mct

1. Dr Sana Khader M
Junior Resident
Department of Community Medicine
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2.  Is a mathematical science pertaining to the collection,
presentation, summarizing, analysis, interpretation or
explanation of data.
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3. CENTRAL TENDENCY
 The tendency of observations to cluster round a value is
known as central tendency
 Measures of central tendency are also usually called as
averages.
 Explains the centrality of data as a single value.
 Give us an idea about values in central part of
distribution.
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4. 1. ARITHMETIC MEAN
2. MEDIAN Average of position
3. MODE
4. HARMONIC MEAN
5. GEOMETRIC MEAN
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Mathematical averages

5. MEASURES OF CENTRAL TENDENCY
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MEDIAN
MEAN
MODE
The middle
value of the data
The average of
the data
Most commonly
occurring value

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7. 1. MEAN
 Mean locate the center of distribution
 Most common measure
 It is simply the sum of values divided by the total number
of items
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8.  The arithmetic mean is a more technical name for what is
more commonly called the mean or average.
 The arithmetic mean is the value that is closest to all the
other values in a distribution.
 The central value of the series is obtained by dividing the
sum of all values by the number of observations, denoted as
𝑥 (𝑥 𝑏𝑎𝑟)
𝑥 =
𝛴𝑥𝑖
𝑛
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9. MEAN
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GROUPED DATA UNGROUPED DATA
𝑥 =
𝑖=1
𝑛
𝑥𝑖
𝑛
𝑥 =
𝑖=1
𝑛
𝑓𝑖𝑥𝑖
𝑛
MEAN =
SUM OF OBSERVATIONS
TOTAL NUMBER OF OBSERVATIONS

10.  Method for calculation:
 Step 1. Add all of the observed values in the distribution.
 Step 2. Divide the sum by the number of observations.
Example
 Find the mean of the following incubation periods for
hepatitis A: 27, 31, 15, 30, and 22 days.
 Answer
 = 125 / 5 = 25
 Therefore, the mean incubation period is 25 days.
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11.  Mean is the arithmetic centre of distribution
 Sometimes also called the centre of gravity of frequency
distribution
 The arithmetic mean is the best descriptive measure for data
that are normally distributed
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12. Example
Birth weights of new borns are :
3.3,4.1,5.8,3.8,2.7,4.1,3.4,3.9,5.1,3.
𝑥 =
𝑖=1
𝑛
𝑥𝑖
𝑛
=
39.2
10
=3.92
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13. Example
Number of teeth erupted at 2 years in 6 children.
3,5,8,9,7,4
𝑥 = 𝑖=1
𝑛
𝑥𝑖
𝑛
= 36/6
= 6
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14.  A survey of 100 families each having five children,revealed the
following distribution.Find the mean of male children.
mean = 𝑥 =
𝑓.𝑥
𝑛
= 200/100=2
N=100 𝑓. 𝑥 = 200
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No of male
children
0 1 2 3 4 5
No: of families 9 24 35 24 6 2
x f f.x
0 9 0
1 24 24
2 35 70
3 24 72
4 6 24
5 2 10

15. Exercise
Find the arithmetic mean of weight(kg) of students of a health
study group.
50,45,39,25,53,33,36,42,39,60
𝑥 =
𝑖=1
𝑛
𝑥𝑖
𝑛
=422/10=42.2 kg
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16. Exercise
Find mean days of confinement after delivery in the following
series.
18 137
𝑥 = 𝑖=1
𝑛
𝑓𝑖𝑥𝑖
𝑛
= 137/18 = 7.61
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Day of confinement No: of patients x*f
6 5 30
7 4 28
8 4 32
9 3 27
10 2 20

17.  Merits
 Easy to understand and calculate
 It is based upon all the observations
 Familiar to common man and rigidlt defined
 Capable of further mathematical treatment
 It is affected by sampling fluctuations,hence more stable.
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18.  Demerits
 It cannot be determined by inspection
 Cant be used for qualitative characters like caste,gender,religion
 It cant be obtained if a single observation is missing
 Affected by extreme values
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19. 2. MEDIAN
 It is the value of middle observation after placing the
observations in either ascending or descending order.
 Half the values lie above it and half below it.
 Median is also the 50th percentile of the distribution
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20.  If N or n is odd, the median is the middle number.
 If N or n is even, the median is the average of the two middle
numbers.
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21.  Step 1. Arrange the observations into increasing or decreasing
order.
 Step 2. Find the middle position of the distribution by using the
following formula: Middle position = (n + 1) / 2
 Step 3. Identify the value at the middle position.
 If the number of observations (n) is odd and the middle
position falls on a single observation, the median equals the
value of that observation.
 If the number of observations is even and the middle position
falls between two observations, the median equals the average
of the two values
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22. Example
To find the median of 4,5,7,2,1.
Count the total numbers given.(ODD)
Arrange them in ascending order.
1,2,4,5,7
The total elements in the distribution is odd(5)
The middle position can be calculated using (n+1)/2
= (5+1)/2 = 6/2 = 3
The number at 3rd position is 4.
So the median is 4.
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23. Example
To find the median of 5,7,2,1,6,4.
Count the total numbers given.
Arrange the numbers in ascending order.
1,2,4,5,6,7
The total number in distribution is 6.(EVEN)
So the average of two numbers which are respectively in
positions n/2 and (n/2)+1 will be the median.
median =
4+5
2
= 4.5
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24.  The median is a good descriptive measure, particularly for data
that are skewed, because it is the central point of the distribution.
 The median is relatively easy to identify.
 The median is not generally affected by one or two extreme
values .
 For example, if the values on the previous page had been 4, 23,
28, 31, and 131 (instead of 31), the median would still be 28.
 The median has less-than-ideal statistical properties. Therefore, it
is not often used in statistical manipulations and analyses.
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25. 3. MODE
 It is the most frequently occurring observation in the data
set.
 Mode of a group of observations, the value around which
the observations tend to be most heavily concentrated.
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26. Example
 The number of doses of diphtheria- pertussis-tetanus (DPT)
vaccine each of seventeen 2-year-old children in a particular
village received:
 0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4
 Two children received no doses;
 Two children received 1 dose;
 Three received 2 doses;
 Six received 3 doses; and
 Four received all 4 doses
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27.  Exercises
 Find the mode of the following incubation periods for
hepatitis A: 27, 31,31, 15, 30, and 22 days.
 Find the mode of the following incubation periods for
Bacillus cereus food poisoning:
2, 3, 3, 3, 3, 3,3, 4, 4, 5, 6, 7, 9, 10, 11, 11, 12, 12, 12,
12, 12, 14, 14, 15, 17, 18, 20, 21 hours
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28.  The mode is the easiest measure of central location to
understand and explain
 A distribution may have more than one mode if two or
more values tie as the most frequent values. It has no
mode if no value appears more than once.
 The mode is not typically affected by one or two
extreme values (outliers).
 The mode is used almost exclusively as a “descriptive”
measure. It is almost never used in statistical
manipulations or analyses
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29.  Two other measures of central tendency
 Geometric mean –of a set of the data comprising n
observations is the n th root of their product.
 Harmonic mean- is the reciprocal of the arithmetic mean of
the given observations.
 These two measures are not much used in biostatistical
work.
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30.  Mean is the measure of choice when data are normally
distributed,
 Median is the measure of choice for data that are not normally
distributed.
 Because epidemiologic data tend not to be normally distributed
(incubation periods, doses, ages of patients), the median is often
preferred.
 Geometric mean is used most commonly with laboratory data,
particularly dilution titers or assays and environmental sampling
data
 In social & psychological studies, qualitative data(SES,
intelligence)median or mode is the better measure.
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