The table below shows the scores of a group of students
on a ten point quiz.
So we have a frequency table with the first column
is the test score.
The second column is the frequency
of the test score.
Member the frequency tells us
how many times that test score occurred.
We ask to determine the mean score
as well as the median score.
Let's first find the mean score.
To find the mean of set of data.
We sum the data values
and divide by the number of values.
The mean can also be called the average.
Let's first determine the number of data values in this case test scores.
By summing the frequency.
The sum of the frequency column
is equal to 22.
So we say n the number of data values or test scores
equal 22.
And therefore to find the mean
often label x bar.
We want to find the sum of all 22 test scores.
And then divide by 22.
We could write out all 22 test scores and then find the sum.
Or we could also write the sum
as the sum of products.
What I mean by that is because three
has a frequency of three.
We know there are three test scores of three.
As a sum that be three, plus three, plus three.
Because we have three three's
we can also write this sum as three times three.
Notice four also has a frequency of three.
So we have the sum of
three four's.
Which we could also write as three times four.
Or if we want it four times three.
So if we find the product of the frequency in the test score
we can avoid having to write out all 22
test scores.
So let's go ahead and do it this way.
So again because three has a frequency of three.
Again with the sum of these test scores as three times three.
Plus
four also has a frequency of three.
So we could write this sum as
three times four.
Plus
five is a frequency of four so we can rewrite the sum of these test scores
as four times five.
Plus
six is the frequency of one so we could just write six or plus one times six. Plus
seven is the frequency of two. Again write that sum as two times seven. Plus
eight is the frequency of four. So four times eight.
Plus
nine is the frequency of two. So we have two nine. So two times nine.
Plus finally we have ten with the frequency of three.
So this sum would be three times ten.
And now let's go and use a calculator.
To find
the mean or average.
So we'll put the numerator in parentheses open parenthesis.
Three times three is nine.
Plus
three times four is 12.
Plus
four times five is 20.
Plus one times six is six.
Plus two times seven 14.
Plus four times eight is 32.
Plus
next we have two times nine that's 18.
Plus
finally we have three times ten that's 30.
Close parenthesis.
And divided by 22.
So the one decimal place
the means approximately
six point four.
And now let's find the median score.
For review
the median of a set of data is the value in the middle
when the date is in order from least to greatest.
So to find the median
we begin by listing the data in order from smallest to largest
or largest to smallest.
In our case
this is already been done.
If the number data values N is odd which is not our case.
Do I do of an example for this.
Than the median is the middle data value.
This value can be found by rounding
and divided by two
up to the next whole number.
If the number data value is even.
Which is our case in this example.
There's no middle value
we need to find the mean or average of the two middle values.
Which should be in the position of N divided by two.
And N divided by two plus one.
So we need to find the two values in the middle.
By using these two formulas here.
So we need to find the position.
Of N divided by two.
And the position
of N divided by two plus one.
And then we'll find the mean of these two test scores.
Well because N is equal to 22.
Here we would have 22 divided by two which equals 11.
And then here we have
22 divided by two plus one which should be equal to 12.
So we need to find the test scores
in the eleventh and twelfth positions.
And then find the mean or average of those two test scores.
Now doesn't matter whether we start from the low test score the high test score
to determine the eleventh and twelfth position.
Let's go and start with the low test score.
And then we use the frequency to determine
the test scores in the eleventh and twelfth position.
So adding the frequency three plus three is six.
Plus four that would be ten.
So the next test score would be the eleventh test score.
Which notice would be six.
So the eleventh test score is six.
And notice the twelfth test score
would be the next test scores of seven.
So the test score in the twelfth position
is seven.
And therefore the median is going to be the mean of these two test scores.
So we have six plus seven divided by two.
13 divided by two is equal to six point five.
So the median of our test scores
is six point five.
Notice the test score of six point five would fall halfway between six and seven.
Or here.
And notice how this test score would be in the middle.
Because there are 11 test scores
looking at the frequencies.
Below six point five
and there also 11 test scores here
above six point five.
So we have the test scores a 50 percent or below six point five.
And have the test scores
one 50 percent or above six point five.
Which verifies our median
is correct.
I hope you found this helpful.