Welcome to Alanis Business Academy.
In this lesson, we're going to talk about the weighted mean.
I you haven't do so already, you may want to watch episode six titled "Population and
Sample Means".
In that lesson I introduce the arithmetic mean, which of course we're going to build
upon in this lesson.
The weighted mean is actually a special type of arithmetic mean, and is commonly used when
a dataset contains several identical values.
First let me put together a dataset.
Say we are attempting to determine the mean or average number of correct responses for
a sample of students on an exam.
Since our data set represents a sample, we would solve for X bar by adding all of the
values together and dividing by the number of values in the sample.
If you'd like to pause this lesson and perform that calculation yourself, go ahead and do
that now.
Upon doing so, you should receive a value of 280.
Next we need to divide that figure by the number of values in our population, which
is 16.
Again, feel free to pause this lesson if you'd like to compute the population mean on your
own.
Dividing 280 by 16 we should receive a sample mean of 17.5, which again represents the average
number of correct responses for our exam.
Now we haven't done anything new at this point, but the purpose of going through this basic
calculation is to show you how the process differs when using the weighted mean method.
First, let me provide an equation for calculating the weighted mean.
That equation is X bar sub w equals sigma w X divided by sigma w.
This might be a little confusing so I'm going to briefly explain the variables that we have
here.
X bar sub w should look a little familiar since X bar refers to a basic sample mean,
like what we calculated above.
The sub w represents weighted, so we know that we're solving for the weighted sample
mean.
Another variable that should appear familiar is sigma, which if you recall from our last
lesson is the sum of.
The little w represents the weight or frequency of a particular value, while the X represents
the values in the data set.
Lastly, sigma w reflects the sum of those weights that were used above.
If this still seems a bit confusing, trust me it will make a lot more sense once we solve
this problem.
We've already listed our values in order from smallest to largest.
So what we're going to do is consolidate those values that are represented more than once.
Prior to summing the values as instructed by sigma, I'm going to list each weight and
the value it corresponds to insure accuracy.
Fifteen is only reflected once so there's no change there, but we're still going to
list it as a weight of 1 with an x of 15.
This will make a little more sense in a few minutes.
Next you'll see that we have more than one sixteen in our data set.
We have exactly four values of sixteen.
Instead of listing them all like we did above, we're going to list our weight, which is four,
and then the value sixteen in parenthesis.
Feel free to pause this lesson and perform the same task for the remaining groups.
Moving on, we have three seventeens, four eighteens, two nineteens, and two twenty values.
Our next task is to list the weights individually, which are one, four, three, four, two, and
two.
Our next step is to summarize or add up each of our values listed in the numerator and
denominator portion of the equation.
For our numerator, we have 280, which is exactly what we received when calculating the regular
sample mean.
And for the denominator we have 16, which again is consistent with our sample mean calculation.
The last step is to divide our numerator by our denominator, which gives us a weighted
sample mean of 17.5.
This figure is identical to the one we received minutes earlier when solving for the sample
mean, and it should be.
Depending upon the way that information is presented, it may make more sense to calculate
the weighted mean as opposed to a sample or population as in our last lesson.
Thank you for watching this lesson on the weighted mean.
In our next lesson we're going to learn about another measure of central tendency, the median.
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