Measures of Variation: Range, Variance, and Standard Deviation

Measures of Variation

Introduction to Variation

In statistics, measures of variation describe how spread out or dispersed the values in a data set are. While measures of central tendency (like the mean) summarize the typical value, measures of variation provide insight into the consistency or variability of the data. Three common measures of variation are the range, variance, and standard deviation.

Range

The range is the simplest measure of variation. It is calculated as the difference between the maximum and minimum values in a data set.

• Definition: The range is the difference between the largest and smallest values.

• Formula:

$\text{Range} = \text{max} - \text{min}$

• Example: For the data set x = [8, 9, 10, 13], the range is 13 - 8 = 5. For y = [2, 5, 11, 22], the range is 22 - 2 = 20.

• Limitations: The range only considers two values and can be misleading if there are outliers.

Deviations from the Mean

To better understand variation, statisticians examine how each value deviates from the mean. The deviation for each value is calculated by subtracting the mean from the value.

• Definition: Deviation is the difference between a data value and the mean.

• Example: If the mean is 10, then for x = 8, the deviation is 8 - 10 = -2.

• Positive deviations indicate values above the mean; negative deviations indicate values below the mean.

Variance

The variance is a measure of how much the values in a data set differ from the mean, on average. It is calculated by squaring each deviation (to eliminate negative signs), summing these squared deviations, and dividing by one less than the sample size (for samples).

• Definition: Variance is the average of the squared deviations from the mean.

• Formula for Sample Variance:

$s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$

• Example: For x = [8, 9, 10, 13] with mean 10:

x	x - mean	(x - mean)2
8	-2	4
9	-1	1
10	0	0
13	3	9

• Sum of squared deviations: 4 + 1 + 0 + 9 = 14

• Sample size n = 4

• Variance: $s^2 = \frac{14}{3} \approx 4.7$

• For y = [2, 5, 11, 22] with mean 10:

y	y - mean	(y - mean)2
2	-8	64
5	-5	25
11	1	1
22	12	144

• Sum of squared deviations: 64 + 25 + 1 + 144 = 234

• Variance: $s^2 = \frac{234}{3} = 78$

• Interpretation: Higher variance indicates greater spread in the data.

• Symbol: Variance is denoted by $s^2$ for samples.

Standard Deviation

The standard deviation is the square root of the variance. It is a widely used measure of variation because it is expressed in the same units as the original data.

• Definition: Standard deviation measures the average distance of data values from the mean.

• Formula for Sample Standard Deviation:

$s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$

• Example: For x, $s = \sqrt{\frac{14}{3}} \approx 2.2$

• For y, $s = \sqrt{78} \approx 8.8$

• Interpretation: Larger standard deviation means more variability in the data.

Comparison of Measures of Variation

Measure	Formula	Considers All Values?	Units
Range	$\text{max} - \text{min}$	No	Same as data
Variance	$\frac{\sum (x - \bar{x})^2}{n - 1}$	Yes	Squared units
Standard Deviation	$\sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$	Yes	Same as data

Summary

• The range is easy to calculate but only uses two values.

• Variance and standard deviation use all data values and are more informative measures of variation.

• Standard deviation is preferred for most statistical analyses because it is in the same units as the data.

Additional info: The reason for dividing by n - 1 for sample variance (instead of n) is to correct for bias in estimating the population variance from a sample. This is known as Bessel's correction.