MEASURES OF CENTRAL TENDENCY OR POSITION
- commonly referred to as an average
- it is a single value that represents a data set
Three different measures of central tendency
1. Mean
2. Median
3. Mode
A.Mean or average value ( X ) bar (for ungrouped data)
- often called as arithmetic mean
-the most frequently used measure of central tendency
Sample Mean :
X bar = ∑ X
N
where;
X = sample mean ( x bar )
X = the value of each item
N= number of items
or
X = X1 + X2 + X3 + . . . + Xn
N
Ex. 1. The prices of certain books are set at P 10, P 15, P 18, P 20, P 24.What is the
arithmetic mean?
Soln;
X = ∑X
N
=
=
X = P10 + P15 + P18 + P20 + P24
5
P 17.40
2. The grades of student A in 5 subjects are : 78, 88, 89, 90 and 95.What is her
mean grade?
Soln: X = 78 + 88 + 89 + 90 + 95
5
= 88
B. Median or positional value / center ( X )
- it is the value of the middle item in an ordered arrangement of data.
- In an ordered distribution, half of the terms are located above the median and
half are below the median.
Note:
a. If the number of items in a distribution is odd, the median value will be an
actual value.
b. If the number of items is even, the median value will be an estimated value.
Ex. 1. Distribution A:
Distribution B:
2, 4, 6, 10, 11, 12, 13
2, 4, 6, 10 , 11, 32, 37
N = 7 ( odd ) , X = 10
2. Find the median of the scores of sophomore students in Chemistry.
12, 34, 23, 14, 16, 33, 41, 35, 10, 45, 25, 24, 50.
Arranged in ascending order:
10, 12, 14, 16, 23, 24, 25 , 33, 34, 35, 41, 45, 50
N= 13 Thus the midpoint is 25
X = 25
3.a ) Number of Employees in Selected Factories ( Ungrouped data)
No. of Employees
50
578
20
65
480
1140
41
305
176
740
562
37
946
245
Arranged in Ascending order
20
37
41
50
65
176
245
305
480
562
578
740
946
1140
N= 14
X = 275
Note:
- With an even number of items there will be two middle items with half of
the distribution located above them and half below them
- To find the median value, we must take the arithmetic mean of these two
middle item.
b.Number of Employees in Selected Factories
No.of Employees
50
578
20
65
480
Arranged in ascending order
20
37
41
50
65
1140
41
305
176
740
562
37
946
176
305
480
562
578
740
946
1140
Note :
C. Mode :
data= odd
N= 13
X = 305
( X ) or “frequency value”
- the simplest measure of central tendency
- it is the value in a data set that appears most frequently
Note:
Unimodal – a distribution with only one mode
Multimodal- a distribution with two or more modes
Bimodal – a distribution which has two modes
No Mode – when a data set values have the same number frequency
Examples:
Set A
Set B
15
13
13
14
17
17
18
12
16
17
16
15
14
15
14
11
X = 13, 17 bimodal
X = 14, 15, 16 multimodal
3. 2 2 4 4 11 11
X= no mode, none
4.
2 , 4, 5, 8, 11
X= no mode , none
5. 12, 10, 10, 10, 10, 15, 15, 15, 8 and 2
X= 10 unimode