Mean Mode and Median
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.