Welcome!
In this video, I’ll be showing how to use the standard normal tables to calculate the
probabilities in a normal distribution. A normal distribution is a symmetric, bell-shaped
distribution where the area under the normal curve is 1 or 100%. The standard normal distribution,
or what is also called the z distribution, is a special normal distribution with a mean
(µ) of 0 and a standard deviation (σ) of1. The formula for transforming a score or observation
x from any normal distribution to a standard normal score is z=(x-μ)/σ
The standard normal score (also known as the z-score or z-value) is the number of standard
deviations a score x is from the mean. The standard normal tables we will be using
are the “Less Than” cumulative tables. They usually have the left tail of the distribution
shaded, and also have positive and negative parts.
Let’s look at an example. Scores on an exam are normally distributed
with a mean of 65 and a standard deviation of 9. We want to find the percent of scores
satisfying a), b), and c) here. In a), we want the probability that x is less
than 54. So for x =54, the corresponding z-score is
54 minus 65 divided by 9. And that gives -1.2222 repeating.
Since the z table is set up to handle only 2 decimal places, we round this to -1.22.
We then go to the z-table and look up the area.
For z = -1.22, we go to the negative side of the table, look for -1.2 in the first column
and 0.02 at the top. The corresponding area here is 0.1112.
That is, the area to the left a z-score of -1.22 is 0.1112.
So on this normal curve, for z = -1.22, the area on the left here is 0.1112 as seen on
the table. Therefore, the probability that x is less
than 54 is the probability that z is less than -1.22 which gives 0.1112 or 11.12%. In b) we want the probability that x is at
least 80. In continuous distributions, like the normal
distribution, there is no distinction between “x is at least 80” and “x is greater
than 80”. We apply the same approach in both cases.
So for z =80, Z equals 80 minus 65 divided by 9. And that
gives 1.67, to 2 decimal places. When we look that up in the z-table by checking
1.6 under .07, we find 0.9525 which is the area to the left of z here.
We then subtract it from 1 to obtain the greater than area, since the total area under the
curve is 1. Therefore, the probability that x is at least
80 is the probability that z is greater than 1.67 which equals 1 - 0.9525, giving 0.0475
or 4.75%. In c) we want the probability that x is between
70 and 86. For x =70, z equals 0.56
And for z =86, z equals 2.33. On the table, the area less than z = 2.33
is 0.9901. While the area less than z = 0.56 is 0.7123.
When finding the area between two z values from the cumulative Less Than tables, we simply
subtract the smaller area from the larger one.
So the probability that x is between 70 and 86 is the probability that z is between .56
and 2.33. That is, 0.9901 minus 0.7123 which gives 0.2778 or 27.78%. In summary, if you’re finding a “less
than” area, using the cumulative Less Than table, the area in the table is the answer.
If you want a greater than area, then do 1 minus the area from the table.
And if you want the area between two z values, then do bigger area (which will correspond
to the larger z value) minus the smaller area (which will correspond to the smaller z-value).
Note that we do not subtract z values, we only subtract areas.
And that concludes this video. Thanks for watching.
