# Normal distribution

A useful continuous distribution is the normal distribution with mean equal to 0 and standard deviation equal to 1. It can be shown that the area under the standard (i.e., mean=0, standard deviation=1) normal distribution from negative infinity to infinity is equal to 1. In order to find the relative frequency of being in an interval (which we shall later call the probability of being in an interval), the area under the normal curve to the left of a cutoff has been tabulated. You will need to use a table, but with a table you will be able to answer the following questions:

## Relative frequency of being in an interval

Tables have tabulated the area under the standard normal curve to the left of a specified z-value (the letter z generally refers to standard normal distribution (mean=0, variance=1); caveat: some tables record the area between 0 and the specified cutoff). We can find the area (or relative frequency or probability) associated with an interval by calculating the difference of the areas associated with the endpoints. Thus to find the area over the interval (-.5, 1) which is shaded in blue in the figure below,

we find in a table of the normal distribution associated with the z-value 1 the area .8413 (which corresponds to the blue and cyan regions in the figure), and we find in the table associated with the z-value -.5 the area .3085 (which corresponds to the cyan area in the figure). Subtracting provides that the area of the blue region is .8413-.3085=.5328. Recall that the total area under the curve is 1, hence the area to the right of a cutoff can be found by subtracting the area to the left of the cutoff from 1. The area to the right of 1 (white in the figure) is 1-.8413=.1587. N.B.: All areas (relative frequencies, probabilities) will be between 0 and 1, inclusive.

## Cutoff that relative frequency is above

If we wanted to know what z-value 10% of the data would be above (or the data would be above 10% of the time), we would want the the area under the curve beyond the z-value to be equal to .10 (the green shaded region in the figure below).

That means that the area under the curve to the left of the z-value (white in the figure) would be .9. Looking for .9 in the body of the table, we find .8997 and .9015, which correspond to the z-values 1.28 and 1.29, respectively. Thus the area to the right of z=1.28 is .1 (actually .1003).

## Caveat

DO NOT BE AFRAID TO DRAW A PICTURE WHEN DOING PROBLEMS OF THIS NATURE. It is important to distinguish between z-values (which may be any real number) and the corresponding areas (probabilities, relative frequencies) which are between zero and one, inclusive).

ALSO, remember that with continuous distributions, it does not matter whether inequalities are strict or weak (the probability of a specific value is 0).

Competencies: If X is normally distributed with mean 0 and standard deviation 1 (N(0,1)), what is the probability that .5 < X < 2 ?
If X is normally distributed with mean 0 and standard deviation 1 (N(0,1)), what value will X be less than 30% of the time?