MATHEMATICS
IN THE
MODERN WORLD
Chapter IV
Statistics
What is a Measure of
Central Tendency?
- A measure of central tendency or measure of central location is summary measure that
describes a whole set of data with a single quantity that represents the middle or center of its
distribution the way in which a group of data that cluster around a central value. In short, this
is a measure that tells where the center of a data set is located.
- The most common commonly used measures of central tendency are the mean, median and
mode.
MEAN
The mean, also called the “average” or “arithmetic average”, is the most commonly used
measure of central tendency. It is said to be the most reliable measure of central tendency and
has the least probable error but does not supply information about the homogeneity of the
distribution.
Exercises:
- Find the mean of the following ungrouped data:
- 1. 37, 37, 24, 28, 43, 44, 36, 41, 33, 27
- 2. 8, 12, 15, 14, 19, 21, 24, 38
- 3. 62, 60.4, 61.8, 61.4, 59.8, 59.2, 59.8, 60.2, 61.1, 62.2, 60.4, 60.3, 60.8, 60.9
- 4. 297, 311, 318, 303, 306, 291, 300, 298, 322, 315, 307, 296, 312, 309, 300, 311
- 5. 74, 73, 77, 77, 71, 68, 65, 77, 67, 66
Mean for Grouped Data
- For grouped data, the midpoints of the classes are used for the values of the x. The following
are the steps in solving for the mean of grouped data.
- Grouped mean:
- 1. find the midpoint for each class. Place them in a column
- 2. multiply the frequency by the midpoint for each class. Place them in another column.
- 3. Find the sum of the resulting column in step 2.
- 4. Divide the sum obtained in step 3 by the total number of frequencies.
Mean = fx / n
Example
- 1. Consider the frequency distribution below:
-
Class
Interval
Frequency
(f)
75 - 79
5
70 - 74
7
65 - 69
8
60 - 64
10
55 - 59
7
50 - 54
9
45 - 49
4
n = 50
Midpoint
x
fx
∑fx = 3100
Example
- 1. Consider the frequency distribution below:
Class
Interval
Frequency
(f)
Midpoint
x
fx
75 - 79
5
77
385
70 - 74
7
72
504
65 - 69
8
67
536
60 - 64
10
62
620
55 - 59
7
57
399
50 - 54
9
52
468
45 - 49
4
47
188
n = 50
∑fx = 3100
MEDIAN
- A median is defined as the middle value/observation in an organized list of numbers and falls in the
middle-most position of the whole data.
- The median value in an ungrouped data is determined by first arranging the numbers in value order
from lowest to highest of vice versa. If there is an odd amount of numbers, the median value is the
middle most number, with the same amount of numbers below and above. If there is an even
amount of numbers in the list, the middle pair must determined, added together and divided by two
to find the median value.
- The median is the midpoint of the data array. Before finding this value, the data must be arranged
in order, from least to greatest of vice versa. The median will either be a specific value or will fall
between two values.
MEDIAN
Example
- 1. Consider the frequency distribution below:
Class
Interval
Frequency
(f)
cf
75 - 79
5
5
70 - 74
7
12
65 - 69
8
20
cf
60 - 64
10
30
Median class
55 - 59
7
37
50 - 54
9
46
45 - 49
4
50
n = 50
MODE
- The number/value/observation in a data set which appears the most number of times. If no
number in the list is repeated, then there is no mode for the list. However, it is also possible to
have more than one mode for the same distribution of data.
- To find the mode for ungrouped data, find the frequency of each number/value/observation in
the given data set. Then, choose the number/value/observation having the highest frequency
as the mode.
MODE
EXAMPLE
- Consider the frequency distribution below:
Class
Interval
Frequency
(f)
75 - 79
5
70 - 74
7
65 - 69
8
60 - 64
10
55 - 59
7
50 - 54
9
45 - 49
4
n = 50
d2 = 2
Modal Class
d1 = 3
EXERCISES
Find the mean, median and mode of the following:
CI
f
0 - 9
44
10 - 19
42
20 - 29
33
30 - 39
30
40 - 49
27
50 - 59
22
60 - 69
18
70 - 79
9
n=
EXERCISES
Find the mean, median and mode of the following:
CI
f
54 - 58
2
59 - 63
5
64 - 68
8
69 - 73
0
74 - 78
4
79 - 83
5
84 - 88
1
n=