The table below shows the scores of a group of students on a 10 point quiz.
Notice how the information is given as a frequency table.
Where the first column with the test scores.
The second column
gives the frequency of each test score.
We want to determined the mean score
as well as the median score.
So for review
let's start with the mean.
The mean of a set of data is the sum of the data values divided by the number of data values.
And sometimes we'll see the formula for the mean
written here where x bar the mean
is equal to the summation of x divided by n.
Where the summation of x is the sum of data values.
And n is the number of data values.
So going back to our frequency table
let's first determine the number data values
by summing these frequencies.
If we sum this column here
we get 19
which means there are 19 test scores.
And therefore to find the mean which we'll label x bar.
We wanna find the sum of the 19 test scores and then divide by 19.
Instead of writing out all 19 test scores though.
Let's look at how we can use multiplication to help us find the sum
of these test scores.
Because the test scores of three has a frequency of two.
There are two test scores of three.
Which is the sum would be three plus three.
We can write three plus three as two times three.
Since we have two three's.
Course we can always write this as three times two.
Notice the test score of four has a frequency of four.
So if we want to find the sum of these test scores we could write out four,
plus four, plus four, plus four.
It be much faster to just write four times four
since we have
four four's.
So we could find the sum of products to determine the total sum
rather than writing out 19 data values.
Let's go ahead and do it this way instead.
So
three has a frequency of two so we have two times three. Plus
four has a frequency of four so four times four.
Plus
five has a frequency of two so plus two times five.
Plus
both six and seven have a frequency of one.
So could just write plus six and plus seven.
But to keep the pattern let's write one time six plus one time seven.
Plus
eight has a frequency of three so three times eight.
Plus
nine has a frequency of five so we have five nine so five times nine. Plus
ten has a frequency of one.
So we have one times ten.
The sum of these products would give us the sum
of all 19 data values.
And now let's go to the calculator.
To determine the mean.
We'll put the numerator in parentheses open parenthesis.
Two times three is six plus four times four is 16.
Plus two times five is ten.
Plus one times six is six plus one times seven.
Plus three times eight that's 24.
And then we have plus
five times nine is 45.
Finally plus
one times ten is ten. Close parenthesis.
Divided by 19.
So the mean is approximately
six point five to one decimal place.
And now let's determine the median.
For review the median of a set of data
is the value in the middle
when the date is in order.
So to find the median we begin by listing the data in order from smallest to largest
or largest to smallest
which in our case is already been done.
So next
if the number data values N is odd
which is our case.
Than the median is the middle data value.
This value can be found by rounding N divided by two
up to the next whole number.
Notice in our book there using capital N rather than lower case n.
But either way N represents a number of data values.
If the number data values happens to be even which is not our case there is no middle value.
So we find the mean or average of the two middle values.
Which should be N divided by two and N divided by two plus one.
Again in our case we wanna find N divided by two
and round up to the next whole number.
So N divided by two would be equal to 19 divided by two which equals nine point five.
So rounding up to the next whole number would be ten.
Which means the median
is the tenth
test score.
Starting any the lowest test score or the highest test score.
So we use the frequency to count down to the tenth
test score.
So adding the frequencies two plus four is six.
Plus two is eight.
Plus one is nine.
So six would be the ninth test score.
And we're looking for the tenth test score. So if we add one more which would be ten.
Notice the tenth test scores is seven.
Which means seven is the median.
Looking at the frequencies notice how there are two plus four, plus two, plus one
are nine test score below seven.
And there also
three plus five plus one
or nine test scores above seven.
Verifying that seven is the median.
Okay I hope you found this helpful.