## Box Plots and Beyond

Last weekend, I watched some statistics videos. Along with the stuff I know, came some new stuff. I also wrestled with some geometry relative to triangles and hyperbolas.

We’ll look at box plots in this post. They tell us what we know. They can also tell us what we don’t know. Tying box plots back to product management, it gives us a simple tool for saying no to sales. “Dude, your prospect isn’t even an outlier!”

So let’s get on with it.

Box Plots

In the beginning, yeah, I came down that particular path, the one starting with the five number summary. Statistics can take any series of numbers and summarize them into the five number summary. The five number summary consists of the minimum, the maximum, the median, the first quartile, and the third quartile.

Boxplots are also known as box and whisker charts. They also show up as candlestick charts. We usually see them in a vertical orientation, and not a horizontal one.

Notice the 5th and 95th percentiles appears in the figure on the right, but not the left. Just ignore it and stick with the maximum and minimum, as shown on the left. Notice that outliers appear in the figure on the left, but not on the right. Outliers might be included in the whisker parts of the notation or beyond the reach of the whiskers. I go with the latter. Where the figure on the left says the outliers are more than 3/2’s upper quartile, or less than 3/2’s the lower quartile. Others say 1.5 * those quartiles. Notice that there are other data points beyond the outliers. We omit or ignore them.

The real point here is that the customer we talk about listening to is somewhere in this notation. Even when we are stepping over to an adjacent step on the pragmatism scale, we don’t do it by stepping outside our outliers. We do it by defining another population and constructing a box and whiskers plot for that population. When sales, through the randomizing processes they use brings us a demand for functionality beyond the outliers of our notations, just say no.

We really can’t work in the blur we call talking to the customer. Which customer? Are they really prospects, aka the potentially new customer, or the customer, as in the retained customer? Are they historic customers, or customers in the present technology adoption lifecycle phase? Are they on the current pragmatism step or the ones a few steps ahead or behind? Do you have a box and whisker chart for each of those populations, like the one below?

This chart ignores the whiskers. The color code doesn’t help. Ignore that. Each stick represents a nominal distribution in a collective normal distribution. Each group would be a population. Here the sticks are arbitrary, but could be laid left to right in order of their pragmatism step. Each such step would have its own content marketing relative to referral bases. Each step would also have its own long tail for functionality use frequencies.

Now, we’ll take one more look at the box plot.

Here the outliers are shown going out to +/- 1.5 IRQs beyond the Q1 and Q3 quartiles. The IRQ includes the quartiles between Q1 and Q3. It’s all about distances.

The diagram also shows Q2 as the median and correlates Q2 with the mean of a standard distribution. Be warned here that the median may not be the mean and when it isn’t, the real distribution would be skewed and non-normal. Going further, keep in mind that a box plot is about a series of numbers. They could be z-scores, or not. Any collection of data, any series of data has a median, a minimum, a maximum, and quartiles. Taking the mean and the standard deviation takes more work. Don’t just assume the distribution is normal or fits under a standard normal.

Notice that I added the terms upper and lower fence to the figure, as that is another way of referring to the whiskers.

The terminology and notation may vary, but in the mathematics sense, you have a sandwich. The answer is between the bread, aka the outliers.

The Normal Probability Plot

A long while back, I picked up a book on data analysis. The first thing it talked about was how to know if your data was normal. I was shocked. We were not taught to check this before computing a mean and a standard distribution. We just did it. We assumed our data fit the normal distribution. We assumed our data was normal.

It turns out that it’s hard to see if the data is normal. It’s hard to see on a histogram. It’s hard to see even when you overlay a standard normal on that histogram. You can see it on a box and whiskers plot. But, it’s easier to see with a normal probability plot. If the data once ordered forms a straight line on a plot, it’s normal.

The following figure shows various representations of some data that is not normal.

Below are some more graphs showing the data to be normal on normal probability plots.

And, below are some graphs showing the data to not be normal on normal probability plots.

Going back to first normal probability plot, we can use it to explore what it is telling us about the distribution.

Here I drew horizontal lines where the plotted line became non-normal, aka where the tails occur. Then, I drew a  horizontal line representing the mean of the data points excluding the outliers. Once I exclude the tails, I’ve called the bulk of the graph, the normal portion, the normal component. I represent the normal component with a normal distribution centered on the mean. I’ve labeled the base axis of the normal as x0

Then, I went on to draw vertical lines at the tails and the outermost outliers. I also drew horizontal lines from the outermost outliers so I could see the points of convergence of the normal with the x-axis, x0. I drew horizontal lines at the extreme outliers. At those points of convergence I put black swans of the lengths equal to the heights or thicknesses of the tails giving me x1 and x2.

Here I am using the notion that black swans account for heavy tails. The distribution representing the normal component is not affected by the black swans. Some other precursor distributions were affected, instead. See Fluctuating Tails I and Fluctuating Tails II for more on black swans.

In the original sense, black swans create thick tails when some risk causes future valuations to fall. Rather than thinking about money here I’m thinking about bits, decisions, choices, functionality, knowledge–the things financial markets are said to price. Black swans cause the points of convergence of the normal to contract towards the y-axis. You won’t see this convergence unless you move the x-axis, so that it is coincident with the distribution at the black swan. A black swan moves the x-axis.

Black swans typically chop off tails. In a sense it removes information. When we build a system, we add information. As used here, I’m using black swans to represent the adding of information. Here the black swan adds tail.

Back to the diagram.

After all that, I put the tails in with a Bezier tool. I did not go and generate all those distributions with my blunt tools. The tails give us some notion of what data we would have to collect to get a two-tailed normal distribution. Later, I realized that if I added all that tail data, I would have a wider distribution and consequently a shorter distribution. Remember that the area under a normal is always equal to 1. The thick blue line illustrates such a distribution that would be inclusive of two tails on x1. The mean could also be different.

One last thing, the fact that the distribution for the normal probability plot I used was said to be a symmetric distribution with thick tails. I did not discover this. I read it. I did test symmetry by extending the  x1 and xaxes. The closer together they are the more symmetric the normal distribution would be. It’s good to know what you’re looking at. See the source for the underlying figure and more discussion at academic.uprm.edu/wrolke/esma3101/normalcheck.htm.

Onward

Always check the normalcy of your data with a normal probability plot. Tails hint at what was omitted during data collection. Box plots help us keep the product in the sandwich.