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All I’ve ever wanted was to be average
It is not ‘normal’ to want to be normal, or average as it were. When was the last time a child came home with a C and everyone was happy for him or her? Even more bizarre, have you ever heard of a child being scolded for getting a perfect score rather than the class average? We live in a very strange world where we all want to be outliers but highly value average.
Not sure what I am talking about? Well, lets break it down. We need two things, a value and a group. In school the group is usually the other students in the class or the other students that took the test. We often call this the population. The other thing we need is the value we are interested it. It could be the value received on a test, or the height or weight of each member of the population. These values become our data set.
In a classroom the average grade is often a C; not by accident but by design. When measuring a physical parameter, the average is whatever it is. Height for example is not assigned a letter grade but has a specific value.
Mean, median and average
Semantics can get you in a lot of trouble (unless you’re a politician where nothing really counts and the points don’t matter). In engineering and mathematics these three words are not the same. Actually, one of these things is not like the others; if you are of a certain age group you will understand.
Let’s start with median. It is synonymous with middle. If you have a class and line the students up according to size, (something my grade school teachers were obsessed with doing for some unknown reason), the median height is the height of the middle child, if there are an odd number of children in the class. If there are an even number of children in the class, then the median height is the mean height of the middle two children.
Mean, the more technical term for average, is the sum of all the values divided by the number of samples. An example is to add up all the heights of the children in a class and divide by then number of students. The number you get is the average, or mean, height.
The answer, when found, will be simple
I worked with an interesting engineer, who was probably three times my age. His favorite saying was that the answer, when found, will be simple. I have found that statement to be true. The trick is to find the answer fast. Sometimes the simplest problems can take a long time to solve. You might be wondering why I am taking the time to talk about such simple and straight forward things like averages. Any engineer who does not know what an average is should not have been given a diploma.
We humans tend to be myopic, we see what is right in front of us and ignore everything else. You think you understand average pretty well? How about the weighted average? In a weighted average you multiply each value by the probability of it occurring. And of course, we must divide by the sum of the probabilities (usually we miss this because it adds up to 1 so often that if we forget we still get the correct answer).
Still think you know everything about the mean? You are making a large assumption, that I am interested in the arithmetic mean. What if I were really interested in the Harmonic Mean, the Power Mean, the Geometric Mean, a Truncated Mean, an Interquartile Mean or the mechanical engineers favorite the Fréchet Mean?
Are you right or left of center?
When we start looking at populations we are entering into the realm of distributions. Distributions have various names. The one people are most familiar will is the Normal Distribution. This is the familiar bell-shaped curve. In this distribution, the mean and median are the same. That is not to say all Normal Distributions look the same. Some will be tall and narrow while others are short and wide. Being left of center in a height distribution, assuming a normal distribution, means you are a bit shorter than average. Conversely, right of center, in the same scenario, means you are slightly taller than normal.
You’re a bit of a deviant
Mathematicians are no different than any other member of the scientific community, (well actually I might like to argue that point, but not here) but they want to nail down their terminology. Describing something as a little left or right of center in a normal distribution is just not going to cut it. Hence the term deviation.
Deviation is a measure of how far a particular datum is from the mean. But this can get really complicated when samples could have units of microns or light years. To make things a bit easier to grasp, we normalize the values. This gives rise to the term standard normal curve. In that case the probability of any particular datum being on the curve is 100%. Another way of saying this is that if you look at the area under a standard normal curve it adds up to 1, or 100%. It covers everything. If you like Venn diagrams, (and let’s face it who doesn’t?), then a distribution means everything is inside the circle.
In a standard normal distribution, one standard deviation (away from the mean) contains 68% of all the items in the population. Two standard deviations contain 95% and three contains 99.7%. Standard deviations are described by the letter sigma. So, you could also say that 3 sigma contains 99.7 % of the population. However, six-sigma is an entirely different animal.
What’s a bi-modal distribution?
The real reason I wanted to write about statistics today (or, I told you all that so I could tell you this) is I recently recounted a conversation I had about resistor tolerances, oh let’s just say it took place before some of you were born. I was told that 10% resistor values (and yes there was a time when people bought 10% resistors all the time) were not normally distributed. They claimed that if I were to measure a box of them I would find that in fact they would be within 10 percent of the value but that they would probably not contain any values that were closer than 5 percent. The postulate being they did not make 1%, 5% and 10 % resistors, but they just made resistors and then sorted them out for various tolerances.
I was dubious. I took about 100 resistors and an analog ohm meter and measured them. Because this was in the days before home computers (but after electricity had be discovered), I then made a histogram by hand and found it to be a camel with two humps.
Sometimes it is very important to have something be very close to average, other times you really want the outlier. I hope the rest of your week is many standard deviations above average.
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