van Gogh, V.

Another quality tool for problem solving is the scatter diagram. Let’s start with an example.
A team of custodians and their supervisors suspected that the higher the humidity, the longer it would take floor wax to dry. To settle the issue, they recorded data on the time it took for freshly waxed floors to dry. Along with the drying times, they recorded the humidity. Then they plotted their data on the chart. As they look at it, they see a relationship between drying time and humidity.
A scatter diagram is a kind of graph that allows us to see how two variables may be related. By plotting our data on a scatter diagram, we can see whether there is a relationship between the variables. If there is a relationship, then by controlling one variable we may be able to control the other. In the drying time and humidity example, if we can control the humidity then may have more control over the drying time. Even if we cannot control the humidity, we may have a good idea of how long drying will take simply from knowing the degree of humidity.

Scatter diagrams are useful when we are trying to determine how one variable relates to another. In our example, we are comparing drying time to humidity. There are other situations where we could use the scatter diagram. In a hospital, we might relate the strength of a antiseptic to bacteria count. In the transport industry, we might want to see how the length of time between maintenance of trucks compares to the frequency of breakdowns. In a secondary school, we could relate the number of hours of remedial study to scores on proficiency exams. These examples help us to see how scatter diagram can apply to our work.

The purpose of the scatter diagram is to show whether there is a relationship between two variables. In our example, the custodians and managers think that the higher the humidity, the longer the drying tie will be. In other words, they suspect that there is a relationship between the two variables of humidity and drying time.
A scatter diagram is a graph where each plotted point represents two variables. If we look at the drying time and humidity example, any one point on the graph represents a particular drying time of 25 minutes. This means that when the floors were waxed that time, they took 25 minutes to dry and the humidity was 42%.
What kind of data do we use in scatter diagrams? It should be variable data. Variable data come from things we can measure, such as time, length, and temperature. We can measure time to the hour, to the minute, or, if necessary, to the second. Time is a variable. Length, too, can be measured to the foot, inch, or fraction of an inch. Temperature is another variable. Variable data are used in scatter diagrams.

Step 1. Is this a scatter diagram kind of problem?

Check whether we have the right kind of problem for a scatter diagram. First, does each point we’re going to plot come from variable data? Second, are there two things we can measure? That is, are there two variables? Third, are we trying to see if there is a relationship between two variables?

Step 2. Collect the data.

Sometimes the data we need are already recorded. Other times we will have to collect them. When collecting data, record any interesting or peculiar things that happen. For instance, if we noticed that different kinds of wax were being used when we were collecting data for the floor wax study, we should record the kind of wax for each data point. This extra information can be useful later on.

Step 3. Determine the scales for the graph and plot the data.

Start with the vertical axis and decide which variable it will represent. Usually, we plot the dependent variable on the vertical axis.
What do we mean by a dependent variable? In our previous example, the different drying times probably resulted from different levels of humidity. That is, the drying times differ depending upon the humidity. When one variable results from or depends upon another variable, this kind of variable is called the dependent variable. In our example, the vertical axis represents the dependent variable “drying time”.
Now find the largest and smallest values of the dependent variable, the one we will plot on the vertical axis. Set the scale for the axis so that the largest and smallest values fit inside, not on, the edges of the chart. In our example, the longest drying time was 55 minutes, and the shortest was 25. We ran the vertical axis from 20 to 60 minutes.
Use scales that are easy to work with. We marked the vertical scale with the number 20, 30, 40, 50, and 60. This makes the diagram easy t construct and to read.
Next, find the largest and smallest values for the other variable. Set the scales of the horizontal axis so that the largest and smallest values of that variable fit inside, but not on, the edges of the chart.
Now plot the data.

Step 4. Do the corner count test.

By doing the corner test a few times, we’ll develop the ability to tell at a glance (1) whether a scatter diagram obviously shows a “straight line relationship” between eth variables; (2) whether it’s quite clear from the diagram that such a relationship does not exist; or (3) when we must do the corner count test because it’s not so clear whether there is a relationship or not. Without doing this step, we could work for a long time without knowing the true situation. That would be a terrible waste of time. The reason for learning this step is to help us become a effective problem solver.
This simple test helps us determine whether there is a straight line relationship between the variables or if there is only a jumble of plotted points with no real relationship. If a relationship does not exist between the variables, it may possibly be a cause and effect one. If there is a true cause and effect relationship, we can predict one variable by knowing the other. Furthermore, if there is a cause and effect relationship, and we are able to control one variable, we will be able to control the other.
What is a straight line relationship? The ideal straight line relationship is one where an increase in one variable results in an increase or decrease in the other variable, and the amount on increase or decrease is always the same. A good example of this is if we plotted temperature in degrees Centigrade versus temperature in degrees Fahrenheit. For every degrees Centigrade temperature, the Fahrenheit temperature always goes up 1.8 degrees. This example is ideal in that all the points fall exactly in a straight line, since degrees Fahrenheit not fit as neatly as this and we will have a situation more like the scatter diagram for drying time versus humidity.
With this introduction in mind, here are the steps to follow to do a corner count test.

A. Do you have enough data?

Are there at least 10 points on the scatter diagram? We need at least 10 for the corner count test to work. If we have less than 10, we will need more help to make the diagram works.

B. Find the medians.

Find the median for the first variable and draw a line so that all the points are above it. (The median is where half the points are above the line and half are below) To do this, set a ruler on the bottom of the chart so that all the points are above it. Move the ruler up slowly. As we move it up, count the points as they disappear beneath the ruler. Stop when we reach half the count.
If the number of points is odd, divide the count in half. This will give us a number with 0.5 behind it. Add 0.5 to round it off. Stop the ruler on the scatter diagram and draw a horizontal line through the rounded number we collected just now. This horizontal line is the median for our process, the vertical variable.
When the number of points is even, say 38, then divide the count in half (38 divided by 2 is 19). Move our ruler until we have covered half, or 19, of the points. Now position the ruler between the nineteenth point and the next one. The horizontal line we draw across the diagram is the median line because half of the point lie above it and the other half are below. Once we have drawn the horizontal median, find the vertical one. Proceed in the same way, but start our ruler at the right hand vertical edge of the scatter diagram. Move the ruler in from the right side of the chart. Stop the ruler when we reach half the count. Then draw a vertical line to represent the median. Once we have drawn these lines, we will find that half the points are below the horizontal line, and half are above. The vertical line represents the median for the horizontal variable – half the points will be to the left of the line and half to the right. Sometimes we will find our median line has more than one point on it. That’s OK.

C. Label each corner of the chart.

The two median lines have divided the scatter diagram into four parts, or quarters. Mark the upper right one as a “+” or write in the word “plus”. Mark the upper left as a “-“ or write in the word “minus”. Mark the lower left as “+” and lower right as “-“.

D. Do the corner counts.

This is the tricky part, so be careful! Place our ruler vertically on the right side of the chart. Slowly move it toward the left side of the chart. Stop at the very first point we come to and note the sign (either plus or minus) of the quarter where the first point lies. Then slowly move the ruler toward the left. If the next point is in the same quarter as the first point, count the point. Continue in this fashion, counting the points as they disappear under our ruler. Move our ruler until we come t a point that is in a different quarter or is on a median line. Stop and write down the number of points we’ve counted so far. Don’t count the stopping point (the one on the median or in a different quarter). In front of this number, write the sign of the quarter where we started counting. In our example, as we move our ruler in from the right, we find one point in the upper right quarter. It’s in a plus quarter. The second point is also in the first quarter. We continue counting until we come to the sixth point, which is on a median line. We stop at a count of 5. Since our first point is in a “plus” quarter, we assign a plus to this count. The first count is +5.
Now place our ruler horizontally at the top of the scatter diagram and move it toward the bottom of the chart. Count as before. In our example, we count two points in the upper right quarter. As we move our ruler further down, we meet two point – both having the same drying time. But one is on the vertical median. So, we stop counting at 2. Since the points we’ve counted are in a “plus” quarter, we write +2. Next, count in the same way, coming in from the left side of the diagram. Then count up from the bottom of the scatter diagram toward the top.
Sometimes we will count a point twice. Some of the points we won’t count at all. This is OK.

E. Total the corner counts.

Add all the counts. If some are pluses and some minuses, subtract the minus values form the plus values. The result is then written down.

F. Compare the total of the counts.

Once we have totalled the counts, remove the sign from the total. If it is +12, as in our example, write 12. If the total is, not whether it is plus or minus.
Now compare our result to the number “11”. The number “11” is a figure that statisticians have worked out as a comparison figure for scatter diagrams (as longs as there are at least 10 data points). If our result is 11 or higher, we probably have a straight line relationship between our two variables. But if our result is smaller than 11, there may not be a straight line relationship. Our data simply don’t tell.
In our example, the total is 12, so we can say there is a straight line relationship between the two variables. The custodians and supervisors have statistical evidence that there is a straight line relationship between drying times and humidity. The data back up their opinion that the higher the humidity, the longer it takes floor wax to dry.

Brainstorming
Cause and Effect Diagram
Pareto Analysis
Flowcharts
Storyboarding

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