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Statistical Basics and Six Sigma Metrics 397

Clearly, the mean is simply a numerical average, which gives a good
sense of where the center is for a data set. It is the most commonly used
measure of central tendency. However, in some cases, it is not a preferred
measure. For example, assume that in a subdivision, there are 20 families.
Most of families have an annual income around $40,000; however, there
is one family with an annual income of $1,000,000. If we use the
arithmetic mean as the measure of central tendency, then y will be in the
neighborhood of $90,000, which is by no means a family’s typical
midincome in this circumstance. The median would be a better measure
in this case.

Median
The median of a set of measurements y , y ,…, y is the middle number
n
2
1
when the measurements are arranged in ascending (or descending) order.
Specifically, let y denote the ith value of the data set when y , y ,…,y are
2
(i)
1
n
arranged in ascending order. Then the median m is the following:
⎧ y [( n+ )/ ] if n is odd
12
⎪
m = ⎨ y + y
2
2
1
/
⎪ n (/ ) ( n + ) if n is even (11.2)
⎩ 2
Example 11.3
The median of the data set in Example 11.1 can be calculated as follows. First,
we arrange the data in ascending order:
y (1) , y (2) ,…,y (n−1) , y (n) = 521, 522, 533,…,610, 612, 616
In this data set, n = 47, and it is an odd number, n + 1/2 = 48/2 = 24. It can be
found that y 24 = 572. Therefore, m = 572.
Measures of Variation
The most commonly used measures of variation are the range, the variance,
and the standard deviation.

Range
The range is equal to the difference between the largest (maximum) and the
smallest (minimum) measurements in a data set, specifically
Range = maximum – minimum (11.3)

Example 11.4
For the data of Example 11.1, the maximum = y (n) = y (47) = 616 and the
minimum = y (1) = 521; therefore, the range = 616 – 521 = 95.