate...just take the square root of the variance.
- Sample Formula: $S = \sqrt{S ^{2}}$
- Population Formula: $\sigma\ = \sqrt{\sigma\ ^{2}}$
Note the use of the word ``Standard'' which you will see often; it refers to standardization, which tends to allow us to compare statistics from different variables or distributions (i.e., apples & oranges).

Formula Smormula: Computational formulas vs. Definitional formulas

Computational: designed to make computing by hand easier.

A matter of opinion these days...

Definitional: designed to make understanding the concept easier, formula follows the definition of the concepts.

Here are both for the standard deviation of a sample.
Definitional Computational

$S = \sqrt{\frac{\sum{\left(X - \overline{X}\right)}^{2}}{n - 1}}$ $S = \sqrt{\frac{\sum{X^{2}} - \left[\left(\sum{X}\right)^{2}/n\right]}{n - 1}}$
Either can be used; both types provide the same answer.

Calculating $SoS$ using example data (X = barrels)

$X_i$	X	$\overline{X}$	$\left(X - \overline{X}\right)$	$\left(X - \overline{X}\right)^{2}$
1	159	192.1	-33.1	1095.61
2	166	192.1	-26.1	681.21
3	176	192.1	-16.1	259.21
4	185	192.1	-7.1	50.41
5	191	192.1	-1.1	1.21
6	194	192.1	1.9	3.61
7	199	192.1	6.9	47.61
8	207	192.1	14.9	222.01
9	216	192.1	23.9	571.21
10	228	192.1	35.9	1288.81
	$\sum{X} = 1921$	$SoS =$	$\sum{\left(X - \overline{X}\right)^{2}}$	$= 4220.90$

Sample mean $= \overline{X} = \sum{X}/n = 1921/10 = 192.1$

Calculating variance & standard deviation using example data (X = barrels)

Taking the information from the last slide...

Sample Variance for `Barrels' is:

$S ^{2} = \frac{\sum{\left(X - \overline{X}\right)}^{2}}{n - 1} = \frac{SoS}{n - 1} = \frac{4220.90}{10 - 1} = \frac{4220.90}{9} = 468.99$

Sample Standard Deviation for `Barrels' is:

$S = \sqrt{\frac{\sum{\left(X - \overline{X}\right)}^{2}}{n - 1}} = \sqrt{\frac{SoS}{n - 1}} = \sqrt{S^{2}} = \sqrt{468.99} = 21.66$

Coefficient of Variation

The Coefficient of Variation (CV) is calculated by dividing the standard deviation by the mean, then multiply the result times 100 to express it as a percentage.
The CV allows us to compare the standard deviation of one distribution to another.

CV $= \frac{S}{\overline{X}} \times100 = \frac{21.66}{192.1} \times100 = 0.1128 \times100 = 11.28$

The CV for `Barrels' tells us that the standard deviation is 11.28% of the mean.
In contrast, the CV of `Costs' was 4.11% of the mean; the mean was 550.

You should be able to work backwards from the information in the lines directly above to get the standard
deviation, variance, & sums of squares for `Costs'.

Next: Shape Up: Classes Previous: Central Tendency Contents

jds0282 2010-10-04