|
Contents
| Quality Encyclopedia
| Discussion Blogs
Box-Whisker Chart Calculations
The calculations of the centerline, box boundaries and whisker boundaries are dependant on
user-specified options.
When Based on Mean & Sigma:
Center Line of Box: Average
Edges of Box: Average ± 1 * Sample Sigma
Whiskers: Average ± 3 * Sample Sigma
Outliers: Data points outside the boundaries of the Whiskers are considered outliers.
When Based on Median & Quartiles*:
Center Line of Box: Median
Box Lower Edge: The first Quartile (Q1), or 25th percentile, of the data.
Box Upper Edge: The third Quartile (Q3), or 75th percentile, of the data.
Upper Whisker: extends to largest data point within the boundary of Q3 + 1.5* (Q3-Q1)
Lower Whisker: extends to smallest data point within the boundary of Q1 - 1.5* (Q3-Q1)
Outliers: Data points outside the upper boundary of Q3 + 1.5 * (Q3-Q1) and the lower
boundary of Q1 - 1.5 * (Q3-Q1) are considered outliers. Extreme outliers, drawn with a clear rather than filled
data point, lie outside Q3 + 3 * (Q3-Q1) and Q1 - 3 * (Q3-Q1).
Calculating Quartiles
Note: There are various mathematical methods that can be used to calculate the Quartiles.
This software uses the n+1 method, which is perhaps the most common approach. In this method, you first calculate
the value p*(n+1), where p is the percentile (in decimal form, from 0 to 1) and n is the number of data
observations within that Box-Whisker category. The calculation results in a decimal number consisting of an
integer component j (the number to the left of the decimal place) and a decimal component g (the number to the
right of the decimal place). The Quartile is calculated as:
Quartile = ((1-g) * xj )+ (g * xj+1)
For example, consider the following set of ten observations, sorted smallest to
largest: 2.1; 2.2; 2.3; 2.4; 2.5; 2.6; 2.7; 2.8; 2.9; 3.0
To find the first Quartile (Q1), we note that the first Quartile corresponds to
the 25th percentile, so p =0.25. Given that there are ten data observations (n=10) in this category,
p * (n+1) = .25 * (10+1) = 2.75 so that j = 2 and g = .75. In the sorted list of the ten observations, the
second (j=2) observation is 2.2 (i.e. x2 = 2.2) and the third (j+1= 3) observation is 2.3 (i.e. x3 = 2.3), so the
first Quartile is calculated as:
Q1 = ((1-.75 )* 2.2) +(0.75 * 2.3) = 2.275
Similarly, the third Quartile is calculated as follows:
p*(n+1) = .75*11 = 8.25
j=8
g=.25
Q3 = ((1-.25 )* 2.8) +(0.25 * 2.9) = 2.825
The Median (Q2) is calculated as follow:
p*(n+1) = .5*11 = 5.5
j=5
g=.5
Q2 = ((1-.5 )* 2.5) +(0.5 * 2.6) = 2.55
Note that this technique is preferred over the approximate method of averaging
between adjacent values in the sorted list to calculate the first and third quartiles.
See also:
SPC Concepts
Box Whisker Charts
When to Use a Box Whisker Chart
Interpreting a Box Whisker Chart
|
