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In everyday life it is often important to collect data, perhaps from
a survey, or a questionnaire. Once we have collected the data it is
then important to arrange it in a way which allows someone to
analyze the information, so that conclusions or decisions can be
made.
Mathematicians have named this study of "data"
Statistics
3 of the most important
statistics we use when we analyze data are called:
The mean, the
median and the
mode.
In this lesson we will define these three terms and demonstrate
how to determine these statistics using some basic arithmetic
skills.
We will also learn when each measure should be used.
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The
Mean
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The
mean
of a set of data is the arithmetic average. To find the
mean,
add all the data values together and divide by the number of
values. |
Let's start with
an easy example.
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In her last 5 games Sara
has scored the following number of points respectively:
12, 16, 11, 17, and 18.
What is the mean of
these scoring totals? |
To
determine the mean all we have
to do is add up all of her points and
then divide by the number of games she
played.
12 + 16 +
11 + 17 + 18 = 74
74/5 =
14.8
Sara's scoring
mean was 14.8 points per game!
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The
Median
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The
median
is the
middle
value after the data has been arranged in either
ascending
(smallest to largest) OR
descending
(largest to smallest) order. If the set of data contains
an
even
number
of values, then the
median
is the
average
(the mean) of the two middle values. |
Let's try a
couple of these...
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Jake received the following test
grades on his last 7 quizzes:
79,88,75,71,89,80,92
What was his median
grade? |
First
we must arrange the scores in either ascending or descending order:
In ascending order: 71,75,79,80,88,89,92
In descending order:
92,89,88,80,79,75,71
In this set we have 7 values, and
because 7 is an odd number we have a
middle value. The middle value
will be the 4th number from either end of the set.
In this case the median would be
71,75,79,80,88,89,92
80! the
median of this set is 80
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Now let's do one
with an
even
number of data values
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Over an
8 day period Bessy, the cow, produced the following total
gallons of milk per day:
16,18,19,15,14,20,17,22
What was the median number
of gallons over this period of time? |
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Again,
the first step is to arrange the values in order.
Let's put them in ascending order
this time:
14,15,16,17,18,19,20,22
Notice that in this problem the number of data values is
even.
There are 8 entries. That
means that there is no value exactly in the middle. In
this case in order to find the median
we will need to find the average (the
mean) of the
two middle values.
14,15,16,17,18,19,20,22
I have highlighted the 2 middle
values.
To average them we simply
add them together
and divide by 2.
17 + 18 = 35
35/2 = 17.5
The median of this set is 17.5 gallons! |
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The
Mode |
The
mode is the data value that
appears the most in the set.
Because no value may appear more often than any other, it is
possible for a set to have no mode. And because it is
possible for a set to contain more than one element which
appears the same number of times as another, it is possible for
a set to have more than one mode. |
Let's look at a
couple examples.....
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The
following is a list of heights (in inches) of a high school
basketball team:
68,67,66,70,74,72,68,71,75,68,72,71
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To
determine the mode all we need
to do is determine which, if any, of the data values appear
the most.
If you look carefully you will see that several of the values appear
more than once, but only one of the values
appears the most...
There are 2-71's, and 2-72's,
BUT
there are 3-68's!
So, the mode of this set is 68!
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Let's look at
one more....
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This is a list of the grades of a class on a recent math test:
85,78,90,91,88,85,73,84,90,95,67,70
Find the mode for this data.
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Again, look carefully for
data values which appear
more than once.
Rewriting the list in numerical order is helpful because you
will clearly see any multiple entries. It is also helpful
because you need to do that to determine the
median anyway.
67,70,73,78,84,85,85,88,90,90,91,95
This set has 2 modes, we
call this set bimodal.
The modes for this set are 85
and 90.
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Now
let's
practice!
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