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BASIC STATISTICS (2)DESCRIPTIVE STATISTICS -Measures of
Central TendencyLecture Delivered At The
INTENSIVE COURSE FOR PART 1 & 2 CANDIDATES
Organised By The
FACULTY OF INTERNAL MEDICINE
NATIONAL POSTGRADUATE MEDICAL COLLEGE OF
NIGERIA
20th 25th February 2012
ByDR. A.O. ABIOLA
Department of Community Health & Primary Care
College of Medicine, University of Lagos,
Idi - Araba, Lagos1
MEASURES OF CENTRAL TENDENCY
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vTwo features of the data which characterize a
distribution are measures of:
1.Location-central or non-central
2.Dispersion (Variation, Spread, Scatter)
vMeasures of location consist of:
Common measures of central tendency-
Arithmetic mean, median, mode
Other measures of central tendency-Weighted
arithmetic mean, Geometric mean, Harmonicmean
Other measures of location-Quartiles, Deciles,
Percentiles
INTRODUCTION
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vThe central tendency of a set of data is measured
by the average.
vThe word average implies a value in the
distribution, around which other values are
distributed.
vIt gives a mental picture of the central value.
vThere are several kinds of averages, of which thecommonly used are
The Arithmetic Mean,
The Median and
The Mode.
COMMON MEASURES OF CENTRAL
TENDENCY
MEASURES OF CENTRAL TENDENCY
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vThe arithmetic mean of a group is the simplearithmetic average of the observations.
vThis is calculated by dividing the total sum of all
the observations by the number of observations.
vIn the case of grouped data (frequency
distribution), arithmetic mean is calculated
assuming that each observation in a class interval
is equal to the midpoint of that class interval.
The Arithmetic Mean
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vIn an ungrouped data, if x represents the
character observed and n the number ofobservations, then all the observations in the data
can be denoted as x1, x2, .xn.
The arithmetic mean is given by
= x1 + x2 + xn = xin n
where denotes summation of values (i.e. xi = x1 +
x2 + xn)
vFor grouped data (frequency distribution) the
arithmetic mean is given by
= fx = fx
f n
where f is the frequency, x the midpoint of the class
interval and n the total number of observations.
The Arithmetic Mean
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Example 1: Calculate the arithmetic mean of thefollowing serum albumin levels (g%) of 4 pre-
school children:
2.90, 3.75, 3.66, 3.57
Solution:The arithmetic mean,
= xi = 2.90+3.75+3.66+3.57 = 13.88 = 3.47 g %
n 4 4
MEASURES OF CENTRAL TENDENCY
DR.A.O. ABIOLA
The Arithmetic Mean
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Protein intake/day (g) No. of families
15-24 325-34 4
35-44 10
45-54 11
55-64 8
65-74 3
75-84 1
Total 40
Example 2: Calculate the arithmetic mean of
protein intake of 40 families given below
MEASURES OF CENTRAL TENDENCY
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The Arithmetic Mean
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ClassInterval
(C.I)
Frequency(f)
Mid-pointof class
interval
(x)
fx
15-24 3 19.5 58.5
25-34 4 29.5 11835-44 10 39.5 395
45-54 11 49.5 544.5
55-64 8 59.5 476
65-74 3 69.5 208.5
75-84 1 79.5 79.5
Total 40 1880
Solution:
Arithmetic mean, m = fx = fx = 1880 = 47.0g
f n 40MEASURES OF CENTRAL TENDENCY
DR.A.O. ABIOLA
The Arithmetic Mean
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vThe arithmetic mean is sometimes simply calledthe mean or the average.
vThe advantages of the mean are that it is easy to
calculate and understand.
vThe disadvantages are that:
It may be unduly influenced by abnormal values inthe distribution.
Sometimes it may even look ridiculous; for
instance, the average number of children born to a
woman was found to be 3.72, which never occurs in
reality.
vNevertheless, the arithmetic mean is by far the
most useful of the statistical averages.
MEASURES OF CENTRAL TENDENCY
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The Arithmetic Mean
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vThe median is an average of different kind,
which does not depend upon the total sum and
number of items.
vTo obtain the median the data is first arranged in
ascending or descending order of magnitude, and
then the value of the middle observation is
located, which is called the median.vIf there are even numbers of values, the median
is worked out by taking the average of the two
middle values.
vThus, for
(i) n odd, median = middle value
(ii) n even, median = arithmetic mean of the
middle two values
The median
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Example 3: Find the median of the following 2, 1, 8, 7, 4,
Solution 3:
Array- 1, 2, 4, 7, 8
x(1) = 1,x(2) = 2,x(3) = 4,x(4) = 7,x(5) = 8
n = 5, odd
MEASURES OF CENTRAL TENDENCY
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The median
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Example 4: Find the median of following
2, 9, 1, 8,, 7, 4,Solution 4:
Array- 1,2,4,7,8,9
x(1) = 1,
x(2) = 2,
x(3) = 4,x(4) = 7,
x(5) = 8,
x(6) = 9,
n = 6, even
median = arithmetic mean of middle two values
MEASURES OF CENTRAL TENDENCY
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The median
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For a grouped data,
Where,
Li = True lower limit of median classr =No. of observations between the last
cumulative frequency before median class and
the median observation
f = No of observations (frequency) of the median
classUi = True upper limit of median class
The class interval that contains the median is
called the median class.MEASURES OF CENTRAL TENDENCY
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The median
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Age Group Frequency
10-19 5
20-29 19
30-39 1040-49 13
50-59 4
60-69 4
70-79 2
Example 5: Calculate the median of the data given
below :
MEASURES OF CENTRAL TENDENCY
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The median
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Age Group Frequenc
y
Cumulative frequency
10-19 5 5
20-29 19 24
30-39 10 34 ***********
40-49 13 47
50-59 4 51
60-69 4 55
70-79 2 57
Solution 5:
Median class (*******)= 30 39;
True limits of median class = 29.5 39.5; r = 2924 = 5; f = 10
MEASURES OF CENTRAL TENDENCY
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The median
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vThe relative merits of median and mean may beexamined from the following example:
vThe income of seven (7) people per day in naira
was :
5, 5,5,7,10,20,102,
Total = 154Mean = 154/7 = 22
Median = 7
vIn this example, the income of the seventh
individual (102) has seriously affected the mean,
whereas it has not affected the median.vIn an example of this kind median is more
nearer the truth and therefore more
representative than the mean.
MEASURES OF CENTRAL TENDENCY
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The median
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vThe Mode is the most frequent item or the
most fashionable value in a series of
observations.
vThe advantages of mode are that it is easy tounderstand and is not affected by the extreme
items.
vThe disadvantages are that the exact location
is often uncertain and is often not clearly
defined. Therefore, mode is not often used in
biological or medical statistics.
MEASURES OF CENTRAL TENDENCY
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The mode
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vThe distribution is unimodal if there is one
maximum (peak).
vIf we have a group of values such as 2,4,5,6,7, it
is apparent that there is no mode.
vFor a moderately asymmetric distribution, the
mode can be calculated using the following
empirical relationship:
Mode = 3 Median 2 Mean
MEASURES OF CENTRAL TENDENCY
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The mode
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For a grouped data,
WhereLm = True lower limit of modal class
d1 = Frequency of modal class minus frequency of
preceding class
d2 = Frequency of modal class minus frequency of
succeeding classUm = True upper limit of modal class
MEASURES OF CENTRAL TENDENCY
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The mode
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Example 6: Calculate the mode of the data given
below.
Age Group Frequency
10-19 5
20-29 1930-39 10
40-49 13
50-59 4
60-69 4
70-79 2
MEASURES OF CENTRAL TENDENCY
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The mode
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modal class(*****) = 20-29
Lm = 19.5; Um = 29.5;
d1= 195=14;
d2=19 10= 9
Age Group Frequency10-19 5
20-29 19****
30-39 10
40-49 13
50-59 460-69 4
70-79 2
Solution 6:
MEASURES OF CENTRAL TENDENCY
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The mode
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1.If the data are symmetrically distributed or are
approximately symmetrical, any one of these measures
may be used because in a symmetrical distribution all
these measures give identical values.
2.When the distribution of the observations is skewed,
the arithmetic mean is usually not suitable. Forpositively skewed series, the mean gives a higher value
than the other two measures; and for a negatively
skewed series, a lower value. It may be preferable to
use the median or the mode which is typical.
3.When there are some observations which relatively
deviate much more than others in the series or when
heterogeneity is suspected in the series, the median
may be used, instead of the mean.
SELECTION OF THE APPROPRIATE MEASURE
OF CENTRAL TENDENCY
MEASURES OF CENTRAL TENDENCY
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4. When subsequent computations involving a
measure are necessary, the arithmetic mean has
certain definite advantages.
5. When the concept of relative standing of the
individual observations in the group is considered,the use of the median is desirable; whereas the
concept of typical observation necessitates the
use of the mode.
Sometimes it may be advisable to use two or allthese measures, since each measure embodies a
different concept. The use of any two, mean and
median, or mean and mode will give us an idea of
the amount of skewness of the distribution of the
series.
SELECTION OF THE APPROPRIATE MEASURE
OF CENTRAL TENDENCY
MEASURES OF CENTRAL TENDENCY
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For a unimodal frequency distribution which is :
vSymmetric,
mean = median = mode
vModerately skewed,
mode = 3 median 2 mean
EMPIRICAL RELATION
BETWEEN MEAN, MEDIAN ANDMODE
MEASURES OF CENTRAL TENDENCY
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