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Box Plots




Box-and-whisker diagrams, or Box Plots, use the concept of breaking a data set into fourths,
or quartiles, to create a display. The box part of the diagram is based on the middle (the
second and third quartiles) of the data set. The whiskers are lines that extend from either
side of the box. The maximum length of the whiskers is calculated based on the length of the
box. The actual length of each whisker is determined after considering the data points in the
first and the fourth quartiles.
Although box-and-whisker diagrams present less information than histograms or dot plots,
they do say a lot about distribution, location and spread of the represented data. They are
particularly valuable because several box plots can be placed next to each other in a single
diagram for easy comparison of multiple data sets.
What can it do for you?
If your improvement project involves a relatively limited amount of individual quantitative
data, a box-and-whisker diagram can give you an instant picture of the shape of variation in
your process. Often this can provide an immediate insight into the search strategies you
could use to find the cause of that variation.
Box-and-whisker diagrams are especially valuable to compare the output of two processes
creating the same characteristic or to track improvement in a single process. They can be
used throughout the phases of the Lean Six Sigma methodology, but you will find box-and-
whisker diagrams particularly useful in the analyze phase.


How do you do it?
1. Decide which Critical-To-Quality (CTQ) characteristic you wish to examine. This CTQ
   must be measurable on a linear scale. That is, the incremental value between units of
   measurement must be the same. For
   example, time, temperature, dimension       Order           Value       Boundary
   and spatial relationships can usually be        1            27.75
   measured in consistent incremental              2            37.35
   units.                                          3            38.35
2. Measure the characteristic and record           4            38.35
   the results. If the characteristic is           5            38.75
   continually being produced, such as            Second Quartile            39.250
   voltage in a line or temperature in an          6            39.75
   oven, or if there are too many items            7            40.50
   being produced to measure all of them,
                                                   8            41.00
   you will have to sample. Take care to
                                                   9            41.15
   ensure that your sampling is random.
                                                  10            42.55
3. Count the number of individual data              Third Quartile           42.725
   points.
                                                  11            42.90
4. List the data points in ascending order.       12            43.60
5. Find the median value. If there are an         13            43.85
   odd number of data points, the median          14            47.30
   is the data point that is halfway between      15            47.90
   the largest and the smallest ones. (For         Fourth Quartile           48.025
   example, if there are 35 data points, the      16            48.15
   median value is the value of the 18th          17            49.86
   data point from either the top or the
                                                  18            51.25
   bottom of the list.) If there is an even
                                                  19            51.50
   number of points, the median is halfway
                                                  20            56.00
   between the two points that occupy the
   centermost position. (If there were 36        Data Table divided into quartiles
points, the median would be halfway between point 18 and point 19. To find the median
   value, add the values of points 18 and 19, and divide the result by 2.) If you think of the
   list of data points being divided into quarters (quartiles), the median is the boundary
   between the second and the third quartile.
6. The next step is to find the boundaries between the first and second and the third and
   fourth quartiles. The first quartile boundary is halfway between the last data point in the
   first quartile and the first data point in the second quartile. (If one data point is on the
   median, that data point is considered to be the last point in the second quartile and the
   first point in the third quartile.) In a similar way, find the third quartile boundary, the
   halfway point between the last value in the third quartile and the first value in the fourth
   quartile.
7. Draw and label a scale line with values. The value of the scale should begin lower than
   your lowest value and extend higher than your highest value. The scale line may be either
   vertical or horizontal.
8. Using the scale as a guideline, create a box above or to the right of the scale. One end of
   the box will be the first quartile boundary; the other will be the third quartile boundary.
   (The width of the box is somewhat arbitrary. Boxes tend to be long and thin. As an option,
   if you have multiple data sets with different numbers of data points in each set, make the
   width of the boxes so that they correspond roughly with the relative quantity of data
   represented in each box.)
9. Draw a line through the box to represent the median (second quartile boundary).
10. The next step is to draw the whiskers on the ends of the box. Find the inter-quartile range
    (IQR) by subtracting the value of the first quartile boundary from that of the third quartile
    boundary.




          a. Smallest data point is bigger than or equal to Q1 -1.5 IQR
          b. Largest data point is less than or equal to Q3 +1.5 IQR
          c. Any points not in the interval [Q1-1.5 IQR; Q3+1.5 IQR] are plotted separately.
11. Multiply the IQR by 1.5. (The use of 1.5 as a multiplier is a convention that has no exact
    statistical basis. Multiplying by this constant helps take into consideration the fact that the
    first and fourth quartiles will naturally have a somewhat wider dispersion than the second
    and third quartiles.)
12. Subtract the value of 1.5(IQR) from the value of the first quartile boundary. Find the
    smallest data point in your list that is equal to or larger than this value. Make a tick mark
    representing this data point to the left of your box (or above, if you used a vertical scale).
    Draw a line, the first whisker, from the side of the box to the tick mark.
13. Add the value of 1.5 (IQR) to the value of the third quartile boundary. Find the largest data
    point in your list that is equal to or smaller than this value. Make a tick mark representing
    this data point to the right of your box (or below, if you used a vertical scale). Draw
    another whisker to this tick mark.
14. It is possible that some data points in your list will lie outside of the ends of the whiskers
    you determined in steps 12 and 13. These points are called outliers. Plot any outliers as
    dots beyond the whiskers.
[Note: steps 3 through 14 happen automatically if you use Excel, Minitab, or JMP to create
your box-and-whisker diagram. If you are familiar with these software packages, their use
can greatly simplify the process of making effective box-and-whisker diagrams.]
15. Title and label your box-and-whisker diagram.
Now what?
The shape that your box-
and-whisker diagram takes
tells a lot about your
process.
One way to help you
interpret box plots is to
imagine that the way a
data set looks as a
histogram is something
like a mountain viewed
from ground level and a
box-and-whisker diagram
is something like a contour
map of that mountain as
viewed from above.




In a Skewed histogram and box plot compared
                                                                  The second-quartile box is
                                                                  considerably larger than
                                                                  the third-quartile box, and
                                                                  the whisker associated with
                                                                  the first quartile extends
                                                                  almost to the end of the 1.5
                                  Skewed histogram                IQR limit. An outlier beyond
                                    and box plot                  the 1.5 IQR limit of the
                                                                  whisker further emphasizes
                                     compared                     the fact that the data is
                                                                  strongly skewed in this
                                                                  direction. On the other side
                                                                  of the distribution, the
                                                                  whisker associated with the
                                                                  fourth quartile is well within
                                                                  the 1.5 IQR. In fact, the
                                                                  fourth-quartile whisker is
                                                                  shorter than the third-
                                                                  quartile box.
A histogram of this data would show a strongly skewed distribution verging on a precipice
that fell off at the high end of the values. This kind of data set often occurs when there is a
natural limit at one end of the distribution or a 100% screening is done for one specification
limit.
Although box-and-whisker diagrams can be oriented horizontally, they are more often
displayed vertically, with lower values at the bottom of the scale.

The next example is what a normal distribution might look like as a box plot.
                                                                  The second- and third-
                                                                  quartile     boxes     are
                                                                  approximately the same
                                                                  size. The whiskers are
                                                                  similar to each other in
              Normal                                              length and extend close to
        distribution curve                                        the 1.5 IQR limit.
           and box plot                                           If the data set were actually
            compared                                              a combination of two
                                                                  different distributions, for
                                                                  example, material from two
                                                                  suppliers or two machines,
                                                                  it might form a histogram
                                                                  that looked like a plateau or
                                                                  a mountain with twin
                                                                  peaks.




                                                                  The box plot would show
                                                                  an even distribution, but
                                                                  would have relatively large
                                                                  boxes and relatively short
                                                                  whiskers.
                                                                  If there were a small
                                                                  amount of data from a
      Plateau histogram                                           different         distribution
      and box plot                                                included in the data set, for
      compared                                                    example, if there were a
                                                                  short-term            process
                                                                  abnormality or a data
                                                                  collection     error,      the
                                                                  histogram formed would
                                                                  look like a mountain with a
                                                                  small isolated peak.
The box plot for that data
                                            set would look like one for
                                            a normal distribution but
                                            with a number of outliers
                                            beyond one whisker.

                            Isolated peak
                            histogram and
                               box plot
                              compared




Some final tips
    A box-and-whisker
diagram is an easy way
 to compare processes
      or to chart the
  improvement process
     in one process.


    Box-and-whisker
 diagrams can quickly
give you a comparative
 feel of the distribution
  of sets of data. They
show the distributional
   spread through the
 length of the box and
      the whiskers.
Some idea of the symmetry of the
distribution can also be gained by
comparing the two segments of the box
and the relative lengths of the whiskers.
The existence and displacement of
outliers gives some indication of the
level of control in the process.
Two or more box-and-whisker diagrams
drawn side by side to the same scale
are an effective way to compare
samples in a way that is compact and
uncluttered. Many box plots can be
added to a diagram without creating
visual overload.
Not only can box-and-whisker diagrams
help you see which processes need
improvement, by comparing initial box-
and-whisker diagrams with subsequent
ones, they can also help you track that
improvement. If specification limits or
improvement targets are involved in
your process, they can be added to the
diagram to help visualize progress.




                   Steven Bonacorsi is the President of the International Standard for Lean Six
                   Sigma (ISLSS) and Certified Lean Six Sigma Master Black Belt instructor and
                   coach. Steven Bonacorsi has trained hundreds of Master Black Belts, Black
                   Belts, Green Belts, and Project Sponsors and Executive Leaders in Lean Six
                   Sigma DMAIC and Design for Lean Six Sigma process improvement
                   methodologies.
                                                 Author for the Process Excellence Network (PEX
                                                 Network / IQPC). FREE Lean Six Sigma and BPM
                                                 content
                    International Standard for Lean Six Sigma
                    Steven Bonacorsi, President and Lean Six Sigma Master Black Belt
                    47 Seasons Lane, Londonderry, NH 03053, USA
                    Phone: + (1) 603-401-7047
                    Steven Bonacorsi e-mail
                    Steven Bonacorsi LinkedIn
                    Steven Bonacorsi Twitter
                    Lean Six Sigma Group

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Box plots

  • 1. Box Plots Box-and-whisker diagrams, or Box Plots, use the concept of breaking a data set into fourths, or quartiles, to create a display. The box part of the diagram is based on the middle (the second and third quartiles) of the data set. The whiskers are lines that extend from either side of the box. The maximum length of the whiskers is calculated based on the length of the box. The actual length of each whisker is determined after considering the data points in the first and the fourth quartiles.
  • 2. Although box-and-whisker diagrams present less information than histograms or dot plots, they do say a lot about distribution, location and spread of the represented data. They are particularly valuable because several box plots can be placed next to each other in a single diagram for easy comparison of multiple data sets. What can it do for you? If your improvement project involves a relatively limited amount of individual quantitative data, a box-and-whisker diagram can give you an instant picture of the shape of variation in your process. Often this can provide an immediate insight into the search strategies you could use to find the cause of that variation. Box-and-whisker diagrams are especially valuable to compare the output of two processes creating the same characteristic or to track improvement in a single process. They can be used throughout the phases of the Lean Six Sigma methodology, but you will find box-and- whisker diagrams particularly useful in the analyze phase. How do you do it? 1. Decide which Critical-To-Quality (CTQ) characteristic you wish to examine. This CTQ must be measurable on a linear scale. That is, the incremental value between units of measurement must be the same. For example, time, temperature, dimension Order Value Boundary and spatial relationships can usually be 1 27.75 measured in consistent incremental 2 37.35 units. 3 38.35 2. Measure the characteristic and record 4 38.35 the results. If the characteristic is 5 38.75 continually being produced, such as Second Quartile 39.250 voltage in a line or temperature in an 6 39.75 oven, or if there are too many items 7 40.50 being produced to measure all of them, 8 41.00 you will have to sample. Take care to 9 41.15 ensure that your sampling is random. 10 42.55 3. Count the number of individual data Third Quartile 42.725 points. 11 42.90 4. List the data points in ascending order. 12 43.60 5. Find the median value. If there are an 13 43.85 odd number of data points, the median 14 47.30 is the data point that is halfway between 15 47.90 the largest and the smallest ones. (For Fourth Quartile 48.025 example, if there are 35 data points, the 16 48.15 median value is the value of the 18th 17 49.86 data point from either the top or the 18 51.25 bottom of the list.) If there is an even 19 51.50 number of points, the median is halfway 20 56.00 between the two points that occupy the centermost position. (If there were 36 Data Table divided into quartiles
  • 3. points, the median would be halfway between point 18 and point 19. To find the median value, add the values of points 18 and 19, and divide the result by 2.) If you think of the list of data points being divided into quarters (quartiles), the median is the boundary between the second and the third quartile. 6. The next step is to find the boundaries between the first and second and the third and fourth quartiles. The first quartile boundary is halfway between the last data point in the first quartile and the first data point in the second quartile. (If one data point is on the median, that data point is considered to be the last point in the second quartile and the first point in the third quartile.) In a similar way, find the third quartile boundary, the halfway point between the last value in the third quartile and the first value in the fourth quartile. 7. Draw and label a scale line with values. The value of the scale should begin lower than your lowest value and extend higher than your highest value. The scale line may be either vertical or horizontal. 8. Using the scale as a guideline, create a box above or to the right of the scale. One end of the box will be the first quartile boundary; the other will be the third quartile boundary. (The width of the box is somewhat arbitrary. Boxes tend to be long and thin. As an option, if you have multiple data sets with different numbers of data points in each set, make the width of the boxes so that they correspond roughly with the relative quantity of data represented in each box.) 9. Draw a line through the box to represent the median (second quartile boundary). 10. The next step is to draw the whiskers on the ends of the box. Find the inter-quartile range (IQR) by subtracting the value of the first quartile boundary from that of the third quartile boundary. a. Smallest data point is bigger than or equal to Q1 -1.5 IQR b. Largest data point is less than or equal to Q3 +1.5 IQR c. Any points not in the interval [Q1-1.5 IQR; Q3+1.5 IQR] are plotted separately.
  • 4. 11. Multiply the IQR by 1.5. (The use of 1.5 as a multiplier is a convention that has no exact statistical basis. Multiplying by this constant helps take into consideration the fact that the first and fourth quartiles will naturally have a somewhat wider dispersion than the second and third quartiles.) 12. Subtract the value of 1.5(IQR) from the value of the first quartile boundary. Find the smallest data point in your list that is equal to or larger than this value. Make a tick mark representing this data point to the left of your box (or above, if you used a vertical scale). Draw a line, the first whisker, from the side of the box to the tick mark. 13. Add the value of 1.5 (IQR) to the value of the third quartile boundary. Find the largest data point in your list that is equal to or smaller than this value. Make a tick mark representing this data point to the right of your box (or below, if you used a vertical scale). Draw another whisker to this tick mark. 14. It is possible that some data points in your list will lie outside of the ends of the whiskers you determined in steps 12 and 13. These points are called outliers. Plot any outliers as dots beyond the whiskers. [Note: steps 3 through 14 happen automatically if you use Excel, Minitab, or JMP to create your box-and-whisker diagram. If you are familiar with these software packages, their use can greatly simplify the process of making effective box-and-whisker diagrams.] 15. Title and label your box-and-whisker diagram.
  • 5. Now what? The shape that your box- and-whisker diagram takes tells a lot about your process. One way to help you interpret box plots is to imagine that the way a data set looks as a histogram is something like a mountain viewed from ground level and a box-and-whisker diagram is something like a contour map of that mountain as viewed from above. In a Skewed histogram and box plot compared The second-quartile box is considerably larger than the third-quartile box, and the whisker associated with the first quartile extends almost to the end of the 1.5 Skewed histogram IQR limit. An outlier beyond and box plot the 1.5 IQR limit of the whisker further emphasizes compared the fact that the data is strongly skewed in this direction. On the other side of the distribution, the whisker associated with the fourth quartile is well within the 1.5 IQR. In fact, the fourth-quartile whisker is shorter than the third- quartile box. A histogram of this data would show a strongly skewed distribution verging on a precipice that fell off at the high end of the values. This kind of data set often occurs when there is a natural limit at one end of the distribution or a 100% screening is done for one specification limit.
  • 6. Although box-and-whisker diagrams can be oriented horizontally, they are more often displayed vertically, with lower values at the bottom of the scale. The next example is what a normal distribution might look like as a box plot. The second- and third- quartile boxes are approximately the same size. The whiskers are similar to each other in Normal length and extend close to distribution curve the 1.5 IQR limit. and box plot If the data set were actually compared a combination of two different distributions, for example, material from two suppliers or two machines, it might form a histogram that looked like a plateau or a mountain with twin peaks. The box plot would show an even distribution, but would have relatively large boxes and relatively short whiskers. If there were a small amount of data from a Plateau histogram different distribution and box plot included in the data set, for compared example, if there were a short-term process abnormality or a data collection error, the histogram formed would look like a mountain with a small isolated peak.
  • 7. The box plot for that data set would look like one for a normal distribution but with a number of outliers beyond one whisker. Isolated peak histogram and box plot compared Some final tips A box-and-whisker diagram is an easy way to compare processes or to chart the improvement process in one process. Box-and-whisker diagrams can quickly give you a comparative feel of the distribution of sets of data. They show the distributional spread through the length of the box and the whiskers.
  • 8. Some idea of the symmetry of the distribution can also be gained by comparing the two segments of the box and the relative lengths of the whiskers. The existence and displacement of outliers gives some indication of the level of control in the process. Two or more box-and-whisker diagrams drawn side by side to the same scale are an effective way to compare samples in a way that is compact and uncluttered. Many box plots can be added to a diagram without creating visual overload. Not only can box-and-whisker diagrams help you see which processes need improvement, by comparing initial box- and-whisker diagrams with subsequent ones, they can also help you track that improvement. If specification limits or improvement targets are involved in your process, they can be added to the diagram to help visualize progress. Steven Bonacorsi is the President of the International Standard for Lean Six Sigma (ISLSS) and Certified Lean Six Sigma Master Black Belt instructor and coach. Steven Bonacorsi has trained hundreds of Master Black Belts, Black Belts, Green Belts, and Project Sponsors and Executive Leaders in Lean Six Sigma DMAIC and Design for Lean Six Sigma process improvement methodologies. Author for the Process Excellence Network (PEX Network / IQPC). FREE Lean Six Sigma and BPM content International Standard for Lean Six Sigma Steven Bonacorsi, President and Lean Six Sigma Master Black Belt 47 Seasons Lane, Londonderry, NH 03053, USA Phone: + (1) 603-401-7047 Steven Bonacorsi e-mail Steven Bonacorsi LinkedIn Steven Bonacorsi Twitter Lean Six Sigma Group