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Mean Mode and Median

By JherellSerju Jan 31, 2015 943 Words
Mean, Mode and Median

Ungrouped and Grouped Data
Ungrouped Data refers to raw data
that has been ‘processed’; so as to
determine frequencies. The data,
along with the frequencies, are
presented individually.

Grouped Data refers to values that
have been analysed and arranged into
groups called ‘class’. The classes are
based on intervals – the range of
values – being used.
It is from these classes, are upper and
lower class boundaries found.

Mean

Mean
The
  ‘Mean’ is the total of all the values in the set of data divided by the total number of values in a set of data.
The arithmetic mean (or simply "mean") of a sample  is the sum the sampled values divided by the number of items in the sample.


x is the value of a member of the set of data

f is the frequency or number of members of the set of data

 Mean=
Therefore: = 6.56

Grades

Frequency (f)

Total Value (x)

1

5

5

2

2

4

3

7

21

4

4

16

5

4

20

6

1

6

7

8

56

8

3

24

9

5

45

10

4

40

11

4

44

12

5

60

TOTALS

52

341

Mean in relation to Grouped Data
Mean in relation to grouped data
emphasizes the usage of class
intervals. Rather than the data being
presented individually, they are
presented in groupings (called
class). It is from there a midpoint is

Grade
Intervals

Frequency (f)

1-3

14

4-6

9

7-9

16

10-12

13

reached (for each interval).

Unlike Ungrouped data, the mean is
estimated using the intervals. It will
prove difficult to gain the most
accurate mean.

Mean in relation to Grouped Data
There several things we must acknowledge before we determine the mean. They are:
1.

Interval width – the number of values in each interval.

2.

Lower class boundaries – the lowest value in each interval.

3.

Upper class boundaries – the highest value in each interval.

4.

Midpoints – the halfway point between the values of each interval.

Keeping all these things in mind, focus on the midpoint. The midpoint is what we must use to estimate the mean.

Mean in relation to Grouped Data

In using the midpoint to determine


the mean, we must assume that each

Therefore:

student in the interval (7-9), received
either seven marks or nine marks. It
is from these two assumptions that
the midpoint will be determined.


 

Do the same for the other classes.
Where: M is the midpoint.
U is the upper class boundary.
L is the lower class boundary.

When this is done, divide the total
frequencies by the sum of the
midpoints of all the classes.

Estimating the Mean using
Grouped Data

 Mean=
Therefore: = 6.61

Midpoints (x)

Frequency (f)

Total Value
(fx)

2

14

28

5

9

45

8

16

128

11

13

143

TOTALS

52

344

Mode

Mode
When selecting the mode, one must
observe the most frequent element
within the data set.
Within the ungrouped data set, an
element may occur numerous times.
The element that occurs the most

3

12

15

3

20

8

20

19

8

15

12

19

9

15

4

2

7

15

10

3

15

9

3

1

4

times is the mode.
* Note: there can be more than one
mode; so long as both elements occur
the same amount of time.

Mode in relation to Grouped Data

Likewise
to the Mean, the Mode in

 

relation to Grouped Data too emphasises
the usage of classes. We easily can
identify the ‘modal group’ by selecting
the class with the highest frequency. We
are allowed to say:
‘the modal class is 1-4’
We further estimate the mode by using
the following formula:

Class

Frequency

1-4

8

5-8

3

9-12

4

13-16

5

17-20

4

Where:
L = the lower class boundary

Median

Median and its relation to Ungrouped Data
Median
refers to the value found in
 
the centre of the numerically
arranged values, beginning from the
lowest to the highest. In the case
where you have two values in the
‘assumed centre’; divide the sum of
these two values by 2.

Given the numbers:
2 ,5, 1, 3, 8, 6, 9, 6, 2, 7, 5, 4
What is the mean?

Where: v1 – value one
v2 - value two

Median in relation to Grouped Data
The
  median is the mean of the two
middle numbers (26th and 27th values),

Class

Frequencies

both within in the ‘7-9’ interval. It would
be foolish to say:

1-3

14

4-6

9

7-9

16

10-12

13

“the median group is 7-9”
Thus we utilise the median value formula
to obtain the median.

Where: L is the lower class boundary,
n is the total number of data, cfbis the
cumulative frequency of the groups
before, Fm is the interval frequency
and W is the group width.

Let’s
  apply the formula:
 L=6.5
 n=52
 cfb = 14+9=23
 fm = 16
 W=3

Grade
Interval

Frequencies

1-3

14

4-6

9

7-9

16

10-12

13
52



 

= 21.5625

Understood?
Case Study:
A class of thirty students had a quiz. At
the end of the class, the teacher posted
the results. From the table on the right:
 Create a frequency table and

calculate the mean.

20

11

9

15

3

12

5

1

18

2

8

15

6

9

7

14

11

19

4

8

18

7

15

5

7

19

12

14

15

2

 Create a class frequency table and

provide the estimated mean – using
a class width of 5.
 Determine the mode and median.
 Show the estimated median using

the grouped data (class frequency
table) method.

Cite This Document

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