In this video we'll take a look at z
scores, or finding the location of a
score in a distribution. Z scores
are given by the following formula, where
we have Z equals x minus mu divided by
sigma or the standard deviation. And here
X is a raw score or a value on some
variable X, mu is just the population
mean, and sigma is the population
standard deviation. And notice here that
in the z score formula, both central
tendency and variability are used to
obtain a z score. Z scores
indicate the number of standard
deviations a score is away from the mean and
we'll be exploring that more in just a
few moments. Z scores can be positive, or
above the mean, they can be negative,
which is below the mean, or they can be 0,
which is equal to the mean. Here are a
few examples, where we have a z of
positive 1, a z of -2, and a z of 0. So a z
of positive 1 indicates that the score,
or X, is 1 standard deviation above the
mean. Now it's above the mean, because
it's positive, and it's one standard
deviation above, because z is equal to 1. A z of
negative 2 indicates that the score is
two standard deviations below the mean.
And it's below the mean, because it's
negative.
Finally, a z of zero indicates that a score
is zero standard deviations away from
the mean, and if it's zero standard
deviations away from the mean, that just
means that it's equal to the mean. So if
you ever see a z of 0, that value, that X,
is just equal to the mean. Now in this
example we see integers but it should be
noted that you also can have non-integer
z-scores, such as a z of negative point
five.
Next let's take a look at some examples
for solving for z. And here's our formula
once again, in the upper right-hand
corner of your screen, and in this example
we have an X of 110, a mean of 100, and a
standard deviation of 10.
So X is going to be 110, mean is 100, and
standard deviation is 10. So that's plugged
into the formula here, we see (110-100)/10
or 10 / 10 is positive 1. And the way we
interpret that, once again, is a score of
110 is one standard deviation above the
mean of 100. We know it's above the mean
because it's positive, it's one standard
deviation above because z is equal to one, and
it's above the mean of 100 because we
saw in this problem that the mean was
equal to 100. Let's go and take a look at
another example. Here we have an X
of 100, a mean of 100, and a standard deviation of 10.
So plugging these values into our z-score
formula, we see 100, that's the X, minus
100, which is the mean, divided by 10, or 0 / 10
= a z of 0. And as we saw before, a z of zero
means that the score is zero standard
deviations away from the mean, or it's
equal to the mean.
Now here's an example where the value is
less than the mean. Notice here we have X
of 90, a mu of 100, and a standard deviation of 10.
So plugging these into the z score
formula once again, we would get z =
(90-100)/10, or negative 10/10, or a z
of negative 1.
And this would be interpreted as follows:
a score of 90 is one standard deviation
below the mean, and once again, it's below
the mean because it's negative. Here we
have an example of X is 120, mu is 100,
and the standard deviation is 10, 120-100
over 10 is 20/10, or a z of 2. So that
means that the score of 120 is two
standard deviations above the mean.
So, for our last example here, we have an
X of 95, mean of 100,
standard deviation of 10, once again. So 95-100 over
10, or negative 5 over ten is a z of
-.5. So we would
interpret this as follows: a score of 95
is one-half or .5 standard deviations
below the mean, or its one-half of a
standard deviation below the mean.
Now here are some practice problems,
where I have seven problems with various
means and standard deviations. I
encourage you to go through and try and
solve all of these. On the next slide the
answers are provided. So you may want to
pause the video, solve these, and check
the answers which I'm going to move to
now.
So, as I said, here are the answers for
the problems on the previous slide.
OK that's it for z scores. Thanks
for watching.