Also called: box-and-whisker plot
Description
The box plot is a graph that summarizes the most important statistical characteristics of a frequency distribution for easy understanding and comparison. Information about where the data falls and how far it spreads can be seen on the plot. The box plot is a powerful tool because it is simple to construct yet yields a lot of information.
6 3 9
4 8
5 7
box plot 121
When to Use
• When analyzing or communicating the most important characteristics of a batch of data, rather than the detail, and . . .
• When comparing two or more sets of data, or . . .
• When there is not enough data for a histogram, or . . .
• When summarizing the data shown on another graph, such as a control chart or run chart
Procedure
1. List all the data values in order from smallest to largest. We will refer to the total number of values, the count, as n. We will refer to the numbers in order like this:
X1is the smallest number; X2is the next smallest number; up to Xn, which is the largest number.
2. Medians. Cut the data in half. Find the median—the point where half the values are larger and half are smaller.
• If the total number of values (n) is odd: the median is the middle one. Count (n + 1)/2 from either end.
median = X(n + 1)/2
• If the total number of values (n) is even: the median is the average of the two middle ones. Count n/2 and n/2 + 1 from either end. Average those two numbers:
median = (Xn/2+ Xn/2+1)/2
3. Hinges. Cut the data in quarters. Find the hinges—the medians of each half.
• If n is even, the median is the average of Xn/2and Xn/2 + 1. Take the values from 1 to Xn/2and find their median just as in step 2. This is the lower hinge.
• If the total number of values is odd, the median is X(n + 1)/2. Take the values from 1 to the median and find their median, just as in step 2. This is the lower hinge.
Do the same with the values at the upper end to find the upper hinge.
4. H-spread. Calculate the distance between the hinges, or H-spread:
H-spread = upper hinge – lower hinge
5. Inner fences. These are values separating data that are probably a predictable part of the distribution from data that are outside the distribution. Inner fences are located beyond each hinge at 11⁄2times the H-spread, a distance called a step.
upper inner fence = upper hinge + 1.5 ×H-spread lower inner fence = lower hinge – 1.5 ×H-spread
6. Outer fences. Data beyond these values are far outside the distribution and deserving of special attention. Outer fences are located one step beyond the inner fences.
upper outer fence = upper inner fence + 1.5 ×H-spread lower outer fence = lower inner fence – 1.5 ×H-spread
7. To draw the box plot, first draw one horizontal axis. Scale it appropriately for the range of data.
• Draw a box with ends at the hinge values.
• Draw a line across the middle of the box at the median value.
• Draw a line at each inner fence value.
• Draw a dashed crossbar at the adjacent value, the first value inside the inner fences.
• Draw whiskers, dashed lines from the ends of the box to the adjacent values.
• Draw small circles representing any outside data points: beyond the inner fences but inside the outer fences.
• Draw double circles to represent far out data points: beyond the outer fences.
8. If you are comparing several data sets, repeat the procedure for each set of data.
9. Analyze the plot. Look for:
• Location of the median
• Spread of the data: how far the hinges and fences are from the median
• Symmetry of the distribution
• Existence of outside points
Example
Suppose two bowling teams, the Avengers and the Bulldogs, have the scores shown in Figure 5.11. Which team is better? We will draw a box plot of each team’s scores and compare the two plots.
1. The scores are already in order from smallest to largest. There are 14 scores for each team, so n = 14.
box plot 123
2. Median. There is an even number of scores, so the median is the average of the two middle ones. We must count n/2 and n/2 + 1 from one end.
n/2 = 14/2 = 7 and n/2 + 1 = 8
Count to the seventh and eighth scores in each group and average them.
3. Hinges. We must find two medians, first of values 1 through 7 and then of values 8 through 14. There are seven values in each half, an odd number, so we count (7 + 1)/2 = 4 from either end.
lower hinge A = 142 upper hinge A = 160 lower hinge B = 152 upper hinge B = 163 4. H-Spread. The distance between hinges is
H-spread = upper hinge – lower hinge
H-spread A = 160 – 142 = 18
H-spread B = 163 – 152 = 11
5. Inner fences.
upper inner fence = upper hinge + 1.5 × H-spread
upper inner fence A = 160 + 1.5 × 18 = 160 + 27 = 187 upper inner fence B = 163 + 1.5 × 11 = 163 + 16.5 = 179.5 lower inner fence = lower hinge – 1.5 × H-spread
lower inner fence A = 142 – 27 = 115
lower inner fence B = 152 – 16.5 = 135.5 Median A
Median B
= + =
= + =
149 150
2 149 5
155 159
2 15
.
7 7
Bowling Scores The Avengers
126 134 137 142 145 148 149 150 155 157 160 165 170 198 The Bulldogs
103 139 147 152 153 154 155 159 161 163 163 165 176 183
Hinge Median Hinge
Figure 5.11 Data for box plot example.
6. Outer fences.
upper outer fence = upper inner fence + 1.5 ×H-spread
upper outer fence A = 187 + 27 = 214
upper outer fence B = 179.5 + 16.5 = 196
lower outer fence = lower inner fence – 1.5 ×H-spread
lower outer fence A = 115 – 27 = 88
lower outer fence B = 135.5 – 16.5 = 119
Figure 5.12 is the box plot of the two teams’ scores. While the Avengers have a star and the Bulldogs have a poor player, overall the Bulldogs tend to score higher than the Avengers. The Bulldogs’ smaller spread also indicates they score more consistently.
Variations
The box plot was created by John W. Tukey. Many variations have been proposed for calculating, drawing, and using box plots. Whenever you use a variation on the basic box plot, draw solid lines beyond the hinges to indicate that you are not conforming to Tukey’s rules. Some variations are:
• Simple box plot. Instead of calculating and drawing fences and outliers, draw lines from the ends of the box (hinge values) to the highest and lowest data values.
• Modified box plot. Calculate the arithmetic average of all the data values and show it with a dot on the box plot. The closer the average is to the median, the more symmetrical the distribution.
• Modified-width box plot. When using two or more box plots to compare several data sets, the widths of the boxes can be drawn proportional to the sample size of the data sets.
box plot 125
100 110 120 130 140 150
Bowling Scores
160 170 180 190 200
Avengers
Bulldogs
Figure 5.12 Box plot example.
• Parentheses can be drawn on the plot to represent 95% confidence limits.
• Ghost box plot or box-plot control chart. A box plot can be drawn with dotted lines directly on a control chart or other graph of individual data points to show a summary of the data. This variation is especially useful if several plots are drawn showing sequential subgroups of the data. For example, draw one ghost box plot in the middle of a set of 15 data points prior to a process change and another in the middle of the next set of 15 data points after the change.
brainstorming
Variations: round robin brainstorming, wildest idea brainstorming, double reversal, starbursting, charette procedure
See also: nominal group technique, brainwriting, contingency diagram, and fishbone diagram
Description
Brainstorming is a method for generating a large number of creative ideas in a short period of time.
When to Use
• When a broad range of options is desired, and . . .
• When creative, original ideas are desired, and . . .
• When participation of the entire group is desired Procedure
Materials needed: flipchart, marking pens, tape, and blank wall space.
1. Review the rules of brainstorming with the entire group:
• No criticism, no evaluation, no discussion of ideas.
• There are no stupid ideas. The wilder the better.
• All ideas are recorded.
• Piggybacking is encouraged: combining, modifying, expanding others’ ideas.
1
6 2 10 9
4 5
2. Review the topic or problem to be discussed. Often it is best phrased as a why, how, or what question. Make sure everyone understands the subject of the brainstorm.
3. Allow a minute or two of silence for everyone to think about the question.
4. Invite people to call out their ideas. Record all ideas, in words as close as possible to those used by the contributor. No discussion or evaluation of any kind is permitted.
5. Continue to generate and record ideas until several minutes’ silence produces no more.
Example
Read the Medrad, St. Luke’s Hospital, and ZZ-400 improvement stories in Chapter 4 for examples of brainstorming used within the quality improvement process.
Variations
There are many versions of brainstorming. The basic one above is sometimes called free-form, freewheeling, or unstructured brainstorming. In addition to the variations described below, see brainwriting and nominal group technique.