Measures of Variation
Measure of variation is an important characteristic
of a data set. It’s a fundamental concept in statistics.
Measure of variation is a measure that describes
how the values in a set of data are spread out around the measure of central
tendency.
It is also known as measures of dispersion or
measures of spread.
The three most commonly used measures of variation
are:
The range, the variance, and the standard deviation
RANGE
The range is the
difference between the extreme values
of a distribution.
In other words, it is the
difference between the greatest and the least numbers in a set of data.
Range = Greatest – Least
The range is the simplest measure
of variation and is the easiest to compute. Though range is not the optimal measure
of variation, it is still widely used in simple descriptions of data.
Example 1
The following are Brian’s Math test scores. Find
the range of his test scores.
79, 63, 74, 81, 77, 67, 84
Solution:
Step 1: First arrange the test scores
in order of their size.
63, 67, 74, 77, 79, 81, 84
Step 2: The lowest score is 63 and
the highest score is 84.
So, the range of these test scores is 84 – 63 = 21 points.
Example 2
The
heights in cm of ten students are:
157, 152, ___,
151, 160, 156, 155, 162, 158, 163
The
maximum height is missing in the data set. Find the maximum height if the range
of the data set is 14 cm.
Solution:
Step 1: First arrange the heights in order from least to
greatest.
151, 152, 155, 156, 157, 158, 160, 162, 163,
___
Step 2: The maximum height is missing in the data set. We’ll call it ‘h’.
Step 3: Minimum height
= 151 cm
Step 4: Range = Maximum
height – Minimum height
Step 5: 14 = h – 151 [Substitute the values.]
Step 6: h = 14 + 151 [Solve the equation for ‘h’.]
Step 7: h = 165
Step 8: So, the maximum height is 165 cm.
VARIANCE
Variance tells us how measured data vary from the
average value of the data set. Variance is the mean of the squares of the
deviations from the arithmetic mean of a data set.
Click the link below or copy/paste the link below
for the formula for variance and a solved example on variance.
http://www.icoachmath.com/SiteMap/Variance.html
The variance is a squared value. It is difficult to
interpret the variance. Hence, statisticians often compute the square root of
the variance which is the ‘standard deviation’. The variance is always the
square of the standard deviation.
STANDARD DEVIATION
The standard deviation is the most popular measure
of variation. It measures the variation around the mean. It takes into account
all values in the data set. The standard deviation is the ideal measure of
variation when the mean is used as a measure of central tendency.
The values of variance and standard
deviation are never negative.
Click the link below or copy/paste the link below
for the formula for standard deviation and a solved example on standard
deviation.
http://www.icoachmath.com/SiteMap/StandardDeviation.html
