There are several tricky things about teaching and understanding the normal distribution, and in this post I’m going to talk about three of them. They are the idea of a model, the limitations of the normal distribution, and the idea of the probability being the area under the graph.

# It’s a model!

When people hear the term distribution, they tend to think of the normal distribution. It is an appealing idea, and remarkably versatile. The normal distribution is an appropriate model for the outcome of many natural, manufacturing and human endeavours. However, it is only a model, not a rule. But sometimes the way we talk about things as “being normally distributed” can encourage incorrect thinking.

This problem can be seen in exam questions about the application of the normal distribution. They imply that the normal distribution controls the universe.

Here is are examples of question starters taken from a textbook:

- “The time it takes Steve to walk to school follows a normal distribution with mean 30 minutes…”.
- Or “The time to failure for a new component is normally distributed with a mean of…”

This terminology is too prescriptive. There is no rule that says that Steve has to time his walks to school to fit a certain distribution. Nor does a machine create components that purposefully follow a normal distribution with regard to failure time. I remember, as a student being intrigued by this idea, not really understanding the concept of a model.

When we are teaching, and at other times, it is preferable to say that things are appropriately modelled by a normal distribution. This reminds students that the normal distribution is a model. The above examples could be rewritten as

- “The time it takes Steve to walk to school is appropriately modelled using a normal distribution with mean 30 minutes…”.
- And “The time to failure for a new component is found to have a distribution well modelled by the normal, with a mean of…”

They may seem a little clumsy, but send the important message that the normal distribution is the approximation of a random process, not the other way around.

# Not everything is normal

It is also important that students do not get the idea that all distributions, or even all continuous distributions are normal. The uniform distribution and negative exponential distributions are both useful in different circumstances, and look nothing like the normal distribution. And distributions of real entities can often have many zero values, that make a distribution far from normal-looking.

The normal distribution is great for things that measure mostly around a central value, and there are increasingly fewer things as you get further from the mean in both directions. I suspect most people can understand that in many areas of life you get lots of “average” people or things, and some really good and some really bad. (Except at Lake Wobegon “where all the women are strong, all the men are good looking, and all the children are above average.”)

However the normal distribution is not useful for modelling distributions that are heavily skewed. For instance, house prices tend to have a very long tail to the right, as there are some outrageously expensive houses, even several times the value of the median. At the same time there is a clear lower bound at zero, or somewhere above it.

Inter-arrival times are not well modelled by the normal distribution, but are well modelled by a negative exponential distribution. If we want to model how long it is likely to be before the next customer arrives, we would not expect there to be as many long times as there are short times, but fewer and fewer arrivals will occur with longer gaps.

Daily rainfall is not well modelled by the normal distribution as there will be many days of zero rainfall. Amount claimed in medical insurance or any kind of insurance are not going to be well modelled by the normal distribution as there are zero claims, and also the effect of excesses. Guest stay lengths at a hotel would not be well modelled by the normal distribution. Most guests will stay one or two days, and the longer the time, the fewer people would stay that long.

# Area under the graph – idea of sand

The idea of the area under the graph being the probability of an outcome’s happening in that range is conceptually challenging. I was recently introduced to the sand metaphor by Holly-Lynne and Todd Lee. If you think about each outcome as being a grain of sand (or a pixel in a picture) then you think about how likely it is to occur, by the size of the area that encloses it. I found the metaphor very appealing, and you can read the whole paper here:

There are other aspects of the normal distribution that can be challenging. Here is our latest video to help you to teach and learn and understand the normal distribution.

They’re good observations. When I’ve taught statistics I’ve tried to emphasize that the normal distribution is only what you’d expect in circumstances where certain criteria apply, and what those criteria mean in real observable quantities. But I’m not sure how much of that communicates. It’s too easy to rush off to the easy-to-test stuff about how they calculate.

Thanks. Maths teachers do like the procedural stuff much better than the conceptual. It’s easier to understand and teach. But we need to be brave and work on the conceptual as well!

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