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
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.
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.