Professor Dave here, let’s compare different kinds of averages.
The word “average” is one we hear a lot in the English language.
We could be talking about average temperature or rainfall, or other statistics.
You could tell your friend that you’re having an average day.
Whatever the context, the word average implies that something is typical or in some way representative
of a larger data set.
But how do we compute an average?
Are there different kinds of averages?
Let’s examine a data set and learn some different ways to look at it.
Let’s say you’re on a baseball team, and you played ten games this season.
You decided to record the number of hits you had in each game, and ordering these from
least to most, you had 0, 1, 1, 2, 2, 2, 3, 5, 5, and 7 hits for your best game.
What are some things we can say about this data set?
First, we might want to mention the range of the data set.
This is how wide the set spans, or just the difference between the smallest number and
the biggest number.
Seven minus zero is seven, so seven is the range of this data set.
Then we might wonder about your average number of hits per game.
When we say average in this way, we are typically referring to the mean of a data set.
To find the mean of a set of numbers, we just add up all the numbers, and then divide by
the number of items in the set.
So for instance, the average of one and two would be one plus two divided by two, because
there are two numbers in the set.
Three halves is one point five, precisely in between one and two, which makes sense
as the average of one and two.
For this particular set, if we add everything up we get twenty-eight.
Dividing by ten since there are ten values in the set, we get two point eight.
That is the average number of hits you had per game.
But there are other values we can identify that could be of use.
The median is the value right in the middle of the data set when the values are listed
in ascending order.
If there are three values, it’s the second one.
If there are five, it’s the third.
For any set with an odd number of values, it’s the one right in the middle.
If there are an even number of values in the set, it’s a little trickier.
This one has ten, so the fifth and sixth terms are equidistant from the two ends.
In this particular case, both of those terms are two, so two is the median of this data
set.
If this was not the case, we would have to take the average of the two middle terms,
so if the fifth and sixth terms were two and three, the median of the set would be two
point five.
Lastly there is the mode.
The mode is the number that shows up with the greatest frequency, or is present the
most times.
For this set, that’s two, because there are three twos, which is more than any other
number in the set.
So we have a range of seven, a mean of two point eight, a median of two, and a mode of
two.
What would we use these values for?
Well they have lots of applications.
Taking the mean of a set is very common practice for obvious reasons, but sometimes the median
will be more statistically significant if we have particularly extraneous data.
Say we are recording the number of fireworks that go off per day in the month of July in
a particular town.
Almost every day would be zero or close to it, but on the fourth of july, for just one
day, this value would be in the hundreds or thousands.
If we were to simply take the mean of this data set, we would get an average of a few
dozen fireworks per day.
But while this is the average value, it certainly does not describe the average day in that
town.
Instead, the median or the mode, which are very likely to be zero, are better descriptors
of the average day in the town, because these types of averages dilute the impact of anomalous
data.
So we can see that there are a variety of ways to summarize data, depending on what
you are trying to communicate.
Now that we know the meanings of range, mean, median, and mode, and how to determine them,
let’s check comprehension.