Download Measures of Central Tendency

MEASURES
OF
CENTRAL TENDENCY
MEASURES OF CENTRAL
TENDENCY
What is a measure of central tendency?
Measures of Central Tendency
Mode
Median
Mean
Shape of the Distribution
Considerations for Choosing an Appropriate
Measure of Central Tendency
WHAT IS A MEASURE OF
CENTRAL TENDENCY?
• Numbers that describe what is average or
typical of the distribution
• You can think of this value as where the
middle of a distribution lies.
WHY CAN’T THE MEAN TELL US
EVRYTHING?
Mean describes Central Tendency, what the average
outcome is.
We also want to know something about how accurate
the mean is when making predictions.
The question becomes how good a representation of the
distribution is the mean? How good is the mean as a
description of central tendency -- or how good is the
mean as a predictor?
Answer -- it depends on the shape of the distribution. Is
the distribution normal or skewed?
THE MODE
• The category or score with the largest
frequency (or percentage) in the
distribution.
• The mode can be calculated for variables
with levels of measurement that are:
nominal, ordinal, or interval-ratio.
MODE AN EXAMPLE
Example: Number of Votes for Candidates for
Mayor. The mode, in this case, gives you the
“central” response of the voters: the most popular
candidate.
Candidate A – 11,769 votes
The Mode:
Candidate B – 39,443 votes
“Candidate C”
Candidate C – 78,331 votes
Measures of Central Tendency
One further parameter of a population that may give some
indication of central tendency of the data is the mode
Define:
mode = most frequently occurring value in the
population
From the previous data we see:
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73,
74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78,
78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81,
81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
That the value 81 occurs 8 times
mode = 81
Note! If two different values were to occur most frequently, the
distribution would be bimodal. A distribution may be multi-modal.
MOST COMMON OUTCOME
Male
Female
The score that divides the distribution into two equal parts, so that half
the cases are above it and half below it.
The median is the middle score, or average of middle scores in a
distribution.
TO COMPUTE THE MEDIAN

first you rank order the values of X from low to
high:  85, 94, 94, 96, 96, 96, 96, 97, 97, 98

then count number of observations = 10.

add 1 = 11.
 divide by 2 to get the middle score  the 5 ½ score
here 96 is the middle score score
EXAMPLE OF MEDIAN (N IS ODD)
Calculate the median for this hypothetical
distribution:
TEMPERATURE
Very High
Frequency
2
High
3
Moderate
5
Low
7
Very Low
4
TOTAL
21
MEDIAN EXERCISE (N IS EVEN)
Calculate the median for this hypothetical distribution:
TEMPERATURE
Frequency
Very High
5
High
7
Moderate
6
Low
7
Very Low
3
TOTAL
28
MEDIAN
Find the Median
4 5 6 6 7 8 9 10 12
Find the Median
5 6 6 7 8 9 10 12
Find the Median
5 6 6 7 8 9 10 100,000
Measures of Central Tendency
A second measure of central tendency is the median
The median of a population of size N is found by
1. Arranging the individual measurements in ascending
order, and
2. If N is odd, selecting the value in the middle of this list as
the median (there will be the same number of values
above and below the median)
3. If N is even find the values at position N/2 and N/2 + 1 in
this list (call them xN/2 and xN/2+1) and let median be given
by the formula median = (xN/2 + xN/2+1)/2 or be the value
halfway between these two measurements.
Note! When N is even the median will usually not be an
actual value in the population
Measures of Central Tendency
We now find the median of the population of temperature
readings
87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,
73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,
84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,
85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68
Arrange these 60 measurements in ascending order
65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73,
74, 74, 75, 75, 75, 75, 76, 76, 77, 77, 77, 77, 77, 77, 78,
78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81, 81, 81, 81,
81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92
Since N/2 = 30 and both the 30th and 31st values in the list are
the same, we obtain median = 78
THE MEAN
The arithmetic average obtained by adding up all the scores and
dividing by the total number of scores.
FORMULA FOR THE MEAN
X
X
“X bar” represents MEAN
N
FINDING THE MEAN
• X = (Σ X) / N
• If X = {3, 5, 10, 4, 3}
X = (3 + 5 + 10 + 4 + 3) / 5
= 25 / 5
=
5
CALCULATING THE MEAN WITH
GROUPED SCORES
X 
 fx
N
where: fx = a score multiplied by its frequency
MEAN: GROUPED SCORES
From the table we obtain
Class
Class Midpoint (x) Total (f)
Frequency
f*x
64.5 - 69.5
67
6
0.100
402
69.5 – 74.5
72
11
0. 183
792
74.5 – 79.5
77
20
0.333
1540
79.5 – 84.5
82
13
0.217
1066
84.5 – 89.5
87
9
0.150
783
89.5 – 94.5
92
1
60
0.0167
92
4675
X 
 fx
N
MERITS AND DEMERITS OF MEAN
MERITS
1.It is easy to calculate
2.It is easy to follow
DEMERITS
1.It is highly effected by extreme values
2.It cannot average the ratios and percentage
properly
MEAN FOR DISCRETE SERIES
Number of People(x)
1
2
3
4
5
6
TOTAL
Frequency(f)
190
316
54
17
2
2
581
SHAPE OF THE DISTRIBUTION
Symmetrical (mean is about equal to median)
Negatively (example: years of education)
mean < median
Positively (example: income)
mean > median
Bimodal (two distinct modes)
Multi-modal (more than 2 distinct modes)
DISTRIBUTION SHAPE
Measures of Central Tendency
Next we show where each of these parameters occur in the
frequency distribution graph for this tabulated data.
Frequency %
42
39
Mean = 79.183
36
33
Median = 78
x
30
Midrange = 78.5
27
24
Mode = 81
x
21
18
x
15
x
12
9
x
6
3
median
mean
x
0
67
72
77
82
87
92
Temperature