The Normal Shape We have said that a normal distribution is bell shaped but we can be much more specific than that. There is actually a very precise shape that represents a perfect normal distribution. The shape is defined by a formula and this page describes the shape and the formula.

Given a sample of data, it can be useful to calculate the shape that its histogram would take if the data were normally distributed. This makes it easier to compare your histogram to the ideal normal shape. You saw this in the game at level one of this topic. It is useful to capture the normal histogram shape in a formula for other reasons that we will meet later. The key points to remember now are:

You can represent the shape of a normal distribution histogram using a formula;

The closest normal curve for a given sample of data is generated based on two measurements from that data:

Its mean,

Its standard deviation.

The formula for generating the curve uses these two values.

Comparing this shape to your data's own histogram tells you how close to normality your data are;

If your data are normally distributed, then the formula can replace the histogram! You can use the formula to tell you how many values you would expect to be above a certain value or in a certain range. You can know all these things from just two numbers: the mean and the standard deviation! That is pretty useful. You will learn how to make these calculations on the page about z scores and probabilities.

Exploration

The shape of the Normal Distribution changes depending on the mean and standard deviation of the data. You can generate the shape of the curve using a formula, which is shown in the help topic above.

The formula above is used in the game to the right. Numbers are generated by the formula using a mean and standard deviation that you choose. These values are plotted across the range from -5 to +5. The higher the curve, the higher the frequency of values in that part of the range.

How does changing the mean affect the normal distribution's shape?

How does increasing the standard deviation affect the normal distribution's shape?

How does decreasing the standard deviation affect the normal distribution's shape?

Application

By generating the closest normal curve to your data, you can see how well the mean and standard deviation reflect the distribution of your data. We have done this for your data in the chart below.

The mean of the data plotted above is 63.43 and the standard deviation is 39.08. You will notice that we have plotted the data in continuous bins, rather than using a bar for each value, as we did in the section on histograms. This is because the normal curve only works over a continuous scale.

What does the highest point along the red line correspond to?

Look at where the mean of the red line is and think about whether this reflects where the bulk of your data lie.