In a previous video we discussed what a normal distribution is, real quick, it's
a graph or curve above the x-axis that is symmetrical about the mean, so if we draw a
line right down the middle representing the mean, both sides are equal in shape and area under the
curve. The biggest thing to understand is that the space under the curve is an area, and this
area actually represents the probability. And this area is equal to 1. The standard normal
distribution makes 2 modifications, the mean is 0, and the standard deviation is 1. The reason
this is important is that it allows us to use z-scores, and z scores are measures of position
that tell us the number of standard deviations of value is from the mean. For instance if we have a
z-score of 2.13, that would be this line here on our standard normal distribution graph.
So it is 2.13 units or standard deviations from the mean. Z-scores are important because they
make it much easier to calculate areas, which are probabilities. The formula for a z-score
is z equals, a value minus the mean, divided by the standard deviation, and statistically it's
written as z equals, x minus mu, divided by Sigma. Let's take a look at an example of how to use
this formula to calculate a z-score and how to use that z-score to calculate probability. So
let's say that you gathered some data for a year, for your commuting time to work. The average time
was 44 minutes, with a standard deviation of 3.5, and we're going to assume the data is normally
distributed. Graphing this, we have the mean of 44 here in the middle, if we add 1 standard
deviation to the mean, we get 47.5 here, and adding 2 standard deviations we
get 51 here. Subtracting 1 standard deviation from the mean, we get 40.5 here, and
subtracting 2 standard deviations from the mean, we get 37 here. Now let's say we want to know the
probability that it takes you less than 40 minutes to get to work tomorrow. 40 on the graph is about
here, so it lies between -1 and -2 standard deviations from the mean. As I stated
earlier, the space under this curve represents an area or probability, so if we can calculate
the area from less than 40 minutes, to the left of the line, that will be the probability that you get
to work in less than 40 minutes. This is where z-scores come in. So first we're gonna convert
the 40, in this case our x value, into a z-score. Using the formula, we have z equals 40, minus 44,
for the mean, divided by 3.5, the standard deviation, which equals -1.14. Now we
can graph this on the standard normal distribution curve like this. And with this z-score of -1.14,
we can use a standard normal distribution z-score table, to look up the area
or probability. You can find this table through an internet search or in any statistics book. So
we find -1.1 in left column, and go over 5 columns to the right, under .04,
which is -1.14 and the value equals 0.1271.
This value is the area, to the left of the line, or the probability. If our z-score was
-1.18, we would go over here to the .08 column, which is 0.1190.
And looking at the standard normal curve with both -1.14 and
-1.18 marked, -1.18 has a smaller area to the
left, that's why it's value is smaller in the table. This is the table for negative z-scores and there
is also a table for positive z-scores as you see here, and we'll use this later in the video. For our
question, we would answer, there is a 12.71% probability that it takes less
than 40 minutes to get to work tomorrow. A couple of notes, when we write this probability, we write
it as probability of x less than 40, we don't use less than or equals to. The reason for this
is that for a continuous variable the probability of an exact value is 0, as vertical lines,
in theory, have no area. So if we wanted to find the probability of between 45 to 50 minutes, we
would write that as probability of 45, less than x, less than 50, or we could have the probability
of x greater than 48. The second thing is that the values in the standard normal distribution z-score
table give the area to the left of the z-score, even the table for positive z-scores gives you
the area to the left. Using the standard normal distribution and z-scores, we have calculated the
area or probability of something lower than, or to the left of a z-score, and there are 2 other
important ways to use this method, for finding the area or probability of something between
2 values, or finding the area or probability of something greater than a value. First let's look at
between 2 values. What is the probability that it takes you between 38 and 45 minutes to get to work?
On the normal distribution graph we're looking for this area here, so we have probability of 38, less
than x, less than 45. First we need to convert 38 and 45 to z-scores. Using our formula we have 38
minus 44, for the mean, divided by 3.5, the standard deviation, which equals -1.71.
And 45 minus 44, divided by 3.5, equals 0.29. Now we need to look up
these z-scores in the table. For -1.71, we have 0.0436,
and for z equals 0.29, the table shows 0.6141. Let's look at
this on the standard normal graph. So we have the area to the left of -1.71
equals .0436, and the area to the left of 0.29, is .6141.
And we need to find the area between these two z-scores. To find this, we subtract the
area to the left of z equals -1.71, from the area of z equals 0.29.
So, 0.6141 minus 0.0436, equals 0.5705.
And that is our probability, now we can say that the probability it takes
you between 38 and 45 minutes to get to work, is 57.05%. The last way
we can use the standard normal distribution and z-scores, is to find the probability of something
greater than a value. Let's stick with the last example, what is the probability that it takes
you longer than 45 minutes to get to work? On the normal distribution graph we're looking for this
area here to the right of the line. From our last problem we know the z-score for 45 minutes is
0.29, and from the table, we know that the area to the left is .6141
on the standard normal graph. Since we know the entire area under the curve is equal to 1, we
can subtract the area to the left of z equals 0.29 from 1, to get the area on
the right. So 1minus 0.6141 equals 0.3859, which is our probability. So there's a
38.59% probability that it takes you longer than 45 minutes to get
to work. Alright my friends, that be the basics on the standard normal distribution. Hopefully
this video helped you out, I do have more videos right there for you, till next time, I am outta here.