Biostatistics Measures of central tendency

1. Measures of central tendency
MEAN, MODE, MEDIAN
Dr. Aswartha Harinatha Reddy
Department of Biotechnology

2. • Some cases the data condensed to a single value, such single
value is known as Central value.
• The central value of the series is also known as central
tendency.
• The measures devised to calculate the Central tendency are
known as Measure of central tendency.

3. Types of measure of central tendency:
• There are three basic measure of central tendency
1. Mean or Mathematical Average
2. Median
3. Mode

4. Mean or Arithmetic Mean:
• The arithmetic mean of a variable is often denoted by a X bar, X
̅
.
• Arithmetic mean of a data is the common average obtained by
dividing the Sum of values of the series by the total number of
items of that series.
• Mean = Sum of observations or values/ Total no of observations or Values
(X
̅ )= ∑X/n

5. For example, let us consider the monthly salary of 10 employees
of a firm:
Calculate mean for following data:
250, 270, 240, 230, 255, 265, 275, 245, 260, 240.
Mean = Sum of observations or values/ Total no of observations
(X
̅ )= ∑X/N
• Mean= 250+270+240+230+255+265+275+245+260+240
10
Mean = 253

6. Ungrouped data:
• The oxygen concentration in four cases was recorded to be:
ABCD: A. 14.9% B. 10.8% C. 12.3% D. 23.3%
• Mean: ?
• 15.325

7. • Discrete series: means where frequencies of a variable are
given but the variable is without class intervals.
• Continuous series: means where frequencies of a variable are
given but the variable is with class intervals.

8. Arithmetic mean of grouped data (Discrete series):
• Discrete series means where frequencies of a variable are
given but the variable is without class intervals.
• Arithmetic mean of grouped data (Discrete series) calculated
by following formula.
Mean (X
̅ )= ∑fx / ∑f
f: frequency
x: is Variable

9. Find the mean form the following data:
Marks (X) 5 10 6 20 ∑x = 41
No of students (f) 10 7 8 6 ∑f = 31
fx 50 70 48 120 ∑fx = 288
Mean (X
̅ )= ∑fx / ∑f
Mean (x)= 288/31
=9.290

10. Calculate Arithmetic mean of Discrete series:
People (x) 10 20 30 40 ∑x = 100
H1N1 (f) 2 2 3 3 ∑f = 10
Mean (X
̅ )= ∑fx / ∑f
=240/10
= 24
20 40 60 120 ∑fx= 240

11. Grouped Data (Continuous series):
• Continuous series means where frequencies of a variable are
given but the variable is with class intervals.
• Mean (x
̅ )= ∑f.m / ∑f
• m= the mid value of various classes.
• f= Total frequency
• ∑f.m= the sum of mid values multiplied by their frequencies.

12. Grouped Data (Continuous data):
Data which consists of the survey done on deaths due to HIV
infection in a community.
Calculate Mean for following continuous data:
HIV Patients Age 20-30 30-40 40-50 50-60 ∑x = ?
No of Death cases 20 25 30 24 ∑f = 99
Mean (x
̅ )= ∑f.m / ∑f
=∑f.m?

13. HIV Patients Age 20-30 30-40 40-50 50-60 ∑x = ?
Mid value (m) 25 35 45 55
No of Death cases (f) 20 25 30 24 ∑f = 99
fm 500 875 1350 1320 ∑f.m = 4045
Mean (x
̅ )= ∑f.m / ∑f
=4045/99
=40.85

14. Merits of Arithmetic mean:
• Arithmetic mean is easy to calculate and simple to understand.
• Arithmetic mean is a relatively stable measure, it is least affected by
fluctuations of sampling.
• Arithmetic mean is based on all the observations of a series.
Therefore it is the most representative measure.
• Arithmetic mean is the best measure for comparing two or more
series of data.
• Arithmetic mean formula is rigid one, therefore the result remains the
same.

15. Demerits of Arithmetic Mean:
• Problem in case of incomplete data: Arithmetic mean cannot
be calculated unless all the items of the series are known.
• Mean value may not figure in the series: Arithmetic mean
value sometimes does not appear in the series.
• For example: the arithmetic mean of 4,8,15, 21 is 12 but it is
not present in the series.

16. • Unreasonable results: Arithmetic average sometimes gives
unreasonable or unacceptable results.
• For example:
• The average number of children per family comes out to be
2,3,4,3,and 6.
• Mean= 18/5 = 3.6 children.
• The result is unreasonable because the children cannot be
divided into fractions.

17. Median
• If the values of a variable are arranged in ascending or
descending order, the median value that divides the whole data
into two equal parts.
• One part having all values smaller than the median value and
other part having all the values greater than the median value.
• The mean value of two middle observations.

18. Median for Ungrouped data
• To calculate the median of ungrouped data, the values of data
are arranged in the order of ascending or descending order.
• The middle most value represent the median (μ or mu).
100, 97, 110, 200, 75, 120,150
Ascending order is:
75,97,100,110,120,150,200
Median is : 110

19. Median formula for ungrouped data:
• Median = Number of observations+1 = N+1
2 2
• 100, 97, 110, 200, 75, 120,150 (Number of observations (N) is ODD)
Ascending order is:
75,97,100,110,120,150,200
• Median = 7+1/2 = 8/2= 4
• Median = 4rth position

20. Calculate median when number of observations (N) is EVEN:
• For example:
75,97,100,120,150,175
3rd observation is = 100
4th observation is = 120
Median = 100+120 = 110
2

21. Calculate median for grouped data (Discrete series):
Discrete series means where frequencies of a variable are given
but the variable is without class intervals.)
Median (μ)= N+1 = Where N = is the Total frequency (∑f) of Data
2
Variable (X) Frequency (f)
2 4
6 10
8 8
9 20
10 8
∑f=50
Median = N+1 = 50+1 = 25.5
2 2

22. Calculate median for following data?
Age 20 30 40 50 60
No of Patients 6 5 20 10 45

23. Calculate median for grouped data (Continuous series):
• Median for continuous series is :
• Where, L1 is the lower limit of that class interval where median
falls,
• ∑f is the total frequency,
• F : Cumulative frequency just above that class interval where
median falls.
• fm is the frequency of that class interval where median falls.
• i is the class width of the class interval.

24. Example: Grouped data with continuous series:
Class interval (N) Frequency (f) Cumulative
Frequency (F)
Class
width (i)
5-10 2 0+2= 2 5
10-15 11 2+11= 13 5
15-20 26 13+26=39 5
20-25 17 39+17=56 5
25-30 8 56+8=64 5
30-35 6 64+6=70 5
35-40 4 70+4= 74 5
∑f: 74 ∑F= 74 i=5
Median= ∑f/2= 74/2= 37
L1: 15,
F: 13,
fm: 26
i: 5
Median; 49.5

25. Example 2: Calculate Median for following data?
Age 10-20 20-30 30-40 40-50 50-60 60-70
HIV patients 12 22 14 50 45 4
L1 : is the lower limit of that class interval where median falls,
∑f : is the total frequency,
F : Cumulative frequency just above that class interval where
median falls.
fm: is the frequency of that class interval where median falls.
i : is the class width of the class interval.

26. Calculate median for following H1N1 patients?
Age 20-25 25-30 30-35 35-40 40-45 45-50
H1N1 50 60 70 50 60 80

27. Merits of Median
 Median is easy to understand and calculate.
 Median is not affected by extreme observations.
 Median best for qualitative data.
 Median can be computed while dealing with a distribution with open
and end class.
Demerits of Median:
 Median cannot be determined in the case of even number of
observations.
 Median is relatively less stable than mean, particularly for small
samples.
 Median is a positional average. It cannot be accepted for each and
every observations.

28. MODE:
• Mode (Mo) is the most frequently occurring value in a data.
• For a given data, mode may exist or may not exist.
• 10,10,9,8,5,4,12,10 : One mode i.e 10.
• 10,10,2,4,6,8,9,9: Two mode i.e 10 and 9.
• 3,2,1,6,5,4,9,8,7: No mode

29. Mode of Individual series or ungrouped data:
Variable X 45 99 45 22 56 26
Step 1: Arrange the data in increasing order i.e:
Variable X 22 26 45 45 56 99
Stpep:2 Value 45 of variable X in this series has occurred twice
while other values are represented just once, therefore mode of
this data is :45.

30. Calculate mode for following data:
Variable X 33 45 33 25 65 89
Variable X 20 23 20 45 23 89
Mode: ?
Mode: ?

31. Mode for Continuous series:
Age 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60
HIV
Patients
5 7 8 18 25 12 7 5
MODAL CLASS: The class having greatest frequency is called Modal class.

32. Mode for Continuous series:
Age
(Intervals)
20-25 25-30 30-35 35-40 40-45 45-50 550-55 55-60
HIV
Patients (f)
5 7 8 18 25 12 7 5
L1: Lower limit of modal class interval: 40
fm: Frequency of modal class or Maximum frequency: 25
f1: Frequency of class just below the modal class: 18
f2: Frequency of class just after the modal class: 12
C: Class interval or class width : 5
Modal class: 40 -45, Mode (Z): 40.78.

33. Example 1: Calculate mode for following data:
Age 20-25 25-30 30-35 35-40 40-45 45-50 550-55 55-60
HBV 8 16 12 50 8 2 10 20
L1: Lower limit of modal class interval:
fm: Frequency of modal class or Maximum frequency:
f1: Frequency of class just below the modal class:
f2: Frequency of class just after the modal class:
C: Class interval or class width :

34. Example 2: Calculate mode for following data:
Age 20-25 25-30 30-35 35-40 40-45 45-50 550-55 55-60
H1N1 16 12 88 55 12 100 18 23
L1: Lower limit of modal class interval:
fm: Frequency of modal class or Maximum frequency:
f1: Frequency of class just below the modal class:
f2: Frequency of class just after the modal class:
C: Class interval or class width :

35. Merits of Mode:
• Mode is easy to calculate and understand.
• It is not affected by extreme observations.
• Mode can be calculated from a grouped frequency distribution with
open end class.
Demerits mode:
• Mode is not defined, if the maximum frequency is repeated more
than one time.
• As compared to mean, mode is affected to a great extent by the
fluctuating of sampling.
• It is not suitable for algebraic treatment.
Example for algebraic methods : (2y+1 ), log 12 (x+5).

36. Types of Mean:
1. Arithmetic mean: is the obtained by dividing the sum of all
observations of the series by the total number of items of that
series. (X
̅ )= ∑X/n.
2. Geometric mean: The geometric mean of a set of data for n
observations is the nth root of their product.
If x1, x2, ..., xn, are the sets of N observations, than
geometric mean is:
GM:
Example:4,8,2,4
𝑛
𝑥1×𝑥2×𝑥3 … . . 𝑥𝑛
4
4×8×2×4 =
4
28 = 28/4 = 4

37. Exercise: 1
The median of the observations is 4,5,6,12, (x+3),(x+2),10,20,25,30
Above data arranged in ascending order is 20.
Find X? and mean for above series.
Median:
𝑋+3+𝑋+2
2
=20
2x+5=40
2x=35
X=17.5
To calculate mean by X value substitute in the above data
4,5,6,12,(x+3),(x+2),10,20,25,30
4,5,6,12,(17.5+3),(17.5+2),10,20,25,30
4,5,6,12, 20.5,19.5, 10,20,25,30
Mean= 4+5+6+12+20.5+19.5+10+20+25+30
10
Mean= 152/10 Mean=15.2

38. Exercise: 2
The median of the observations is 2,3,6, (y+4),(y+5),11,21,25
Above data arranged in ascending order is 10.
Find y? and mean for above series.

39. THANK YOU