Welcome to lesson on z-scores. In this lesson
we will determine z-scores in a normal distribution, and also determine a score
given a z-score mean, and standard deviation.
A z-score describes how far a data value
is above, or below the mean. The units used to describe the distance above, or
below the mean
are standard deviations. So we can calculate the z-score
by taking the deviation, and dividing by the center deviation,
which can also be expressed as x minus mu,
divided by sigma, where x is the data value.
Mu is the mean, and sigma is the standard deviation.
Let's take a look at our first example.
The results of a common standardized test
used in psychology research is designed so that the population mean
is one hundred thirty, and the standard deviation is twenty.
A subject earns a score of one, fifty-two.
What is a z-score for this raw score?
Which means were asking how many standard deviations
above one hundred thirty is one hundred fifty-two.
So we know that mu, the population mean is equal to one hundred thirty.
Sigma, the standard deviation, is equal to twenty.
And our score x is one hundred fifty-two.
So the z-score is going to be equal to x minus mu,
divided by sigma, or one hundred fifty-two,
minus one hundred thirty, divided by
twenty. This would be twenty-two, divided by twenty,
which is equal to one, point, one. Notice how the z-score is positive,
and this tells us that one hundred fifty-two is one, point, one standard deviations
above the mean at one hundred thirty.
Let's take a look at a second example.
We have the same situation, but now the population mean is one hundred twenty-five.
The standard deviation is forty and the score is twenty-nine.
Our goal is to determine the z-score.
Notice how here the score is below the mean, so our z-score will be negative.
So we have mu. The mean is equal to one hundred twenty-five.
Sigma, the standard deviation is forty,
and the score x is equal to twenty-nine.
So the z-score it's going to be equal to twenty-nine minus one hundred twenty-five,
divided by forty.
So we'll have negative ninety-six divided by forty,
which is equal to negative two, point, four.
So this tells us that the score of twenty-nine is two, point, four standard deviations
below the mean of one hundred twenty-five.
Now let's take a look at a different type of question.
The given information is similar. The mean is one hundred.
The standard deviation is forty. But now we want to determine the score
that corresponds to the given z-score.
Then determine what percent of the data is above and below the score.
Well if we take our z-score formula and solve for x,
which is the score, we would have x equals mu plus z times sigma,
or the test score is equal to the mean, plus z-score,
times the standard deviation. So again we know mu equals one hundred,
and we know sigma equals forty,
and we know for part a, the z-score is one, point five.
So we have x equals the mean of one hundred plus the z-score, which is one, point, five
times the standard deviation of forty. Well one, point, five times forty would be sixty.
One hundred plus sixty equals one hundred sixty,
which would be the test score if the z-score is one, point, five,
which again tells us that one hundred sixty is one, point, five standard deviations.
above the mean of one hundred.
But now we're also asked determine what percent of the data is above, or below this score.
There are several ways to do this using technology, but for this lesson
we'll go old school and use a z-score table.
And there are typically two types of z-score tables.
The cumulative table gives us the percent of data below a particular z-score,
and the zero through z table gives us the percent of data between the mean
and a particular z-score.
So for our first example, since we have a z-score of one, point, five,
let's use the cumulative table. So we'll look for one, point, five
in the cumulative table. So go down the first column, look for one, point five, which is here.
Notice if we go across this row, this gives us
a second decimal place for the z-score.
Since we have one, point, five, or one, point, five, zero,
notice how they cumulative value is zero, point, nine, three, three, two,
which means ninety-three, point, three, two percent of the data
would be below the z-score of one, point, five.
So we can say that ninety-three, point, three, two percent
of the scores will be below
one hundred sixty, which means one hundred percent
minus ninety-three, point, three, two, or six, point, six, eight percent
of the scores would be above one hundred sixty.
Let's take a look at our second example. The information is the same,
but now the z-score is negative two, point, one. So the score that correspond to this z-score
would be x equals one hundred.
And then minus two, point, one times the standard deviation of forty,
which give a score of sixteen.
Now we'll determine the percent of data that is above and below the score of sixteen.
This will be slightly more challenging
because our z-score table only gives positive values of z.
So we'll have to use the symmetry of the normal curve to determine our percentages.
Again we have a choice of the cumulative table, or the zero through z table.
And just to show the difference, in this example, let's use the zero through z table.
So we're not going to a find negative two, point, one, so we'll have to use positive two, point, one.
Notice how if z is positive two, point, one,
the percent of the data that would be between the mean and z equals two, point, one,
would be approximately forty-eight, point, two, one percent here.
So that would give us the percent of data that's actually, let's say in this region here.
But we're looking for the percent of data that's above and below the score that corresponds to
z equals negative, two, point, one.
But if forty-eight, point, two, one percent of the data is in this region here,
then forty-eight, point, two, one percent of the data would also be in this region here,
because of the symmetry.
And we also know that fifty percent of the data would be above the mean,
so the amount of data in this region here all the way to the end
would be fifty percent. And therefore, forty-eight, point, two, one plus fifty percent
would be ninety-eight, point, two, one percent of the data
would have to be above the z-score of negative two, point, one.
So ninety-eight, point, two, one percent of the scores will be
above sixteen,
which means one hundred percent minus ninety-eight, point, two, one,
or one, point, seven, nine percent
of the scores will be
below sixteen.
So sometimes using the tables, it can be a little trickier if the z-score is negative.
I hope you found this helpful.