2.descriptive statistics-measures of central tendency

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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


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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


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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


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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.


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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


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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


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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 :


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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


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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.


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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.


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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


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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


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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


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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:


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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


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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


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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


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DR.A.O. ABIOLA

2.descriptive statistics-measures of central tendency

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