Measures of Central Tendency Nursing Research nursing Statistics Descriptive Statistics

1. UNIT: IV
DESCRIPTIVE
STATISTICS
Mrs. D. Melba Sahaya Sweety RN,RM
PhD Nursing , MSc Nursing (Pediatric Nursing),
BSc Nursing
Associate Professor
Department of Pediatric Nursing
Enam Nursing College, Savar,
Bangladesh.
1

2. MEASURES OF CENTRAL
TENDENCY
2

3. INTRODUCTION
 Summarization of Data set in a single value is necessary. Such a value usually
somewhere in the center and represent the entire data set and hence it is called
measure of central tendency or averages.
 Since a measure of central tendency (i.e. an average) indicates the location or
the general position of the distribution on the X-axis therefore it is also
known as a measure of location or position.
 There are several statistical measures of central tendency or “averages”. They
are 1. Arithmetic Mean, 2. Geometric Mean, 3. Harmonic Mean,
4. Mode, 5. Median
3

4. CLASSIFICATION OF CENTRAL TENDENCY
4

5. ARITHMETIC MEAN
• Arithmetic mean is the most commonly used
measure of central tendency. “A value obtained by
dividing the sum of all the observations by the
number of observation is called arithmetic Mean”
and is usually denoted by
In general, if there are N observations as X1 + X2 +
X3 + X4+.........+ XN
5

6. ARITHMETIC MEAN
• In general, if there are N observations as X1 + X2 +
X3 + X4+.........+ XN then the Arithmetic Mean is
given by
Thus where X = sum of all observations
And n = total number of observations.
The calculation of arithmetic mean can be studied
under two broad categories:
1. Arithmetic Mean for Ungrouped Data.
2. Arithmetic Mean for Grouped Data. 6

7. ARITHMETIC MEAN
Arithmetic Mean for Ungrouped Data
Calculate Arithmetic Mean from discrete data
:(Direct Method)
• Example 1: Calculate Arithmetic Mean from the data
showing marks of students in a class in an economics test:
40, 50, 55,78, 58.
The average mark of students in the economics test is 56.2.
7

8. ARITHMETIC MEAN
• Example 2: The haemoglobin levels of the 10
women are given here, i.e. 12.5,
13,10,11,5,11,14, 9,7.5, 10, 12
• Example 3: The marks scored by the 10
students are given here, i.e. 75,58,
62,84,63,76,75,69,60,64 calculate the mean
8

9. Assumed Mean Method to Calculate Arithmetic Mean from
discrete data (Short cut Method) :
• In order to save time in calculating mean from a data set containing a large
number of observations as well as large numerical figures, you can use assumed
mean method.
Let, A = assumed mean , X = individual observations, N = total numbers of
observations d = deviation of assumed mean from individual observation, i.e. d
= X – A Then sum of all deviations is taken as Σd= Σ (X-A)
ARITHMETIC MEAN
9

10. ARITHMETIC MEAN
Exercise : The following data shows the weekly income of 10
families.
Family A B C D E F G H I J
Weekly income in taka 850 700 100 750 5000 80 420 2500 400 360
10

11. X d = X – A ( A = 700)
80 - 620
100 - 600
360 - 340
400 - 300
420 - 280
700 0
750 50
850 150
2500 1800
5000 4300
Total 4160
= 700 + 4160/10
= 700 + 416
= 1116
11

12. ARITHMETIC MEAN
Arithmetic Mean for Grouped Data
Calculate Arithmetic Mean from discrete data
with frequency :(Direct Method)
In discrete frequency table the mean is calculated
using the following formula
Where x = corresponding value variable, f = frequency
12

13. ARITHMETIC MEAN
Arithmetic Mean for Grouped Data
Calculate Arithmetic Mean from discrete data with
frequency :(Direct Method)
For an example the following data gives the age of 100
adolescent girls . Find the mean age
Age in
years (x)
No. of
students (f)
16 30
17 25
18 14
19 12
20 19 13

14. Age in
years
(x)
No. of
students
(f )
fx
16 30 480
17 25 425
18 14 252
19 12 228
20 19 380
Total Σf =
100
Σfx =
1765
= 1765
100
= 17.65
The mean age of 100
students is 17.65
14

15. ARITHMETIC MEAN
Sum No : 1 Calculate the mean protein level of 100 patients
Protein
level (x)
No of
patients(f)
7.5 7
12.5 13
15 20
17.5 10
20 35
22.5 15
15

16. ARITHMETIC MEAN
Arithmetic Mean for Grouped Data
Calculate Arithmetic Mean from Class interval with frequency :(Direct
Method)
Example, Find Mean from the table
Marks
Scored
0 - 5 5 - 15 15 - 30 30 - 40 40 - 60
No. of
Students
6 9 11 14 10
16

17. Marks
scored (x)
No. of
students
(f )
Mid x fx
0 - 5 6 2.5 15
5 - 15 9 10 90
15 - 30 11 22.5 247.5
30 - 40 14 35 490
40 - 60 10 50 500
Total Σf = 50 Σfx =
1342.5
= 1342.5
50
= 26.9
The average
marks scored
by the students
is 26.9
17

18. ARITHMETIC MEAN
Sum No : 1 Calculate the Mean from the following data
Class
interval (x)
Frequency(f)
0 - 10 3
10 - 20 14
20 - 30 6
30 - 40 7
40 - 50 5
50 - 60 10
18

19. Assumed Mean Method to Calculate Arithmetic Mean from
class interval with Frequency (Short cut Method) :
• In order to save time in calculating mean from a data set containing a large
number of observations as well as large numerical figures, you can use assumed
mean method.
Let, A = assumed mean , X = individual observations, Σf = total numbers of
observations d = deviation of assumed mean from individual observation, i.e. d
= X – A Then sum of all deviations is taken as Σfd= Σ (X-A)
ARITHMETIC MEAN
19

20. ARITHMETIC MEAN
Arithmetic Mean for Grouped Data
Calculate Arithmetic Mean from Class interval with frequency :(short cut
Method)
Example, Find Mean from the table
Marks
Scored
0 - 5 5 - 15 15 - 30 30 - 40 40 - 60
No. of
Students
6 9 11 14 10
20

21. Marks
scored
(x)
No. of
studen
ts (f )
Mid
x
d = (X – A)
A = 22.5
fd
0 - 5 6 2.5 -20 -120
5 - 15 9 10 - 12.5 -112.5
15 - 30 11 22.5 0 0
30 - 40 14 35 12.5 175
40 - 60 10 50 27.5 275
Total Σf = 50 Σfd =
217.5
217.5
= 22.5 + 50
= 22.5 + 4.35
The average
marks scored
by the
students is
26.9
21

22. ARITHMETIC MEAN
Arithmetic Mean for Grouped Data
Calculate Arithmetic Mean from Class interval with frequency :(step
deviation method)
Formula using to calculate step deviation method is
A = Assumed mean , h = class width
Example, Find Mean for the following frequency distribution
Age in years 50 - 55 45- 50 40 - 45 35 - 40 30 - 35 25 - 30
No . Of
Labors
28 29 31 47 51 70
22

23. Age of
the years
(x)
No. of
labors
(f )
Mid
x
d = (X– A)
h
A = 42.5
h = 5
fd
50 - 55 28 52.5 2 56
45 - 50 29 47.5 1 29
40 - 45 31 42.5 0 0
35 - 40 47 37.5 -1 - 47
30 - 35 51 32.5 -2 - 102
25 - 30 70 27.5 -3 - 210
Total Σf =
256
Σfd =
- 274
= 42.5 + 5 X -274
256
= 42.5 + 5 X (-1.07)
= 42.5 + (- 5.35)
= 37.15
The average of the
labor is 37.15
23

24. ARITHMETIC MEAN
Merits of Mean :
1) Arithmetic mean rigidly defined by Algebraic
Formula therefore the result will be same.
2) It is easy to calculate and simple to
understand.
3) It is based on all observations of the given
data.
4) It is capable of being treated mathematically
hence it is widely used in statistical analysis.
6) It is least affected by the fluctuation of
sampling.
7) For every kind of data mean can be
calculated.
Demerits of Arithmetic mean :
1) Arithmetic mean can not be computed for
qualitative data like data on intelligence honesty and
smoking habit etc.
2) It is too much affected by extreme observations
and hence it is not adequately represent data
consisting of some extreme point.
3) Arithmetic mean can not be computed when class
intervals have open ends.
4) If any one of the data is missing then mean can not
be calculated.
5) It cannot be located graphically.
24

25. GEOMETRIC MEAN
• The Geometric Mean (GM) is the average of a set of products, It is
technically defined as the nth root of the product of observation.
• In the arithmetic mean, data values are added and then divided by the
total number of values. But in geometric mean, the given data values are
multiplied, and then you take the root with the radical index for the final
product of data values.
• For example, if you have two data values, take the square root, or if you
have three data values, then take the cube root, or else if you have four
data values, then take the 4th root, and so on.
• GM = (X1 x X2 x X3 x X4 x.........x Xn )1/n
25

26. GEOMETRIC MEAN
Geometric Mean for Ungrouped Data
Calculate Geometric Mean from discrete data :
Example 1: Calculate Geometric Mean from the data
10, 5, 25,100,7,50
x Log x
10 1.0000
5 0.6989
25 1.3979
100 2.0000
7 0.8450
50 1.6989
Total 7.6407
GM = Antilog (7.6407/ 6)
= Antilog 1.2734
GM = 18.767
26

27. GEOMETRIC MEAN
Geometric Mean for Ungrouped Data
Calculate Geometric Mean from discrete data :
• Example 2: Calculate Geometric Mean from the data showing marks
of students in a class in an economics test: 40, 50, 55,78, 58.
27

28. GEOMETRIC MEAN
Geometric Mean for grouped Data
Calculate Geometric Mean from discrete data with
frequency :
Example 1: Calculate Geometric Mean from the data
x f
7 3
10 2
13 6
15 7
6 9
5 5
28

29. GEOMETRIC MEAN
x f Log x f log x
7 3 0.8450 2.5350
10 2 1.0000 2.0000
13 6 1.1139 6.6834
15 7 1.1760 8.232
6 9 0.7781 7.0029
5 5 0.6989 3.4945
Total Σf = 32 Σf log x =
29.9478
GM = Antilog (29.9478/ 32)
= Antilog (0.9358)
GM = 8.6258
29

30. GEOMETRIC MEAN
Calculate Geometric Mean from discrete data with
frequency :
Example 2: Calculate Geometric Mean from the data
Age in
years (x)
No. of
students (f)
16 30
17 25
18 14
19 12
20 19 30

31. GEOMETRIC MEAN
Geometric Mean for grouped Data
Calculate Geometric Mean from class interval with
frequency :
Example 1: Calculate Geometric Mean from the data
CI f
0 - 4 3
4 - 8 13
8 - 12 7
12 - 16 27
16 - 20 10
31

32. GEOMETRIC MEAN
CI Mid x f Log x f log x
0 - 4 2 3 0.3010 0.9030
4 - 8 6 13 0.7781 10.1153
8 - 12 10 7 1.0000 7.0000
12 - 16 14 27 1.1461 30.9447
16 - 20 18 10 1.2552 12.552
Total Σf = 60 Σf log x =
61.515
GM = Antilog (61.515/
60)
= Antilog (1.02525)
GM = 10.5986
32

33. GEOMETRIC MEAN
Calculate Geometric Mean from class interval with
frequency :
Example 2: Calculate Geometric Mean from the data
Marks
Scored
0 - 5 5 - 15 15 - 30 30 - 40 40 - 60
No. of
Students
6 9 11 14 10
33

34. GEOMETRIC MEAN
Merits of Geometric Mean :
 Geometric Mean is calculated based on all observations
in the series.
 Geometric Mean is clearly defined.
 Geometric Mean is not affected by extreme values in
the series.
 Geometric Mean is amenable to further algebraic
treatment.
 Geometric Mean is useful in averaging ratios and
percentages.
 Logarithm of GM for a set of observation, is the
Arithmetic mean of the logarithm of the observation.
Demerits of Geometric mean :
Geometric Mean is difficult to
understand.
We cannot compute geometric
mean if there are both positive
and negative values occur in the
series.
We cannot compute geometric
mean if one or more of the values
in the series is zero.
34

35. HARMONIC MEAN
• Harmonic Mean (HM) is defined as the reciprocal of Arithmetic mean of
the reciprocal of the observation. The harmonic mean is often used to
calculate the average of the ratios or rates of the given values. It is the most
appropriate measure for ratios and rates because it equalizes the weights of
each data point.
Note the following:
• Arithmetic mean is used when the data values have the same units.
• The geometric mean is used when the data set values have differing units.
• When the values are expressed in rates we use harmonic mean.
35

36. HARMONIC MEAN
Harmonic Mean for Ungrouped Data
Calculate Harmonic Mean from discrete data :
Example 1: Calculate Harmonic Mean from the data 2,5, 9,7,6
HM = 5
1 + 1 + 1 + 1 + 1
2 5 9 7 6
HM = 5
0.5 + 0.2+0.111 + 0.142 + 0.166
HM = 5 = 4.46
1.119 36

37. HARMONIC MEAN
Harmonic Mean for grouped Data
Calculate Harmonic Mean from discrete data with
frequency :
Example 1: Calculate Harmonic Mean from the data
x f
7 3
10 2
13 6
15 7
6 9
5 5
HM = 32
3 + 2 + 6 + 7 + 9 + 5
7 10 13 15 6 5
HM = 32
0.4285 + 0.2+0.4615 + 0.4666 + 1.5 + 1
HM = 32 = 7.88
4.0566 37

38. HARMONIC MEAN
Harmonic Mean for grouped Data
Calculate Harmonic Mean from class interval with frequency :
Example 1: Calculate Harmonic Mean from the data
CI f
0 - 4 3
4 - 8 13
8 - 12 7
12 - 16 27
16 - 20 10
HM = 60
3 + 13 + 7 + 27 + 10
2 6 10 14 18
HM = 60
1.5 + 2.166 +0.7 + 1.9285 + 0.5555
HM = 60
6.8506
HM = 8.75
38

39. HARMONIC MEAN
Merits of Harmonic Mean :
 It will always be the lowest as compared to the
geometric and arithmetic mean. HM will have the lowest
value, geometric mean will have the middle value and
arithmetic mean will have the highest value.
 The products of the harmonic mean (HM) and the
arithmetic mean (AM) will always be equal to the square
of the geometric mean (GM) of the given data set so
GM2 = HM × AM. Thus, HM = GM2 / AM
Demerits of Harmonic mean :
 It cannot be used on a data set
consisting of negative or zero
rates.
 The method to calculate the
harmonic mean can be lengthy and
complicated.
 The extreme values in a series
greatly affect the harmonic mean.
39

40. COMBINED MEAN
A combined mean is a mean of two or more separate groups.The formula
for calculating combined mean is
40

41. Example: 1, Calculate the combined
mean for M.sc first year and
second year students There are 13
M.Sc 2nd year and 12 M.sc 1st
year students and their mean are
17 and 24 respectively.
= 13 x 17 + 12 x 24
13 + 12
= 221 + 288
25
= 509
25
CM = 20.36
41

42. Example: 2, Calculate the combined mean for
two classes
.
CM = 5 x 22 + 7 x 4.85
5 + 7
= 110 + 33.95
12
= 143.95
12
CM = 11.99
Class
A
13 26 15 25 30
Class
B
7 6 4 4 3 9 1
.
= 13 + 26+ 15+ 25+ 30 = 22
5
= 7+6+4+4+3+9+1
7
= 34
7
= 4.85
42

43. MEDIAN
Median, in statistics, is the middle value of the given list
of data when arranged in an order.
To Calculate the Median: Arrange the n measurements in
ascending (or descending) order.
We denote the median of the data by M.
1. If n is odd, M is the middle number.
2. If n is even, M is the average of the two middle numbers.
• The calculation of median can be studied under two
broad categories:
• 1. Median for Ungrouped Data.
• 2. Median for Grouped Data.
43

44. MEDIAN
• 1. Median for Ungrouped Data.
• In this case, the data is arranged in either ascending or
descending order of magnitude.
I. If the number of observations n is an odd number, then the
median is represented by the numerical value of x, corresponds
to the positioning point of n+1 / 2 in ordered observations. That
is,
II. If the number of observations n is an even number, then the
median is defined as the arithmetic mean of the middle values
in the array That is,
44

45. 1. Median for Ungrouped Data.
Example : 1
The number of rooms in the seven girls hotel in Dhaka city is 71,
30, 61, 59, 31, 40 and 29. Find the median number of rooms
Solution:
Arrange the data in ascending order 29, 30, 31, 40, 59, 61, 71
n = 7 (odd)
Median = 7+1 / 2 = 4th positional value
Median = 40 rooms
MEDIAN
45

46. 1. Median for Ungrouped Data.
Example : 2
The export of agricultural product in million dollars from a
country during eight quarters in 1974 and 1975 was recorded as
29.7, 16.6, 2.3, 14.1, 36.6, 18.7, 3.5, 21.3
Find the median of the given set of values
Solution:
We arrange the data in descending order
36.6, 29.7, 21.3, 18.7, 16.6, 14.1, 3.5, 2.3
MEDIAN
46

47. (8)th term + (8 + 1)th term
2 2
2
MEDIAN
M =
47

48. 1. Median for grouped Data with frequency.
In case of Discrete grouped data, first we find the
cumulative frequency and then use the following
formula for median.
MEDIAN
48

49. 1. Median for grouped Data with frequency.
Example : 3 Calculate the median for the following frequency
with values
MEDIAN
x 10 20 30 40 50 60 70
f 3 6 5 9 7 10 20
X f Lcf
10 3 3
20 6 9
30 5 14
40 9 23
50 7 30
60 10 40
70 20 60
N = 60
M = 60 + 1 th term
2
= 30.5 th term
M = 60
49

50. 1. Median for grouped Data with frequency.
Example : 5 Calculate the median for the following frequency with values
MEDIAN
x 10 20 30 40 50 60 70 80
f 3 6 5 9 7 10 20 24
X f Lcf
10 3 3
20 6 9
30 5 14
40 9 23
50 7 30
60 10 40
70 20 60
80 24 84
N = 84
M = 84 + 1 th
2 term
= (85/2)th term
= (42.5) th term
M = 70 50

51. 1. Median for grouped Data with frequency.
Example : 4 Calculate the median for the following frequency
with values
MEDIAN
x f
7 3
10 2
13 6
15 7
6 9
5 5
51

52. MEDIAN
1. Median for grouped Data class interval
with frequency.
The formula for computing median is
Where
l = Lower class interval of
the median class
N = sum of frequency
m = cumulative frequency
of the class preceding
the median class
c = width of the median
class
52

53. 1. Median for grouped Data with frequency.
Example : 6 Calculate the median for the following frequency
with values
MEDIAN
x f
0 - 10 3
10 - 20 5
20 - 30 4
30 - 40 8
40 - 50 2
Where
N/2 = 22/2 = 11
l = 20
m = 8
c = 10
M = 20 + (11 – 8)/4 x 10
M = 20 + (3/4) x 10
M = 20 + 30/4
M = 20+ 7.5 = 27.5
x f Lcf
0 - 10 3 3
10 - 20 5 8
20 - 30 4 12
30 - 40 8 20
40 - 50 2 22
N = 22
53

54. 1. Median for grouped Data with frequency.
Example : 7 Calculate the median for the following frequency
with values
MEDIAN
x f
0 - 100 5
100 - 200 17
200 - 250 33
250 - 300 40
300 - 400 5
54

55. MEDIAN
Merits of Median :
 Median can be calculated in all distributions.
 Median can be understood even by common
people.
 Median can be ascertained even with the extreme
items.
 It can be located graphically
 It is most useful dealing with qualitative data
Demerits of Median:
 It is not based on all the values.
 It is not capable of further
mathematical treatment.
 It is affected fluctuation of
sampling.
 In case of even no. of values it
may not the value from the
data.
55

56. MODE
Mode is the most frequent value or score in the distribution. Or A
mode is defined as the value that has a higher frequency in a given
set of values. It is the value that appears the most number of times.
It is denoted by the capital letter Z.
• When there are two modes in a data set, then the set is
called bimodal
• For example, The mode of Set A = {2,2,2,3,4,4,5,5,5} is 2 and 5,
because both 2 and 5 is repeated three times in the given set.
• When there are three modes in a data set, then the set is
called trimodal
• For example, the mode of set A = {2,2,2,3,4,4,5,5,5,7,8,8,8} is 2, 5
and 8
• When there are four or more modes in a data set, then the set is
called multimodal
56

57. 1. Mode for Ungrouped Data.
Example: {19, 8, 29, 35, 19, 28, 15}
• Arrange them in order: {8, 15, 19, 19, 28, 29,
35}
19 appears twice, all the rest appear only once,
so 19 is the mode.
MODE
57

58. 2. Mode for grouped Data ( Discreate value with Frequency).
Example : 1 Example : 2
MODE
x f
2 3
7 8
10 3
19 9
25 2
32 5
Mode = 19
x f
5 4
10 8
15 5
20 8
25 4
30 2
Mode = 10 and 20 58

59. 3.Mode for grouped Data ( class interval with Frequency).
Formula to calculate mode for grouped data is
MODE
• Where,
• l = lower limit of the modal class
• h = size of the class interval
• f1 = frequency of the modal class
• f0 = frequency of the class preceding the modal class
• f2 = frequency of the class succeeding the modal class
59

60. 3.Mode for grouped Data ( class interval with Frequency).
Example : 1
MODE
x f
20 -25 7
25 - 30 3
30 - 35 11
35 - 40 5
40 - 45 2
45 - 50 9
l = 30
f1 = 11
f2 = 5
f0 = 3, h = 5
Mode = 30 + (11 – 3) x 5
(2 x 11 – 3 – 5)
Mode = 30 + 8 x 5 = 30 + 40
22 -8 14
Mode = 30 + 2.857 = 32.857 60

61. 3.Mode for grouped Data ( class interval with Frequency).
Example : 2
MODE
x f
20 - 39 6
40 - 59 4
60 - 79 3
80 - 99 7
100 - 119 10
120 - 139 5
l = 99.5
f1 = 10
f2 = 5
f0 = 7, h = 20
Mode = 99.5 + (10 – 7) x 20
(2 x 10 – 7 – 5)
Mode = 99.5 + 3 x 20 = 99.5 + 60
20 -12 8
Mode = 99.5 + 7.5 = 107
x f
19.5 – 39.5 6
39.5 – 59.5 4
59.5 – 79.5 3
79.5 – 99.5 7
99.5 – 119.5 10
119.5 – 139.5 5
61

62. MODE
Merits of Mode :
 Mode is readily comprehensible and easily
calculated
 It is the best representative of data
 It is not at all affected by extreme value.
 The value of mode can also be determined
graphically.
 It is usually an actual value of an important
part of the series.
Demerits of Mode :
 It is not based on all
observations.
 It is not capable of further
mathematical manipulation.
 Mode is affected to a great
extent by sampling fluctuations.
 Choice of grouping has great
influence on the value of mode.
62

63. FORMULA TO CALCULATE CENTRAL TENDENCY
Methods Ungrouped Data Grouped Data ( Discrete
data with Frequency)
Grouped Data (class
interval with frequency)
Short cut method
Arithmetic
Mean
Geometric
Mean
Harmonic
Mean
Median
Mode
63

64. 64