amity global business (bangalore)

1. 1
Measures of
Central Tendency
AGBS Bangalore | 2013

2. Central Tendency
• In general terms, central tendency is a
statistical measure that determines a
single value that accurately describes the
center of the distribution and represents
the entire distribution of scores.
• The goal of central tendency is to identify
the single value that is the best
representative for the entire set of data.

3. 3
Central Tendency (cont.)
• By identifying the "average score," central
tendency allows researchers to summarize or
condense a large set of data into a single
value.
• Thus, central tendency serves as a descriptive
statistic because it allows researchers to
describe or present a set of data in a very
simplified, concise form.
• In addition, it is possible to compare two (or
more) sets of data by simply comparing the
average score (central tendency) for one set
versus the average score for another set.

5. 5
The Mean, the Median,
& the Mode
• It is essential that central tendency be
determined by an objective and well-defined
procedure so that others will understand exactly
how the "average" value was obtained and can
duplicate the process.
• No single procedure always produces a
good, representative value. Therefore,
researchers have developed three commonly
used techniques for measuring central tendency:
the mean, the median, and the mode.

6. 6
The Mean
• The mean is the most commonly used
measure of central tendency.
• Computation of the mean requires scores
that are numerical values measured on
an interval scale.
• The mean is obtained by computing the
sum, or total, for the entire set of scores,
then dividing this sum by the number of
scores.

7. 7
The Mean (cont.)
Conceptually, the mean can also be defined as:
1.The mean is the amount that each individual
receives when the total (ΣX) is divided equally
among all N individuals.
2.The mean is the balance point of the
distribution because the sum of the distances
below the mean is exactly equal to the sum of
the distances above the mean.

10. Calculate the Mean number of credit hours

11.  Calculate the Mean Salary
 Calculate the Median Salary

13. 13
Changing the Mean
• Because the calculation of the mean involves
every score in the distribution, changing the
value of any score will change the value of
the mean.
• Modifying a distribution by discarding scores or
by adding new scores will usually change the
value of the mean.
• To determine how the mean will be affected for
any specific situation you must consider: 1) how
the number of scores is affected, and 2) how the
sum of the scores is affected.

14. 14
Changing the Mean (cont.)
• If a constant value is added to every score
in a distribution, then the same constant
value is added to the mean.
• Also, if every score is multiplied by a
constant value, then the mean is also
multiplied by the same constant value.

15. 15
When the Mean Won’t Work
• Although the mean is the most commonly used
measure of central tendency, there are
situations where the mean does not provide a
good, representative value, and there are
situations where you cannot compute a mean at
all.
• When a distribution contains a few extreme
scores (or is very skewed), the mean will be
pulled toward the extremes (displaced toward
the tail). In this case, the mean will not provide a
"central" value.

16. 16
When the Mean Won’t Work (cont.)
• With data from a nominal scale it is
impossible to compute a mean, and when
data are measured on an ordinal scale
(ranks), it is usually inappropriate to
compute a mean.
• Thus, the mean does not always work as a
measure of central tendency and it is
necessary to have alternative procedures
available.

17. 17
The Median
• If the scores in a distribution are listed in order
from smallest to largest, the median is
defined as the midpoint of the list.
• The median divides the scores so that 50% of
the scores in the distribution have values
that are equal to or less than the median.
• Computation of the median requires scores that
can be placed in rank order (smallest to
largest).

18. 18
The Median (cont.)
Usually, the median can be found by a
simple counting procedure:
1.With an odd number of scores, list the
values in order, and the median is the
middle score in the list.
2.With an even number of scores, list the
values in order, and the median is half-
way between the middle two scores.

20. 20
The Median (cont.)
• One advantage of the median is that it is
relatively unaffected by extreme
scores.
• Thus, the median tends to stay in the
"center" of the distribution even when
there are a few extreme scores or when
the distribution is very skewed. In these
situations, the median serves as a good
alternative to the mean.

21. Median for Grouped Frequency Distribution
21

22. Median for Grouped Frequency Distribution
22

23. Median for Grouped Frequency Distribution
23

24. The Mode
• The most common observation in a group of scores.
– Distributions can be unimodal, bimodal, or multimodal.
• If the data is categorical (measured on the nominal scale)
then only the mode can be calculated.
• The most frequently occurring score (mode) below is
Vanilla.
0
5
10
15
20
25
30
Vanilla
C
hocolate
Straw
berryN
eapolitanButterPecanR
ocky
RoadFudge
R
ipple
fFlavor f
Vanilla 28
Chocolate 22
Strawberry 15
Neapolitan 8
Butter Pecan 12
Rocky Road 9
Fudge Ripple 6

25. Chap 3-25
The Characteristics of the Mode
• Value that occurs most often
• Not affected by extreme values
• Used for either numerical or categorical
(nominal) data
• There may be no mode
• There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode

26. The Mode
• The mode can also be calculated with
ordinal and higher data, but it often is not
appropriate.
– If other measures can be calculated, the
mode would never be the first choice!
• 7, 7, 7, 20, 23, 23, 24, 25, 26 has a mode
of 7, but obviously it doesn’t make much
sense.

27. Mode for Grouped Frequency Distribution
27

28. Calculate the Mode
28

29. What is the mean rate of return here?
An investment of $100,000 declined to $50,000 at the end of
year one and rebounded to $100,000 at end of year two:
The overall two-year return is zero, since it started and ended
at the same level.
000,100$X000,50$X000,100$X 321 
50% decrease 100% increase

30. The Geometric Mean & The Geometric Rate of Return
 Geometric mean
 Used to measure the rate of change of a variable over time
 Geometric mean rate of return
 Measures the status of an investment over time
 Where Ri is the rate of return in time period i
n/1
n21G )XXX(X  
1)]R1()R1()R1[(R n/1
n21G  

31. The Geometric Mean Rate
of Return: Example
Use the 1-year returns to compute the arithmetic mean
and the geometric mean:
%2525.
2
)1()5.(


X
Arithmetic
mean rate
of return:
Geometric
mean rate of
return:
%012/1112/1)]2()50[(.
12/1))]1(1())5.(1[(
1/1)]1()21()11[(


 n
nRRRGR 
Misleading result
More
representative
result
(continued)

32. Measures of Central Tendency:
Summary
Central Tendency
Arithmetic
Mean
Median Mode Geometric Mean
n
X
X
n
i
i
 1
n/1
n21G )XXX(X  
Middle value
in the ordered
array
Most
frequently
observed
value
Rate of
change of
a variable
over time

33. Which measure to use?
Mean is generally considered the best measure of central tendency and
the most frequently used one. However, there are some situations where
the other measures of central tendency are preferred.
Median is preferred to mean when
 There are few extreme scores in the distribution.
 Some scores have undetermined values.
 There is an open ended distribution.
 Data are measured in an ordinal scale.
Mode is the preferred measure when data are measured in a nominal
scale.
A geometric mean is often used when comparing different items – finding
a single "figure of merit" for these items – when each item has multiple
properties that have different numeric ranges
33

34. Quiz Time
1) A teacher gives a 10 point quiz to a class of 9 students. All
the scores are whole numbers and nobody got a 0 or a
perfect core of 10. If the median is 7, what is the lowest
possible mean? What is the highest possible mean?
2) A poll reports that out of 100 families surveyed, the mean
number of children per family was 2.038, the median was 1.9,
and the mode was 1.82. Which of these values must be
wrong (independently)?
3) What is the mode here?
34

35. Solve this!
Wages 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total
Frequency 4 16 ? ? ? 6 4 230
35
Median = 33.5; Mode = 34. Calculate the missing frequencies.

36. Now solve this!
Variable 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Total
Frequency 12 30 ? 65 ? 25 18 229
36
Median = 46. Calculate the missing frequencies.