GRAPHIC DISPLAYS: WHEN YOU'VE GOT IT, FLAUNT IT

What's a graph for, anyway?

When you've collected all the data you need, and it's already in a format you can understand, why bother to transfer all of that to some kind of graph? Good question.

First, you may not have to put it into a graph. Your data collection format might speak very well all by itself. Then again, it might not. Remember, you've been studying this information, you're close to it.

You will want to present your findings in more graphic form when you have to try to make it understandable to someone else. When your picture, in other words, can save you a thousand words. When your graphic image will speak more clearly and convincingly than you can. When you want to impress somebody.

Before you start drawing up a graph, you should ask yourself these four questions:

1.	Are the data reliable? You're not ready to start making pictures if the "facts" are questionable.

2.	What message do we want to convey?

3.	Which graphic form is best for conveying this message?

4.	Would any special visual effects enhance and sharpen this message? Consider using color, shading and so forth.

There are five basic kinds of graphs to choose from. With variations, these are bar/column graphs, line graphs, pie charts, scattergrams and pictograms. Each has its own peculiarities and advantages.

Bar charts and column graphs

You've probably seen a zillion bar charts and column graphs in your lifetime. They are among the most common formats for illustrating comparative data. And they're neither hard to understand nor hard to construct.

A column graph builds its bars up from the horizontal, like trees growing in the forest. A bar chart is simply a column on its side, with bars built horizontally, like the branches of a tree coming out from its trunk.

A column graph tends to lead the "reader's" eye from left to right, across the shape formed by the top of the column bars. In this sense, it is like a line graph, emphasizing a trend.

A bar chart, on the other hand, tends to draw the reader's eye to the individual bars that stick out the most, or other ones that are unusually short. The emphasis—even though the information may be the same—is one the exception, rather than on the overall trend.

In both kinds of graphs, you may choose to "double-up," pairing two similar graphs on one chart instead of two. This is possible with a column graph when two charts have the same information on the horizontal axis. On a bar chart, of course, the common information would have to lie on the vertical axis.

The examples show how the same information might be displayed in each of the possible formats. Which chart would you choose, in this case, to get across your message?

Line Graphs

The beginnings of a simple line graph are drawn in exactly the same way as the first few steps of building a column graph. The non-variable data (like "days" in the examples below) are labeled on the horizontal axis. The vertical axis is labeled for the variable data, in this example, "number of errors." Plotting the number of errors by day produces a graph like the one below.

Draw bars up to each of these spots, and you have a column graph. Connect the dots, and it's a line graph!

Again, like column graphs, you may want to display two sets of similar information on the same graph—to have two lines. This is okay, and very useful for showing different aspects of the same variable and non-variable items. In other words, the non-variable axis has to be identical for both lines charted, and the variable items (as opposed to measures) have to be the same.

The example, left, shows a paired line graph. This graph tells not only the errors per day, but also the percent of errors per day (number of errors divided by total number produced).

The paired line graph highlights a fact that's not easily seen from the raw data. It tells the reader that the percent of errors is nearly constant, regardless of the total production. A cash register that mistakenly rings "no sale" every fourth time a sale is rung up, will have the same 25 percent level of error regardless of how many times the machine is used each day.

In addition to the simple line graph and the paired line graph, there are three other varieties you may use. One is the surface chart. This kind of line graph helps highlight a trend, by coloring-in or shading the portion of the graph above or below the line.

Another kind of line graph is used to keep track of something in process. Control limits are shown and the line graph is updated, say daily. If the line jumps outside of the control limits, that's your signal that there's a problem to be identified and solved. If the line stays within the control limits, the assumption is that everything's okay.

And, finally, there's one more kind of line graph you may find useful: the target-trend chart. This kind of trend chart is also another variety of control chart. In this graph, you first draw the desired or predicted trend, the target you will measure against as the data come in. The trend desired is usually drawn across the chart as a straight line, tilted up or down, according to your target. Then, as the facts roll in, you build an actual line graph on the same chart. If your line graph, as it's updated, shows the same trend as your target line, you're in fat city. If it varies too much from your target, as the example below does, then you know something's not going according to plan.

Pie Charts and Pictograms

Use a pie chart when you want to show how 100 percent of something is divided up among its parts. A pie chart gives a quick visual message about the proportional differences among the parts.

Corporate annual reports use pie charts to show, for example, that by far the largest pieces of the company's income is spent on materials and wages, with only tiny slices left for, say, capital investments.

If you were a hospital administrator, you might want to use a pie chart to show what proportions of your patient population are doctor-referred, self-referred, or emergency cases.

A group of telephone switchboard operators could use a pie chart to show, from all the calls they handle, what proportions of calls are for time/weather, emergency assistance, long distance, and local numbers.

When you build a pie chart, remember that four or five categories—or slices—are enough. Many more, and the chart gets hard to read. You can always lump together a collection of little pieces under one category labeled "other."

In the examples to the right, two pie charts are used to illustrate a "before and after" situation. In this case, each chart highlights only one slice, the number of errors as a proportion of total production. It's easy to see that the 25 percent error rate was greatly reduced following implementation of the work team's proposed solution.

Pictograms take graphic presentations very close to the field of entertainment. Pictograms are fun to build, and often more pleasing than "ordinary graphs" to look at. Nigel Holmes, who does many of the pictograms in TIME magazine, is a chartmaker who's practically made an art form of his craft. Look at his work for examples of all the possibilities. His pictograms make information jump right off the page!

Unless you're an artist, though, you'll want to stick with simple pictograms. The idea, remember, is to get a message across, quickly and clearly. That means paying attention to the basics.

Basic number one: Do not confuse the reader. The pictogram shown, below-left, violates this rule. It seems to say that the number of volunteers recruited in 1980 was four times the number recruited in 1976, even though the scale on the left indicates the actual increase to be double. The pictogram at right conveys this information without misleading the reader.

Your use of pictograms is limited only by your imagination. The "good" pictogram to the right, for example, could have been a regular bar chart. The one below was made from a regular column graph.

Scattergrams

Scatter diagrams—scattergrams for short—have a variety of uses.

Scattergrams are used to correlate two sets of different data on the same graph. You will want to use a scattergram to find out—or to demonstrate—that two things have either a positive or a negative correlation.

In the example to the right, the scattergram is intended to show the correlation between conveyor speed and percent or errors.

An experiment was set up, so that the conveyor speed would be raised by the same amount each day for six days. Then the graph was drawn, with one axis marked for the different conveyor speeds, and the other axis marked for the different error percentages.

Each day's error percentage was then calculated. That day's performance is shown on the graph as a single dot. The dot is placed at the intersection of that day's error rate

and that day's line speed. One dot, in other words, shows both speed and errors at that speed.

The "scatter" pattern in this graph shows that the error rate does indeed have a positive correlation to the conveyor speed.

Truth in graphics

Figures can lie,

and so can graphs

Labeling

When you make a graph of any kind, don't forget to include the labels! You need to show what each axis represents and what each "interval" measure along each axis represents. In a pie chart, you should label each slice.

On the complex charts, you will also need to take care to provide a " legend" or some other way for your reader to understand your code symbols.

And, finally, you should make sure your legend contains historical information: the period covered, your name(s), date the graph was drawn, source(s) of the data, and whatever else might be helpful to the reader.

Honest graphs

Figures can lie, and so can graphics. Your job—whatever kind of graph you build—is to present an accurate picture of the information you want to convey. You don't want to make a graph that misses the point, misleads, or otherwise confuses the issue. You want a graph that communicates honestly.

Look at these two graphs, below:

The graph on the left seems to say that things are running along fairly smoothly. The one on the right says: "Hold it! We've got problems!" But both graphs are actually built from the same information.

The problem, here, is in the scales used. The graph on the left uses the entire lot size in its scale. This is "honest," but also leads to the wrong conclusion, that things are running smoothly. The graph on the right highlights what's happening at the upper range of the scale, raising the flag of alarm. This is the "honest" message, all right, but it obscures the full truth.

One answer would be to leave the top of the chart at right as it is, in order to convey the urgency of the problem, but to extend the bars their full length, in order take in the whole truth. That's what Figure 3 does.

Figure 4 is your other option, when a large part of the "full story" can be left out, and still make the point, or when space is limited.

In this section, "truth in graphics," all four of the examples shown tell the truth, don't they? But only one tells the whole truth and nothing but the truth. Which one? Which is the best chart, for telling the truth simply and clearly?

When is a graph not a graph?

Up to this page, you've been reviewing the make-up and uses of various graphs, primarily for the display of information. The idea has been to turn your raw data into pictures. To make visuals that can communicate clearly what the raw data have to say.

Now, we're switching gears. A graph is still a graph, of course, but certain "special purpose" graphs are used not primarily to make a point, but mostly to discover the point.

These graphs are a little like the drawings in children's books that challenge the readers to find out "what's wrong in this picture."

You won't find a fish hidden in the clouds, or a schoolgirl wearing only one shoe, but specialty graphs are still worth studying anyway! Pert charts, process flow charts, organization charts, and histograms can all be very useful in your search for the "true causes" or possible solutions of problems.

"Specialty graphs: Can you find the problem in this picture?

"Every problem contains within itself

the seeds of its own solution. "

—Stanley Arnold

Pert Charts

If your organization uses pert charts, you may find these very helpful to analyzing problems.

PERT, by the way, stands for Program Evaluation and Review Technique. This kind of chart came into wide usage after the US space program demonstrated the value of pert charts in planning, scheduling, and controlling complex projects.

Networks of connecting arrows show all the things that have to go on in sequence, or at the same time, and how these relate to each other in time.

Here's an example:

As you can see, Task Y cannot be done until both Task W and Task X are completed. Simply shortening the time it takes to complete Task W will not allow you to start Task Y earlier. To do that, you'd also have to shorten the time required for Task X, or find a way to start Task X earlier.

A pert chart can show you the value, or futility, of concentrating on certain areas of the whole process.

Process flow charts

A process flow chart shows the step-by-step sequence in a process of events. In this regard, it is somewhat like a pert chart. But a process flow chart does not usually show a time line, nor does it have to work-in a great number of parallel events outside of the straight-line process. If your organization uses—or could use—process flow charts, you'll find these charts are really more like an elaborate checklist than a simplified pert chart.

As a problem-solving tool, a process flow chart can be examined to determine if the sequence of events is truly correct or necessary. Perhaps an improvement could be made by re-ordering certain events. Process flow charts can also be used in a presentation, for example, to show where a bottle-neck is occurring and how the process might be changed to correct the problem.

If you're having a problem with some process for which no process flow chart exists, it might be a good idea to construct one. Then you can study it for possible improvements in the sequence.

Here's an example of a process flow chart:

Organization Charts

What good is an organization chart?

In the hands of good problem analyzer like you, an organization chart becomes more than just a Christmas tree picture of who's who around the place.

An organization chart shows the formal structure of the decision making process in the organization. Is this structure organized properly? Maybe not.

Maybe what actually happens is considerably different. Or should be different.

Below is a blank organization chart. Can you mentally fill-in the tasks/functions for each position in this chart? Are they grouped in ways that make sense? If not, what's actually happening, or what should be the structure of decision making in your organization?

An organizational chart can help you spot problems in the way tasks of functions are organized. If these problems get in the way of getting the tasks done, you can be darn sure that "somebody up there" wants to see your suggestions for organizing things in a better way!

Histograms

A histogram is a special kind of graph that you will find extremely helpful in problem-solving. When a histogram is complete, it looks like a "funny" column chart.

What's so funny about histograms? The bars are supposed to fit into a "bell curve," that's what, instead of jumping up and down or instead of showing a constant trend up, down, or across.

The bell shape, according to the mathematicians, represents a normal distribution. If you were ever "graded on the curve" at school, that meant your teacher believed her classroom was populated by a normal mix of students. Accordingly, she used a bell-shaped curve to help decide how to distribute A, B, C, D, and E letter grades.

At the top of the curve is the average or standard performance. The biggest share of students, therefore, got Cs. Next, on either side, were somewhat better and somewhat worse than average. They got the Bs and Ds. At the bottom ends of the curve were the exceptional few: the As and Es.

It didn't matter to that teacher whether your test scores, as a class, were widely dispersed or very close. She simply found the worst score and best score and let the curve do the rest!

Now, that kind of grading systems may or may not be fair, in a school setting. In solving a problem at work, however, your search for the curve will do wonders!

A histogram is used three ways:

One, it can help you spot a problem in a process. In other words, if the process doesn't produce a normal bell shaped histogram, you know there may be a problem. And you get some clues as to what it could be.

Two: A histogram can help you see if the process is accurate. If the bell does not fit inside your specified limits, the process is not accurate.

Three: A double-hump or other non-bell shape can tell you that there's more than one contributing cause or that more data will be necessary.

How to make a histogram

"What in the world are you going to do with all those numbers?"

A histogram, like every other kind of chart or graph, starts with raw data. So let's start building one from scratch.

Follow along closely now. You'll be doing your own—on your own—before you know it! Ready?

Okay, suppose you've been collecting data on a certain process in order to find out how close that process is to the specifications. You decided on a sample size of 30 observations, then measured 30 pieces that came out of the process.

Here's what your raw data look like:

What in the world are you going to do with all those numbers? At a glance, these numbers don't tell much of a story. It's not easy to spot the high or the low, much less to estimate the average. So, do this:

Put a circle around the largest number and a square around the smallest. Do that before you continue reading.

You're back so soon? Great. If you circled .231 as the high, and put a square around .226 as the low, you're on target. Your next task is to compute the Range (R) between these two numbers. Simply subtract the low number (L) from the high number in the circle (H).

R=H-L. Your answer for this example should be .005. This number (.005) will now be used to help order your raw data into classes, or groupings that will be useful for building a manageable histogram. Hang on to R for a minute.

The next thing you need to know is how many classes are the right number. How many groupings will produce a manageable number of classes. Here's a little rule-of-thumb chart to help you decide:

As you can see, 5-to-7 classes would be right for a histogram built from 30 observations. Choose the actual number on the basis of what would be most convenient and sensible. How about having six classes? That seems about right.

Now you're ready to determine the exact size of each class. This is also called the "interval" measured by each class, or, when it's finally blocked out in column graph form, as the "cell" size. To compute your cell size, divide the Range by the number of classes: R/K = Cell

Using this example, your cell, or interval, size is .005 + 6 = .00083. Let's round that up to .001, just to keep things simple.

Now you can draw up a tally sheet or frequency chart to organize your data into these six classes. In this case, as it turns out, no measured observation is more or less than .001 from the next. Each is already in a class by itself!

The only problem with the chart to the left is that the actual observations fall right on the boundaries between intervals. Right on the line between the bars of the graph! You don't want that to happen. The observation .227, for example: is it to be included in Class 1 or Class 2? To avoid confusion—especially in more difficult cases, where more than one measured observation has to be included in each class—you should adjust the interval boundaries. This is to make sure than no actual observation falls exactly on the boundary.

The observations in this example are accurate to .001. To adjust the intervals (which will later be represented by the width of the bars in your histogram), simply take half the accuracy number and adjust your intervals accordingly: .001 x .5 = .0005.

The chart above, then, could be adjusted to look like this:

You now have enough organized information to draw your histogram. Show the six classes across the horizontal axis, and the number of observations for each class up the vertical axis. Like this:

You can see that this process is pretty well "centered. " That is, it produces a shape that'll fit within a single normal bell curve. But there's a problem here, also.

The specifications for this process (you just looked them up!) are .220-to-.230. Your histogram shows Class Six (.231) out of specification! Not to worry, though. Since the bell is the right shape, and there's plenty of room to "move left" before reaching the lower specification, your analysis-by-histogram tells you the process is running smoothly, it needs but a small adjustment in order to fall entirely within the normal specification—with room to spare.

Make sense? It will soon; you're about to get a chance to put all of this into practice.

Naturally, this next problem is a little more difficult too. Nobody promised you a rose garden.

Try this "hard case"

"Here 's as tough a histogram as you're likely to come across. Can you handle it?"

Here's as tough a problem as you're likely to come across. It's real, too, and has already been solved by a work team in a large manufacturing company. Let's see if you can practice what you've learned to come up with the solution yourself.

The "right answers" are on the next page. Don't peek, though, until you've given it your best shot. Here's the situation.

An audit has been conducted on three machines that produce the same unit. The inner diameters of the units were measured. The standard, or specifications, for this inner diameter are 1.0000 inches +\- 0.0006 inches (or between 0.9994 and 1.0006 inches).

At right is a table of the raw data that was collected. Following the steps outlined here, construct a histogram and "read" its meaning. Actually write down what you think the histogram tells you.

9 Steps to the histogram:

1.	Count the number of observations (N)

2.	Determine the number of classes (K)

3.	Divide the data range (R) by K, to get the estimated class size

4.	Round off the class size to a convenient number

5.	Compute the accuracy of the class boundaries

6.	Determine one-half the accuracy number

7.	Adjust the class boundaries

8.	Tabulate the data

9.	Draw the histogram on grid paper

Step One:	N = __________________ (Number of observations)

Step Two:	K = __________________ (Number of classes)

Step Three:	R + K = ___________________ (Estimated interval size for each class)

Step Four:	= __________________ (Round off class size, if necessary, for convenience)

Step Five:	= __________________

Step Six:	Step 5 x 0.5 = ___________________

Step Seven:	Adjust class boundaries and post in chart at right (make sure the adjusted intervals do not mix out-of-spec observations with in-spec observations)

Step Eight:	Tabulate the data in the frequency chart, on the next page (number of classes may have to be altered, up or down, to take in all measured observations)

Final Exam: In your own words, what does this histogram say?

________________________________________________

________________________________________________

________________________________________________

________________________________________________

________________________________________________

Step Nine:	Draw the histogram on the grid provided, adding heavy tick marks of dotted lines to indicate the upper and lower limits of the specifications for bearing housing inner diameters.

The "hard case:"

What's the meaning of this?

"The shape of the histogram has a story to tell. "

How did you do on the "hard case?" Compare your answers to what the actual work team did.

1.	N= 108

2.	K = 9 (You could have selected any number of classes from 7 to 12 and still be "right." The actual circle took "9" as a number that felt right to them.)

3.	R (1.0008 - 0.9993 = 0.0015) ÷ K (9) = 0.00016

4.	Rounding off, the quality circle decided on a class size of 0.0002

5.	Class boundaries are accurate to 0.0001

6.	One-half the boundary accuracy is 0.0001 x 0.5 = 0.00005

7.	Adjusted class boundaries are shown as "intervals" in the tally sheet opposite. Class 1, for example, is adjusted from 0.9992-0.9994 to 0.99915-0.99935.

8.	The raw data were tabulated (i.e., organized) as shown in the tally sheet.

9.	The histogram shown was the result.

Interpretations

In the "hard case" histogram, the problem-solving team concluded that the bi-modal shape had a story to tell. To wit: that the histogram showed more than one influence on the process. Next question was to decide which of the three machines used was producing out of specification parts. So the team produced more histograms—one for each of the three machines.

Machine X turned out to be fine. Machine Y was found to be off-center. And Machine Z was obviously way out of adjustment, and needed to be thoroughly checked by the manufacturer.

How to "read" a histogram's shape

"Sometimes differences are significant.

Sometimes not."

Histograms are very valuable in presenting a picture of how a process is working. Mere tabulation of data does not always present such a clear picture.

Repetitive operations in nature, human activities and machine operations can be counted on to yield slightly different results from one "repeat" to the next. Sometimes these differences are significant, other times not. Use a histogram to help determine if these differences are significant.

The perfect histogram, remember, shows a normal, bell-shaped distribution.

The other examples shown touch on some of the many various histogram "shapes" you might encounter:

A wide flat histogram like this one, on a machine, might indicate the machine is worn out and won't hold close tolerances.

A bi-modal or tri-modal histogram like this one suggests mixed data. Are there two or three operators, shifts, or gauges being mixed together here? Break down the data into two or more separate histograms.

Skewed histograms, whether they "tilt" to the left or right, are an indication of some problem with the data or the process itself. Further investigation is called for. (Have the out-of-spec units already been discarded?)

Incomplete histograms, like this one, also suggest a data problem. Perhaps the person doing the measuring is rounding off the numbers toward a certain value, or is tending to record, say, even numbers rather than odd.

Condensed histograms show that the scale of the measuring device is too large, If you follow the guidelines in this booklet for making histograms, you should be able to avoid building a condensed histogram

Perfect!

Review questions:

Circle the correct answers.

1.	Which of the following graphs depend upon first collecting data?

(a) Pareto diagrams

(b) Bar charts

(c) Histograms

(d) All of the above

2.	A "normal" distribution will produce a histogram with which shape?

(a) Bar

(b) Bell

(c) Hour glass

(d) Two humps

3.	The right number of columns in a histogram is:

(a) 5 to 7

(b) Based on the number of observations

(c) 7 to 12

(d) 10 to 20

4.	The word "class" means:

(a) Some columns are higher than others

(b) The groupings of histogram data

(c) Letter grades are given

5.	Boundaries between classes are adjusted so that:

(a) The diagram won't go off the grid paper

(b) Actual observations won't fall on the line between classes

(c) No teacher will have too many students

6.	Frequency is measured on the vertical axis of the histogram:

(a) True

(b) False

7.	Histograms can be:

(a) Skewed or bi-modal

(b) Useful in analyzing the nature of a problem

(c) Used in management presentations

(d) All of the above