Box Plots
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Steven Bonacorsi
Aug 17, 2011, 12:25:21 AM8/17/11
to Lean Six Sigma
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 and spatial relationships can
usually be measured in consistent incremental units.
2. Measure the characteristic and record the results. If the
characteristic is continually being produced, such as voltage in a
line or temperature in an oven, or if there are too many items being
produced to measure all of them, you will have to sample. Take care to
ensure that your sampling is random.
3. Count the number of individual data points.
4. List the data points in ascending order.
5. Find the median value. If there are an odd number of data points,
the median is the data point that is halfway between the largest and
the smallest ones. (For example, if there are 35 data points, the
median value is the value of the 18th data point from either the top
or the bottom of the list.) If there is an even number of points, the
median is halfway between the two points that occupy the center most
position. (If there were 36 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.
Order Value Boundary
1 27.75
2 37.35
3 38.35
4 38.35
5 38.75
Second Quartile 39.250
6 39.75
7 40.50
8 41.00
9 41.15
10 42.55
Third Quartile 42.725
11 42.90
12 43.60
13 43.85
14 47.30
15 47.90
Fourth Quartile 48.025
16 48.15
17 49.86
18 51.25
19 51.60
20 56.00
Data table divided into quartiles
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 IQR limit. An outlier beyond the 1.5 IQR limit
of the whisker further emphasizes 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.
Normal distribution curve and box plot compared
The second- and third-quartile boxes are approximately the same size.
The whiskers are similar to each other in length and extend close to
the 1.5 IQR limit. If the data set were actually 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.
Plateau histogram and box plot compared
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 different distribution included in the
data set, for 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.
Isolated peak histogram and box plot compared
The box plot for that data set would look like one for a normal
distribution but with a number of outliers beyond one whisker.
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 a 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. Bought to you by the Process
Excellence Network the world leader in Business Process Management
(BPM)
Author for the Process Excellence Network (PEX Network / IQPC)
Process Excellence Network
Steven Bonacorsi, President of International Standard for Lean Six
Sigma(ISLSS)
Certified Lean Six Sigma Master Black Belt
47 Seasons Lane
Londonderry, NH 03053
Phone: +(1) (603) 401-7047
E-mail: sbona...@islss.com
Process Excellence Network: http://bit.ly/n4hBwu
ISLSS: http://www.islss.com
Article Source: http://EzineArticles.com/?expert=Steven_Bonacorsi
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 and spatial relationships can
usually be measured in consistent incremental units.
2. Measure the characteristic and record the results. If the
characteristic is continually being produced, such as voltage in a
line or temperature in an oven, or if there are too many items being
produced to measure all of them, you will have to sample. Take care to
ensure that your sampling is random.
3. Count the number of individual data points.
4. List the data points in ascending order.
5. Find the median value. If there are an odd number of data points,
the median is the data point that is halfway between the largest and
the smallest ones. (For example, if there are 35 data points, the
median value is the value of the 18th data point from either the top
or the bottom of the list.) If there is an even number of points, the
median is halfway between the two points that occupy the center most
position. (If there were 36 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.
Order Value Boundary
1 27.75
2 37.35
3 38.35
4 38.35
5 38.75
Second Quartile 39.250
6 39.75
7 40.50
8 41.00
9 41.15
10 42.55
Third Quartile 42.725
11 42.90
12 43.60
13 43.85
14 47.30
15 47.90
Fourth Quartile 48.025
16 48.15
17 49.86
18 51.25
19 51.60
20 56.00
Data table divided into quartiles
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 IQR limit. An outlier beyond the 1.5 IQR limit
of the whisker further emphasizes 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.
Normal distribution curve and box plot compared
The second- and third-quartile boxes are approximately the same size.
The whiskers are similar to each other in length and extend close to
the 1.5 IQR limit. If the data set were actually 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.
Plateau histogram and box plot compared
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 different distribution included in the
data set, for 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.
Isolated peak histogram and box plot compared
The box plot for that data set would look like one for a normal
distribution but with a number of outliers beyond one whisker.
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 a 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. Bought to you by the Process
Excellence Network the world leader in Business Process Management
(BPM)
Author for the Process Excellence Network (PEX Network / IQPC)
Process Excellence Network
Steven Bonacorsi, President of International Standard for Lean Six
Sigma(ISLSS)
Certified Lean Six Sigma Master Black Belt
47 Seasons Lane
Londonderry, NH 03053
Phone: +(1) (603) 401-7047
E-mail: sbona...@islss.com
Process Excellence Network: http://bit.ly/n4hBwu
ISLSS: http://www.islss.com
Article Source: http://EzineArticles.com/?expert=Steven_Bonacorsi
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