Welcome! In this video,
I’ll be showing how to use the mean to Z standard normal tables to calculate
normal distribution probabilities. I will also be showing software
formulas along the way. Note that the table we’ll be
using in this video is the mean to Z table. That is, the areas provided in the table
are 0 to Z areas under the curve. A normal distribution is a symmetric,
bell-shaped distribution where the area under the normal curve is 1 or 100%.
Because it is symmetric, the mean and median are equal
and are at the centre of the distribution. It means, if we cut it in half at the centre,
we will have 50% of the area below the mean and 50% above the mean.
The standard normal distribution, or what is also called
the Zee or Zed distribution, is the standardized normal distribution
with a mean (µ) of 0 and a standard deviation (σ) of 1.
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.
Again, the standard normal tables we will be using is the Mean to Z table.
This table contains areas (or probabilities) from Z = 0 to the specified Z value,
as shaded under the curve here. This table comes in positive Z values only.
To obtain areas for negative Z values, we will use the same table and
apply the symmetry property of the normal distribution.
In the first column here, we have the first decimal place of the Z score,
and at the top, we have the second decimal place. For example, if we want the area
between Z = 0 and Z = 1.32, we will go to 1.3 in the row
here and 0.02 on the column. So the area is 0.4066.
Now because of symmetry, the area between Z = 0 and Z = -1.32 will be found in the exact same way.
And it will also be 0.4066. However,
if we want the probability that Z < -1.32, that is, the probability of this lower tail,
we will subtract this area from 0.5 since the total area below the mean is 0.5.
So, this less than area will be 0.5 minus 0.4066 which gives 0.0934.
But if we want the probability that Z > -1.32, we will add 0.5 to this area to obtain 0.9066.
Now, note that the normal distribution is a continuous distribution so there is
no distinction between Z > -1.32) and Z ≥ -1.32. They are both treated the same way.
Furthermore, if we want the probability that Z is between -1.32 and +1.32,
we will add the two areas or double the single area to obtain 0.8132.
Now let’s consider further examples. Scores on an exam are normally
distributed with a mean (mu) of 70 and a standard deviation (sigma) of 7.
We want to find the probability of obtaining scores in the intervals listed in a) to d) below.
In a), we want the probability that a score X is less than 84.
So for X = 84, the corresponding Z-score is 84 minus 70 divided by 7.
And that gives Z = 2 or 2.00.
Now on the Z-table, we look up Z = 2.0 under 0.00. to obtain an area of 0.4772.
That is, the area from Z = 0 to Z = 2 is 0.4772. Therefore, this shaded area
is the area we found in the table. However, the probability that X is less than 84
is the probability that Z is less than 2 which is the total area below Z = 2.
Since the area to the left of the centre here is 0.5,
the total area below Z = 2 is 0.4772 + 0.5 which gives 0.9772 or 97.72%.
For part b), we want the probability that X is at least 84.
Recall that the normal distribution is a continuous distribution so there
is no distinction between “X is at least 84” and “X is greater than 84.”
We apply the same approach in both cases. Also recall from our previous calculation
that when X = 84, Z = 2. So we will be correct in
saying that P(X ≥ 84) = P(Z > 2). And one way to find this “greater
than” probability is to subtract the “less than” probability from 1.
If we don’t have the less than Area, since the area above the mean is 0.5,
we can do 0.5 minus the 0.4772 area to obtain 0.0228.
In part c) we want the probability that X is between 60 and 74.
For X = 60, Z equals -1.4286 which becomes -1.43 when rounded to 2 decimal places.
We have to ensure we round correctly to 2 decimal places if we’re using the table.
Also, for X = 74, the standard normal score equals 0.57.
Recall that we do not have negative Z values on the Mean to Z tables.
Therefore, for Z = -1.43, we will look up 1.4 under 0.03 which gives 0.4236;
It’s the same area from 0 to -1.43 because of symmetry.
Similarly, for Z = 0.57, we look up 0.5 under 0.07 and that gives 0.2157.
Thus, the probability of obtaining a score between 60 and 74
is found by adding the two areas. And that gives 0.6393.
Finally, we find the probability of a score lying between 74 and 84.
When X = 74, the Z value is 0.57 And when X = 84, the Z value is 2.
The area between 0 and 0.57 is 0.2157 And the area for 2.00 is 0.4772.
The required probability is the area between the two Z scores, shaded in blue here.
And it can be obtained by subtracting the smaller area from the larger area.
And that gives 0.2615.
In summary, to find the area from Z = 0 to any Z value in the mean to Z table,
just lookup the area in the table. And if you have a negative Z value,
the area from 0 is the same as the area for the positive Z, due to symmetry.
You can also obtain the area in the tail of the curve,
whether for a negative or positive value of Z, by subtracting the 0 to Z area from 0.5.
If you want the area above a negative Z value or below a positive Z value,
then add 0.5 to the area in the table.
Next if you want to find the area or probability between a negative Z and a positive Z value,
then simply add the two areas obtained directly from the 0 to Z table.
If you want the area beyond them, then subtract this total area from 1.
Finally, if the two Z values are on the same side of the centre.
That is, either both positive or both negative, then subtract
the smaller area from the larger one. Note that we only add or subtract areas.
We do not add or subtract Z values. Note also that you cannot obtain a negative area.
Areas or probabilities have to be between 0 and 1 or between 0 and 100%.
And that’s it for this video. Thanks for watching.