# 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.

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.