Title: The standard normal curve
Course: UGQuants (SSPS) > Section: Probability

The standard normal curve

The standard normal curve or standard normal distribution has a mean of 0 and a standard deviation of 1. From just these two properties we can calculate the area under the curve between any two points on the X axis.

We can think of the proportion of the total area under the curve between any two points as corresponding to the proportion of all the observations of X that occur between these two values. Since proportions are also probabilities, we can also see such an area as representing the probability that X takes this range of values for any given observation. If you find this concept difficult to grasp, then you may find Fielding and Gilbert’s description (pp149-151) helpful.

Statistical tables, or Standard Normal tables (such as those found in the appendices of many statical books) tell us the proportion of the area under a standard normal curve that lies between the mean (0) and each unit of standard deviation. Units of standard deviation are referred to with the letter z.

If z equals 1, (that is one standard deviation) these tables give us the result 0.341. (Check yourself here. Some z tables calculate the whole area under the curve up to the z-score and arrive at a value of 0.841 [0.341 + the other half of the curve leading up to the mean] - don't get confused by this.). This means that just over one third of the area under the standard normal curve – 34.1% – lies between the mean and a point one standard deviation above that mean. This is shown in the figure below.

If z equals 2, we get the result 0.477: that is 47.7% of the total area under the curve lies between the mean and a point two standard deviation units above it.

Since the normal curve is symmetrical we also know that the areas above and below the mean must be the same, each representing half the total area under the curve.

We can know, therefore, that the proportion of the area under the curve that lies beyond a point two standard deviations above the mean on the X axis must be 0.5-0.477 = 0.023 of the area under the curve. The same proportion lies beyond a point two standard deviations below the mean.

That is to say about 2.3 out of every hundred observations would have a value more than two standard deviations above the mean. Now let us remember that we can use proportions as probabilities. This allows us to say that if we pick any value of X at random it has a probability of .023 or 2.3% of being more than two standard deviation units above the mean. The value of X, remember, refers to each individual result of a number of infinitely repeated trials, as in a series of coin tosses, or the values of some characteristic obtained in a randomly drawn sample.

If we wanted to know the probability of a value lying two or more standard deviations units away from the mean, whether above or below it, we can simply double this number: 2.3% of observations will lie at least this far below the mean, and another 2.3% this far above it, as shown in the figure below. This diagram illustrates the regions (coloured red) two standard deviations and more from the mean, indicating that each of these two regions represents 0.023 of the total area under the curve.

Reversing the arithmetic, we can also say that altogether about 68% of observations will be within one standard deviation unit above or below the mean, while about 95.4% of our observations will be within two standard deviation units above or below the mean. This is a very useful property that we will exploit in inferential statistics, but first we need to deepen our knowledge of the standard normal distribution.