SlideShare a Scribd company logo
Measures of
Central Tendency
Harshit Jadav
M.S.Pharm, PGDRA
• Measures of central tendency are also usually
called as the averages.
• They give us an idea about the concentration
of the values in the central part of the
distribution.
• The following are the five measures of
central tendency that are in common use:
• (i) Arithmetic mean, (ii) Median, (iii) Mode,
(iv) Geometric mean, and (v) Harmonic mean
(vi) weighted mean
MEASURE OF CENTRAL TENDANCY
MEAN
MEDIAN MODE
The average of the data
The middle value of the
data
most commonly
occurring value
Mean (Average)
Mean locate the centre of distribution.
Also known as arithmetic mean
Most Common Measure
The mean is simply the sum of the values divided by
the total number of items in the set.
Affected by Extreme Values
XX
XX
nn
XX XX XX
nn
ii
ii
nn
nn
== ==
++ ++ ++==
∑∑
11 11 22 
Merits:
• It is easy to understand and easy to calculate
• It is based upon all the observations
• It is familiar to common man and rigidly
defined
• It is capable of further mathematical
treatment.
• It is affected by sampling fluctuations. Hence
it is more stable.
Demerits
• It cannot be determined by inspection.
• Arithmetic mean cannot be used if we are
dealing with qualitative characteristics,
which cannot be measured quantitatively
like caste, religion, sex.
• Arithmetic mean cannot be obtained if a
single observation is missing or lost
• Arithmetic mean is very much affected by
extreme values.
MEAN
UNGROUPED DATA GROUPED DATA
n
x
n
i
∑=
= 1
ix
n
x
n
i
∑=
= 1
ii xf
MEAN =
nsobservatioofno.total
nsobservatioofSum
Birth weight of new borns are :
3.3, 6.1, 5.8, 3.8, 2.7, 4.1, 3.4, 3.9, 5.1, 3
n
x
n
i
∑=
= 1
ix =41.2/10
=4.12 kg
Q. A Survey of 100 families each having five children,
revealed the following distribution
No. of male children 0, 1, 2, 3, 4, 5
No. of Families 9, 24, 35, 24, 6, 2
Find the Mean of male children.
x f f . X
0 9 0
1 24 24
2 35 70
3 24 72
4 6 24
5 2 10
N=100 Σ f.x = 200
Mean = X =Σ f.x / Σf
X =200/ 100 = 2
Find mean days of confinement after
delivery in the following series:-
Day of
confinement
No. of
patients
6 5
7 4
8 4
9 3
10 2
Day of
confinement
(x)
No. of
patients
(f)
X*f
6 5 30
7 4 28
8 4 32
9 3 27
10 2 20
Total 18 137
Solution:-
n
x
n
i
∑=
= 1
ii xf =137/18
=7.61
Median
1.Measure of Central Tendency.
2. The median is determined by sorting the data set
from lowest to highest values and taking the data
point in the middle of the sequence.
3.Middle Value In Ordered Sequence
• If Odd n, Middle Value of Sequence
• If Even n, Average of 2 Middle Value
4.Not Affected by Extreme Values
Merits:
• It is rigidly defined
• It is easy to understand and easy to
calculate.
• It is not at all affected by extreme values.
• It can be calculated for distributions with
open-end classes.
• Median is the only average to be used
while dealing with qualitative data.
• Can be determined graphically.
Demerits:
• In case of even number of observations
median cannot be determined exactly.
• It is not based on all the observations.
• It is not capable of further mathematical
treatment
For ungrouped data:-
Step-1
Arranged data in ascending or descending order.
Step:-2
If total no. of observations ‘n’ is odd then used the
following formula for median
Step:-3
If total no. of observations ‘n’ is even then used
the following formula for
median = arithmetic mean of two middle
observations.
.
2
1
nobservatioth
n +
=
Median
If X1, X2, X3,... …. ,Xn are n observations
arranged in ascending order of magnitude.
X(n+1)/2 if n is odd
Median = {
Xn/2 + X(n/2)+1 if n is even
------------------
2
Calculate the median for the following
series :-
2,3,5,1,4,5,8
1,2,3,4,5,5,8.
Median .
2
1
nobservatioth
n +
=
= 7+1/2
=4th
number
74+75
Median = ---------- = 74.5
2
The data on pulse rate per minute of 10 healthy
individuals are 82, 79, 60, 76, 63,81, 68, 74, 60, 75.
n= 10
60, 60, 63, 68, 74,75, 76, 79, 81, 82
Xn/2 + X(n/2)+1 / 2
Find out the median for number of
sports injuries happened in cricket in all teams
37, 57, 65, 46, 12, 14, 19, 23, 56, 78, 5, 33
Median:-
f
hc
n
l






−
+
2
l = lower limit of class interval where the
median occurs
f = Frequency of the class where median occurs
h = Width of the median class
C= Cumulative frequency of the class
preceding the median class (PCF)
For Grouped data:-
Weight of infant in kg No of infants
2.0-2.5 37
2.5-3.0 117
3.0-3.5 207
3.5-4.0 155
4.0-4.5 48
4.5 and above 26
Find the median weight of 590 infants born in a
hospital in one year from the following table.
Weight of infants in
kg
No of infants Cumulative frequency
2.0-2.5 37 37
2.5-3.0 117 37+117=154
3.0-3.5 207 154+207=361
3.5-4.0 155 361+155=516
4.0-4.5 48 516+48=564
4.5 and above 26 564+26=590
N/2 =590/2 =295
Median Class = 3.0-3.5, so L=3.0, f= 207
Cf = 154 h=0.5
f
hc
n
l






−
+
2
= 3.0 + (295-154) * 0.5
207
= 3.0 + 0.34
= 3.34 Median Weight
For grouped Data:-
Class interval Frequency
5-9 2
10-14 11
15-19 26
20-24 17
25-29 8
30-34 6
35-39 3
40-44 2
45-49 1
Calculate the median for the following data
series:-
Solution:-
Class interval
Frequency
cumulative
frequency
5-9 2 2
10-14 11 13
15-19 26 39
20-24 17 56
25-29 8 64
30-34 6 70
35-39 3 73
40-44 2 75
45-49 1 76
n=76
l = lower limit of class interval where the median
occurs
= 15
h = Width of the median class
= 4
f = Frequency of the class where median occurs
= 26
C = Cumulative frequency of the class
preceding the median class
=13
f
hc
n
l






−
+
2
Mode
1.Measure of Central Tendency
2.The mode is the most frequently
occurring value in the data set.
3.May Be No Mode or Several Modes
Merits:
• Mode is readily comprehensible and easy to
calculate.
• Mode is not at all affected by extreme values.
• Mode can be conveniently located even if the
frequency distribution has class intervals of
unequal magnitude
• Open-end classes also do not pose any
problem in the location of mode.
• Mode is the average to be used to find the
ideal size.
Demerits:
• Mode is ill defined.
• It is not based upon all the observations.
• It is not capable of further mathematical
treatment.
• As compared with mean, mode is affected
to a great extent by fluctuations of
sampling.
Mode Example
No Mode
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
One Mode
Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9
More Than 1 Mode
Raw Data: 21 28 28 41 43 43
Mode for ungrouped data:-
2,2,3,4,6,7,4,4,4,4,8,9,0 mode is 4
10,10,3,3,4,2,1,6,7 mode is 10 and 3
10,34,23,12,11,3,4 no mode
Mode for Group Data
Fm-F1
Mode = L1 + ------------------ * c
2Fm– F1- F2
Where
•Fm is mode freq.
• F1 is freq. just lower mode class.
•F2 is freq. after mode class.
• L1 is lower limit of mode class
•C is class difference.
C I FREQU. (F)
20 - 30 3
30 - 40 20
40 - 50 27
50 - 60 15
60 - 70 9
Q. Find the Mode for group data
Age group 20-30 30-40 40-50 50-60 60-70
No. of persons 3 20 27 15 9
Q. Find the Mode for group data
Age group 20-30 30-40 40-50 50-60 60-70
No. of persons 3 20 27 15 9
C I FREQU. (F)
20 - 30 3
30 - 40 20
40 - 50 27
50 - 60 15
60 - 70 9
L1
Fm
F1
F2
FFmm -- FF11
Mode = LMode = L11 + ------------------ * c+ ------------------ * c
2F2Fmm – F– F11- F- F22
2727 –– 20 7020 70
Mode = 40 + ------------------ * 10 = 40 + -----Mode = 40 + ------------------ * 10 = 40 + -----
2*272*27 – 20- 15 19– 20- 15 19
Mode = 43.68Mode = 43.68
Calculate the mode for the following
frequency distribution:-
IQ Range Frequency
90-100 11
100-110 27
110-120 36
120-130 38
130-140 43
140-150 28
150-160 16
160-170 1
Modal class by inspection is
130-140
fm= 43
f1= 38
f2= 28
C=10
l = 130
c
fff
ff
l
m
m






−−
−
+
)(2
)(
21
1
=130.6579
Relationship between
Mean, Median and Mode
Mode = 3 Median – 2 Mean
Summary of
Central Tendency Measures
MeasureMeasure DescriptionDescription
MeanMean Balance PointBalance Point
MedianMedian Middle ValueMiddle Value
When OrderedWhen Ordered
ModeMode Most FrequentMost Frequent
Ex. Calculate Mean, Median, Mode.
Age Group
No. of
Patients
25-30 4
30-35 3
35-40 2
40-45 3
45-50 4
50-55 8
55-60 6
Age Group No. of Patients (F) X F*X C.F
25-30 4 27.5 110 4
30-35 3 32.5 97.5 7
35-40 2 37.5 75 9
40-45 3 42.5 127.5 12
45-50 4 47.5 190 16
50-55 8 52.5 420 24
55-60 6 57.5 345 30
30 1365
MEAN =1365/30 = 45.5
MEDIAN =45+ (15-12)*5/4 = 48.75
MODE=50 + [(8-4)*5/(2*8-4-6)]=53.34
Ex. The following table gives the frequency
distribution of marks obtained by 2300 medical
students of Gujarat in MCQ of PSM exam. Find
Mean, Median and Mode.
Marks
No. of
students
11-20 141
21-30 221
31-40 439
41-50 529
51-60 495
61-70 322
71-80 153
Geometric Mean
• Geometric mean is defined as the positive root of the
product of observations. Symbolically,
• It is also often used for a set of numbers whose values are
meant to be multiplied together or are exponential in nature,
such as data on the growth of the human population or
interest rates of a financial investment.
• Find geometric mean of rate of growth: 34, 27, 45, 55, 22, 34
n
nxxxxG /1
321 )( =
Geometric mean of Group
data
• If the “n” non-zero and positive variate-values occur
times, respectively, then the geometric mean of the
set of observations is defined by:
[ ] Nn
i
f
i
Nf
n
ff in
xxxxG
1
1
1
21
21






== ∏=

∑=
=
n
i
ifN
1
Where
nxxx ,........,, 21 nfff ,.......,, 21
Geometric Mean (Revised Eqn.)
)( 321 nxxxxG =








= ∑=
n
i
ixLog
N
AntiLogG
1
1








= ∑=
n
i
ii xLogf
N
AntiLogG
1
1
)( 321
321 n
fff
xxxxG =
Ungroup Data Group Data
Measures of Central Tendency - Biostatstics
What is the geometric mean of 4,8.3,9
and 17?
First, multiply the numbers together
and then take the 5th root (because
there are 5 numbers)
G = (4*8*3*9*17)(1/5)
G = 6.81
Harmonic Mean
• Harmonic mean (formerly sometimes called the
subcontrary mean) is one of several kinds of
average.
• The harmonic mean is a very specific type of
average.
• It’s generally used when dealing with averages of
units, like speed or other rates and ratios.
Harmonic Mean Group Data
• The harmonic mean H of the positive real numbers x1,x2, ..., xn
is defined to be
∑=
= n
i i
i
x
f
n
H
1
∑=
= n
i ix
n
H
1
1
Ungroup Data Group Data
What is the harmonic mean of 1,5,8,10?
Here,
N=4
H = 4 / (1/1) +(1/5) + (1/8) + (1/10)
H = 4/ 1.425
H = 2.80
∑=
= n
i ix
n
H
1
1
Rahul drives a car at 20 mph for the
first hour and 30 mph for the second.
What’s his average speed?
We need the harmonic
mean:
= 2/(1/20 + 1/30)
= 2(0.05 + 0.033)
= 2 / 0.083
= 24.09624 mph.
Weighted Mean
• The Weighted mean of the positive real numbers
x1,x2, ..., xn with their weight w1,w2, ..., wn is defined
to be
∑
∑
=
=
= n
i
i
n
i
ii
w
xw
x
1
1
Σ = the sum of (in other words…add them up!).
w = the weights.
x = the value.
A weighted mean is a kind of average.
Instead of each data point contributing
equally to the final mean, some data points
contribute more “weight” than others.
If all the weights are equal, then the
weighted mean equals the arithmetic mean
(the regular “average” you’re used to).
Weighted means are very common in
statistics, especially when studying
populations.
Steps:
1.Multiply the numbers in your data set
by the weights.
2.Add the numbers in Step 1 up. Set this
number aside for a moment.
3.Add up all of the weights.
4.Divide the numbers you found in Step 2
by the number you found in Step 3.
You take three 100-point exams in your
statistics class and score 80, 80 and 95. The last
exam is much easier than the first two, so your
professor has given it less weight. The weights
for the three exams are:
•Exam 1: 40 % of your grade. (Note: 40% as a
decimal is .4.)
•Exam 2: 40 % of your grade.
•Exam 3: 20 % of your grade
1.Multiply the numbers in your data set
by the weights:
.4(80) = 32
.4(80) = 32
.2(95) = 19
2. Add the numbers up. 32 + 32 + 19 = 83.
3. (0.4+0.4+0.2) = 1
4. 83/1= 83
The arithmetic mean is best used when the sum
of the values is significant. For example, your
grade in your statistics class. If you were to get
85 on the first test, 95 on the second test, and 90
on the third test, your average grade would be
90.
Why don't we use the geometric mean
here?
What about the harmonic mean?  
What if you got a 0 on your first test and
100 on the other two?
The arithmetic mean would give you a
grade of 66.6.
The geometric mean would give you a grade
of 0!!!
The harmonic mean can't even be applied at
all because 1/0 is undefined.  

More Related Content

What's hot

Measures of dispersion
Measures of dispersionMeasures of dispersion
Measures of dispersion
Sachin Shekde
 
Correlation and Regression
Correlation and RegressionCorrelation and Regression
Correlation and Regression
Sir Parashurambhau College, Pune
 
Student t test
Student t testStudent t test
Student t test
Dr Shovan Padhy, MD
 
Anova ppt
Anova pptAnova ppt
Anova ppt
Sravani Ganti
 
Analysis of variance (anova)
Analysis of variance (anova)Analysis of variance (anova)
Analysis of variance (anova)
Sadhana Singh
 
Standard deviation
Standard deviationStandard deviation
Standard deviation
Abdelrahman Alkilani
 
Chi squared test
Chi squared testChi squared test
Chi squared test
Ramakanth Gadepalli
 
Testing of hypothesis
Testing of hypothesisTesting of hypothesis
Testing of hypothesis
Jags Jagdish
 
Non parametric tests
Non parametric testsNon parametric tests
Non parametric tests
Raghavendra Huchchannavar
 
Test of significance in Statistics
Test of significance in StatisticsTest of significance in Statistics
Test of significance in Statistics
Vikash Keshri
 
Regression Analysis
Regression AnalysisRegression Analysis
Regression Analysis
Birinder Singh Gulati
 
Parametric tests
Parametric testsParametric tests
Parametric tests
Ananya Sree Katta
 
Statistics
StatisticsStatistics
Statistics
Pranav Krishna
 
Parametric and nonparametric test
Parametric and nonparametric testParametric and nonparametric test
Parametric and nonparametric test
ponnienselvi
 
Measures of dispersion
Measures of dispersionMeasures of dispersion
Measures of dispersion
Jagdish Powar
 
T test statistics
T test statisticsT test statistics
T test statistics
Mohammad Ihmeidan
 
Standard error
Standard error Standard error
Standard error
Satyaki Mishra
 
Introduction to kurtosis
Introduction to kurtosisIntroduction to kurtosis
Introduction to kurtosis
Amba Datt Pant
 
Central tendency
Central tendencyCentral tendency
Central tendency
Tauseef Jawaid
 
Regression
RegressionRegression
Regression
Buddy Krishna
 

What's hot (20)

Measures of dispersion
Measures of dispersionMeasures of dispersion
Measures of dispersion
 
Correlation and Regression
Correlation and RegressionCorrelation and Regression
Correlation and Regression
 
Student t test
Student t testStudent t test
Student t test
 
Anova ppt
Anova pptAnova ppt
Anova ppt
 
Analysis of variance (anova)
Analysis of variance (anova)Analysis of variance (anova)
Analysis of variance (anova)
 
Standard deviation
Standard deviationStandard deviation
Standard deviation
 
Chi squared test
Chi squared testChi squared test
Chi squared test
 
Testing of hypothesis
Testing of hypothesisTesting of hypothesis
Testing of hypothesis
 
Non parametric tests
Non parametric testsNon parametric tests
Non parametric tests
 
Test of significance in Statistics
Test of significance in StatisticsTest of significance in Statistics
Test of significance in Statistics
 
Regression Analysis
Regression AnalysisRegression Analysis
Regression Analysis
 
Parametric tests
Parametric testsParametric tests
Parametric tests
 
Statistics
StatisticsStatistics
Statistics
 
Parametric and nonparametric test
Parametric and nonparametric testParametric and nonparametric test
Parametric and nonparametric test
 
Measures of dispersion
Measures of dispersionMeasures of dispersion
Measures of dispersion
 
T test statistics
T test statisticsT test statistics
T test statistics
 
Standard error
Standard error Standard error
Standard error
 
Introduction to kurtosis
Introduction to kurtosisIntroduction to kurtosis
Introduction to kurtosis
 
Central tendency
Central tendencyCentral tendency
Central tendency
 
Regression
RegressionRegression
Regression
 

Similar to Measures of Central Tendency - Biostatstics

Measures of central tendency
Measures of central tendency Measures of central tendency
Measures of central tendency
Jagdish Powar
 
Central tendency
Central tendencyCentral tendency
Central tendency
keerthi samuel
 
Central tendency and Variation or Dispersion
Central tendency and Variation or DispersionCentral tendency and Variation or Dispersion
Central tendency and Variation or Dispersion
Johny Kutty Joseph
 
Group 3 measures of central tendency and variation - (mean, median, mode, ra...
Group 3  measures of central tendency and variation - (mean, median, mode, ra...Group 3  measures of central tendency and variation - (mean, median, mode, ra...
Group 3 measures of central tendency and variation - (mean, median, mode, ra...
reymartyvette_0611
 
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES Arithmetic mean Median Mode ...
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES  Arithmetic mean   Median Mode ...MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES  Arithmetic mean   Median Mode ...
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES Arithmetic mean Median Mode ...
Muhammad Amir Sohail
 
Measures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.pptMeasures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.ppt
GrandeurAidranMamaua
 
Biostatistics community medicine or Psm.pptx
Biostatistics community medicine or Psm.pptxBiostatistics community medicine or Psm.pptx
Biostatistics community medicine or Psm.pptx
eastmusings
 
Arithmetic Mean in Business Statistics
Arithmetic Mean in Business StatisticsArithmetic Mean in Business Statistics
Arithmetic Mean in Business Statistics
muthukrishnaveni anand
 
MEASURE OF CENTRAL TENDENCY
MEASURE OF CENTRAL TENDENCY  MEASURE OF CENTRAL TENDENCY
MEASURE OF CENTRAL TENDENCY
AB Rajar
 
Calculation of Median
Calculation of MedianCalculation of Median
Calculation of Median
Dr. Sunita Ojha
 
Central Tendancy.pdf
Central Tendancy.pdfCentral Tendancy.pdf
Central Tendancy.pdf
MuhammadFaizan389
 
First term notes 2020 econs ss2 1
First term notes 2020 econs ss2 1First term notes 2020 econs ss2 1
First term notes 2020 econs ss2 1
OmotaraAkinsowon
 
Measures of Central Tendency.pptx
Measures of Central Tendency.pptxMeasures of Central Tendency.pptx
Measures of Central Tendency.pptx
Melba Shaya Sweety
 
B.Ed.104 unit 4.1-statistics
B.Ed.104 unit 4.1-statisticsB.Ed.104 unit 4.1-statistics
B.Ed.104 unit 4.1-statistics
GANGOTRIROKADE1
 
Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
Srinivasan Padmanaban
 
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit SinghMeasures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
Vikramjit Singh
 
central tendency.pptx
central tendency.pptxcentral tendency.pptx
central tendency.pptx
NavnathIndalkar
 
Mean Mode Median.docx
Mean Mode Median.docxMean Mode Median.docx
Mean Mode Median.docx
Sameeraasif2
 
Measures of central_tendency._mean,median,mode[1]
Measures of central_tendency._mean,median,mode[1]Measures of central_tendency._mean,median,mode[1]
Measures of central_tendency._mean,median,mode[1]
Samuel Roy
 
Descriptive statistics
Descriptive statisticsDescriptive statistics
Descriptive statistics
Murugesan Kandan
 

Similar to Measures of Central Tendency - Biostatstics (20)

Measures of central tendency
Measures of central tendency Measures of central tendency
Measures of central tendency
 
Central tendency
Central tendencyCentral tendency
Central tendency
 
Central tendency and Variation or Dispersion
Central tendency and Variation or DispersionCentral tendency and Variation or Dispersion
Central tendency and Variation or Dispersion
 
Group 3 measures of central tendency and variation - (mean, median, mode, ra...
Group 3  measures of central tendency and variation - (mean, median, mode, ra...Group 3  measures of central tendency and variation - (mean, median, mode, ra...
Group 3 measures of central tendency and variation - (mean, median, mode, ra...
 
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES Arithmetic mean Median Mode ...
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES  Arithmetic mean   Median Mode ...MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES  Arithmetic mean   Median Mode ...
MEASURE OF CENTRAL TENDENCY TYPES OF AVERAGES Arithmetic mean Median Mode ...
 
Measures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.pptMeasures-of-Central-Tendency.ppt
Measures-of-Central-Tendency.ppt
 
Biostatistics community medicine or Psm.pptx
Biostatistics community medicine or Psm.pptxBiostatistics community medicine or Psm.pptx
Biostatistics community medicine or Psm.pptx
 
Arithmetic Mean in Business Statistics
Arithmetic Mean in Business StatisticsArithmetic Mean in Business Statistics
Arithmetic Mean in Business Statistics
 
MEASURE OF CENTRAL TENDENCY
MEASURE OF CENTRAL TENDENCY  MEASURE OF CENTRAL TENDENCY
MEASURE OF CENTRAL TENDENCY
 
Calculation of Median
Calculation of MedianCalculation of Median
Calculation of Median
 
Central Tendancy.pdf
Central Tendancy.pdfCentral Tendancy.pdf
Central Tendancy.pdf
 
First term notes 2020 econs ss2 1
First term notes 2020 econs ss2 1First term notes 2020 econs ss2 1
First term notes 2020 econs ss2 1
 
Measures of Central Tendency.pptx
Measures of Central Tendency.pptxMeasures of Central Tendency.pptx
Measures of Central Tendency.pptx
 
B.Ed.104 unit 4.1-statistics
B.Ed.104 unit 4.1-statisticsB.Ed.104 unit 4.1-statistics
B.Ed.104 unit 4.1-statistics
 
Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
 
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit SinghMeasures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh
 
central tendency.pptx
central tendency.pptxcentral tendency.pptx
central tendency.pptx
 
Mean Mode Median.docx
Mean Mode Median.docxMean Mode Median.docx
Mean Mode Median.docx
 
Measures of central_tendency._mean,median,mode[1]
Measures of central_tendency._mean,median,mode[1]Measures of central_tendency._mean,median,mode[1]
Measures of central_tendency._mean,median,mode[1]
 
Descriptive statistics
Descriptive statisticsDescriptive statistics
Descriptive statistics
 

More from Harshit Jadav

Cardiac cycle I Harshit Jadav
Cardiac cycle  I Harshit JadavCardiac cycle  I Harshit Jadav
Cardiac cycle I Harshit Jadav
Harshit Jadav
 
COVID-19 I Coronavirus Disease I Harshit Jadav
COVID-19 I Coronavirus Disease I Harshit JadavCOVID-19 I Coronavirus Disease I Harshit Jadav
COVID-19 I Coronavirus Disease I Harshit Jadav
Harshit Jadav
 
Reversible cell injury I Pathology
Reversible cell injury I PathologyReversible cell injury I Pathology
Reversible cell injury I Pathology
Harshit Jadav
 
Cell injury and Cellular Adaptation: Pathology
Cell injury and Cellular Adaptation: PathologyCell injury and Cellular Adaptation: Pathology
Cell injury and Cellular Adaptation: Pathology
Harshit Jadav
 
Receptor - Pharmacology
Receptor - PharmacologyReceptor - Pharmacology
Receptor - Pharmacology
Harshit Jadav
 
Routes of Drug administration - Pharmacology
Routes of Drug administration - PharmacologyRoutes of Drug administration - Pharmacology
Routes of Drug administration - Pharmacology
Harshit Jadav
 
How to write a review article
How to write a review articleHow to write a review article
How to write a review article
Harshit Jadav
 
Ebola virus disease (EVD)
Ebola virus disease (EVD)Ebola virus disease (EVD)
Ebola virus disease (EVD)
Harshit Jadav
 
UV Spectroscopy
UV SpectroscopyUV Spectroscopy
UV Spectroscopy
Harshit Jadav
 
Classification of anti cancer agents
Classification of anti cancer agentsClassification of anti cancer agents
Classification of anti cancer agents
Harshit Jadav
 
Agarose Gel Electrophoresis
Agarose Gel ElectrophoresisAgarose Gel Electrophoresis
Agarose Gel Electrophoresis
Harshit Jadav
 
agarose gel electrophoresis
agarose gel electrophoresisagarose gel electrophoresis
agarose gel electrophoresis
Harshit Jadav
 

More from Harshit Jadav (12)

Cardiac cycle I Harshit Jadav
Cardiac cycle  I Harshit JadavCardiac cycle  I Harshit Jadav
Cardiac cycle I Harshit Jadav
 
COVID-19 I Coronavirus Disease I Harshit Jadav
COVID-19 I Coronavirus Disease I Harshit JadavCOVID-19 I Coronavirus Disease I Harshit Jadav
COVID-19 I Coronavirus Disease I Harshit Jadav
 
Reversible cell injury I Pathology
Reversible cell injury I PathologyReversible cell injury I Pathology
Reversible cell injury I Pathology
 
Cell injury and Cellular Adaptation: Pathology
Cell injury and Cellular Adaptation: PathologyCell injury and Cellular Adaptation: Pathology
Cell injury and Cellular Adaptation: Pathology
 
Receptor - Pharmacology
Receptor - PharmacologyReceptor - Pharmacology
Receptor - Pharmacology
 
Routes of Drug administration - Pharmacology
Routes of Drug administration - PharmacologyRoutes of Drug administration - Pharmacology
Routes of Drug administration - Pharmacology
 
How to write a review article
How to write a review articleHow to write a review article
How to write a review article
 
Ebola virus disease (EVD)
Ebola virus disease (EVD)Ebola virus disease (EVD)
Ebola virus disease (EVD)
 
UV Spectroscopy
UV SpectroscopyUV Spectroscopy
UV Spectroscopy
 
Classification of anti cancer agents
Classification of anti cancer agentsClassification of anti cancer agents
Classification of anti cancer agents
 
Agarose Gel Electrophoresis
Agarose Gel ElectrophoresisAgarose Gel Electrophoresis
Agarose Gel Electrophoresis
 
agarose gel electrophoresis
agarose gel electrophoresisagarose gel electrophoresis
agarose gel electrophoresis
 

Recently uploaded

slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
MANIVALANSR
 
5. Postharvest deterioration of fruits and vegetables.pptx
5. Postharvest deterioration of fruits and vegetables.pptx5. Postharvest deterioration of fruits and vegetables.pptx
5. Postharvest deterioration of fruits and vegetables.pptx
UmeshTimilsina1
 
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
Alvaro Barbosa
 
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
ALBERTHISOLER1
 
INSIDE OUT - PowerPoint Presentation.pptx
INSIDE OUT - PowerPoint Presentation.pptxINSIDE OUT - PowerPoint Presentation.pptx
INSIDE OUT - PowerPoint Presentation.pptx
RODELAZARES3
 
2 Post harvest Physiology of Horticulture produce.pptx
2 Post harvest Physiology of Horticulture  produce.pptx2 Post harvest Physiology of Horticulture  produce.pptx
2 Post harvest Physiology of Horticulture produce.pptx
UmeshTimilsina1
 
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdfPRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
nservice241
 
A beginner’s guide to project reviews - everything you wanted to know but wer...
A beginner’s guide to project reviews - everything you wanted to know but wer...A beginner’s guide to project reviews - everything you wanted to know but wer...
A beginner’s guide to project reviews - everything you wanted to know but wer...
Association for Project Management
 
Open and Critical Perspectives on AI in Education
Open and Critical Perspectives on AI in EducationOpen and Critical Perspectives on AI in Education
Open and Critical Perspectives on AI in Education
Robert Farrow
 
MVC Interview Questions PDF By ScholarHat
MVC Interview Questions PDF By ScholarHatMVC Interview Questions PDF By ScholarHat
MVC Interview Questions PDF By ScholarHat
Scholarhat
 
2024 Winter SWAYAM NPTEL & A Student.pptx
2024 Winter SWAYAM NPTEL & A Student.pptx2024 Winter SWAYAM NPTEL & A Student.pptx
2024 Winter SWAYAM NPTEL & A Student.pptx
Utsav Yagnik
 
React Interview Question PDF By ScholarHat
React Interview Question PDF By ScholarHatReact Interview Question PDF By ScholarHat
React Interview Question PDF By ScholarHat
Scholarhat
 
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
Nguyen Thanh Tu Collection
 
ASP.NET Core Interview Questions PDF By ScholarHat.pdf
ASP.NET Core Interview Questions PDF By ScholarHat.pdfASP.NET Core Interview Questions PDF By ScholarHat.pdf
ASP.NET Core Interview Questions PDF By ScholarHat.pdf
Scholarhat
 
View Inheritance in Odoo 17 - Odoo 17 Slides
View Inheritance in Odoo 17 - Odoo 17  SlidesView Inheritance in Odoo 17 - Odoo 17  Slides
View Inheritance in Odoo 17 - Odoo 17 Slides
Celine George
 
RDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEWRDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEW
Murugan Solaiyappan
 
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
Dr. Nasir Mustafa
 
6. Physiological Disorder of fruits and vegetables.pptx
6. Physiological Disorder of fruits and vegetables.pptx6. Physiological Disorder of fruits and vegetables.pptx
6. Physiological Disorder of fruits and vegetables.pptx
UmeshTimilsina1
 
SQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHatSQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHat
Scholarhat
 
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdfFINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
maritescanete2
 

Recently uploaded (20)

slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
slidesgo-mastering-the-art-of-listening-insights-from-robin-sharma-2024070718...
 
5. Postharvest deterioration of fruits and vegetables.pptx
5. Postharvest deterioration of fruits and vegetables.pptx5. Postharvest deterioration of fruits and vegetables.pptx
5. Postharvest deterioration of fruits and vegetables.pptx
 
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
Benchmarking Sustainability: Neurosciences and AI Tech Research in Macau - Ke...
 
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
Brigada Eskwela 2024 PowerPoint Update for SY 2024-2025
 
INSIDE OUT - PowerPoint Presentation.pptx
INSIDE OUT - PowerPoint Presentation.pptxINSIDE OUT - PowerPoint Presentation.pptx
INSIDE OUT - PowerPoint Presentation.pptx
 
2 Post harvest Physiology of Horticulture produce.pptx
2 Post harvest Physiology of Horticulture  produce.pptx2 Post harvest Physiology of Horticulture  produce.pptx
2 Post harvest Physiology of Horticulture produce.pptx
 
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdfPRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
 
A beginner’s guide to project reviews - everything you wanted to know but wer...
A beginner’s guide to project reviews - everything you wanted to know but wer...A beginner’s guide to project reviews - everything you wanted to know but wer...
A beginner’s guide to project reviews - everything you wanted to know but wer...
 
Open and Critical Perspectives on AI in Education
Open and Critical Perspectives on AI in EducationOpen and Critical Perspectives on AI in Education
Open and Critical Perspectives on AI in Education
 
MVC Interview Questions PDF By ScholarHat
MVC Interview Questions PDF By ScholarHatMVC Interview Questions PDF By ScholarHat
MVC Interview Questions PDF By ScholarHat
 
2024 Winter SWAYAM NPTEL & A Student.pptx
2024 Winter SWAYAM NPTEL & A Student.pptx2024 Winter SWAYAM NPTEL & A Student.pptx
2024 Winter SWAYAM NPTEL & A Student.pptx
 
React Interview Question PDF By ScholarHat
React Interview Question PDF By ScholarHatReact Interview Question PDF By ScholarHat
React Interview Question PDF By ScholarHat
 
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
 
ASP.NET Core Interview Questions PDF By ScholarHat.pdf
ASP.NET Core Interview Questions PDF By ScholarHat.pdfASP.NET Core Interview Questions PDF By ScholarHat.pdf
ASP.NET Core Interview Questions PDF By ScholarHat.pdf
 
View Inheritance in Odoo 17 - Odoo 17 Slides
View Inheritance in Odoo 17 - Odoo 17  SlidesView Inheritance in Odoo 17 - Odoo 17  Slides
View Inheritance in Odoo 17 - Odoo 17 Slides
 
RDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEWRDBMS Lecture Notes Unit4 chapter12 VIEW
RDBMS Lecture Notes Unit4 chapter12 VIEW
 
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
Dr. Nasir Mustafa CERTIFICATE OF APPRECIATION "NEUROANATOMY"
 
6. Physiological Disorder of fruits and vegetables.pptx
6. Physiological Disorder of fruits and vegetables.pptx6. Physiological Disorder of fruits and vegetables.pptx
6. Physiological Disorder of fruits and vegetables.pptx
 
SQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHatSQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHat
 
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdfFINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
 

Measures of Central Tendency - Biostatstics

  • 1. Measures of Central Tendency Harshit Jadav M.S.Pharm, PGDRA
  • 2. • Measures of central tendency are also usually called as the averages. • They give us an idea about the concentration of the values in the central part of the distribution. • The following are the five measures of central tendency that are in common use: • (i) Arithmetic mean, (ii) Median, (iii) Mode, (iv) Geometric mean, and (v) Harmonic mean (vi) weighted mean
  • 3. MEASURE OF CENTRAL TENDANCY MEAN MEDIAN MODE The average of the data The middle value of the data most commonly occurring value
  • 4. Mean (Average) Mean locate the centre of distribution. Also known as arithmetic mean Most Common Measure The mean is simply the sum of the values divided by the total number of items in the set. Affected by Extreme Values XX XX nn XX XX XX nn ii ii nn nn == == ++ ++ ++== ∑∑ 11 11 22 
  • 5. Merits: • It is easy to understand and easy to calculate • It is based upon all the observations • It is familiar to common man and rigidly defined • It is capable of further mathematical treatment. • It is affected by sampling fluctuations. Hence it is more stable.
  • 6. Demerits • It cannot be determined by inspection. • Arithmetic mean cannot be used if we are dealing with qualitative characteristics, which cannot be measured quantitatively like caste, religion, sex. • Arithmetic mean cannot be obtained if a single observation is missing or lost • Arithmetic mean is very much affected by extreme values.
  • 7. MEAN UNGROUPED DATA GROUPED DATA n x n i ∑= = 1 ix n x n i ∑= = 1 ii xf MEAN = nsobservatioofno.total nsobservatioofSum
  • 8. Birth weight of new borns are : 3.3, 6.1, 5.8, 3.8, 2.7, 4.1, 3.4, 3.9, 5.1, 3 n x n i ∑= = 1 ix =41.2/10 =4.12 kg
  • 9. Q. A Survey of 100 families each having five children, revealed the following distribution No. of male children 0, 1, 2, 3, 4, 5 No. of Families 9, 24, 35, 24, 6, 2 Find the Mean of male children. x f f . X 0 9 0 1 24 24 2 35 70 3 24 72 4 6 24 5 2 10 N=100 Σ f.x = 200 Mean = X =Σ f.x / Σf X =200/ 100 = 2
  • 10. Find mean days of confinement after delivery in the following series:- Day of confinement No. of patients 6 5 7 4 8 4 9 3 10 2
  • 11. Day of confinement (x) No. of patients (f) X*f 6 5 30 7 4 28 8 4 32 9 3 27 10 2 20 Total 18 137 Solution:- n x n i ∑= = 1 ii xf =137/18 =7.61
  • 12. Median 1.Measure of Central Tendency. 2. The median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. 3.Middle Value In Ordered Sequence • If Odd n, Middle Value of Sequence • If Even n, Average of 2 Middle Value 4.Not Affected by Extreme Values
  • 13. Merits: • It is rigidly defined • It is easy to understand and easy to calculate. • It is not at all affected by extreme values. • It can be calculated for distributions with open-end classes. • Median is the only average to be used while dealing with qualitative data. • Can be determined graphically.
  • 14. Demerits: • In case of even number of observations median cannot be determined exactly. • It is not based on all the observations. • It is not capable of further mathematical treatment
  • 15. For ungrouped data:- Step-1 Arranged data in ascending or descending order. Step:-2 If total no. of observations ‘n’ is odd then used the following formula for median Step:-3 If total no. of observations ‘n’ is even then used the following formula for median = arithmetic mean of two middle observations. . 2 1 nobservatioth n + =
  • 16. Median If X1, X2, X3,... …. ,Xn are n observations arranged in ascending order of magnitude. X(n+1)/2 if n is odd Median = { Xn/2 + X(n/2)+1 if n is even ------------------ 2
  • 17. Calculate the median for the following series :- 2,3,5,1,4,5,8 1,2,3,4,5,5,8. Median . 2 1 nobservatioth n + = = 7+1/2 =4th number
  • 18. 74+75 Median = ---------- = 74.5 2 The data on pulse rate per minute of 10 healthy individuals are 82, 79, 60, 76, 63,81, 68, 74, 60, 75. n= 10 60, 60, 63, 68, 74,75, 76, 79, 81, 82 Xn/2 + X(n/2)+1 / 2
  • 19. Find out the median for number of sports injuries happened in cricket in all teams 37, 57, 65, 46, 12, 14, 19, 23, 56, 78, 5, 33
  • 20. Median:- f hc n l       − + 2 l = lower limit of class interval where the median occurs f = Frequency of the class where median occurs h = Width of the median class C= Cumulative frequency of the class preceding the median class (PCF) For Grouped data:-
  • 21. Weight of infant in kg No of infants 2.0-2.5 37 2.5-3.0 117 3.0-3.5 207 3.5-4.0 155 4.0-4.5 48 4.5 and above 26 Find the median weight of 590 infants born in a hospital in one year from the following table.
  • 22. Weight of infants in kg No of infants Cumulative frequency 2.0-2.5 37 37 2.5-3.0 117 37+117=154 3.0-3.5 207 154+207=361 3.5-4.0 155 361+155=516 4.0-4.5 48 516+48=564 4.5 and above 26 564+26=590 N/2 =590/2 =295 Median Class = 3.0-3.5, so L=3.0, f= 207 Cf = 154 h=0.5
  • 23. f hc n l       − + 2 = 3.0 + (295-154) * 0.5 207 = 3.0 + 0.34 = 3.34 Median Weight
  • 24. For grouped Data:- Class interval Frequency 5-9 2 10-14 11 15-19 26 20-24 17 25-29 8 30-34 6 35-39 3 40-44 2 45-49 1 Calculate the median for the following data series:-
  • 25. Solution:- Class interval Frequency cumulative frequency 5-9 2 2 10-14 11 13 15-19 26 39 20-24 17 56 25-29 8 64 30-34 6 70 35-39 3 73 40-44 2 75 45-49 1 76
  • 26. n=76 l = lower limit of class interval where the median occurs = 15 h = Width of the median class = 4 f = Frequency of the class where median occurs = 26 C = Cumulative frequency of the class preceding the median class =13 f hc n l       − + 2
  • 27. Mode 1.Measure of Central Tendency 2.The mode is the most frequently occurring value in the data set. 3.May Be No Mode or Several Modes
  • 28. Merits: • Mode is readily comprehensible and easy to calculate. • Mode is not at all affected by extreme values. • Mode can be conveniently located even if the frequency distribution has class intervals of unequal magnitude • Open-end classes also do not pose any problem in the location of mode. • Mode is the average to be used to find the ideal size.
  • 29. Demerits: • Mode is ill defined. • It is not based upon all the observations. • It is not capable of further mathematical treatment. • As compared with mean, mode is affected to a great extent by fluctuations of sampling.
  • 30. Mode Example No Mode Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 One Mode Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9 More Than 1 Mode Raw Data: 21 28 28 41 43 43
  • 31. Mode for ungrouped data:- 2,2,3,4,6,7,4,4,4,4,8,9,0 mode is 4 10,10,3,3,4,2,1,6,7 mode is 10 and 3 10,34,23,12,11,3,4 no mode
  • 32. Mode for Group Data Fm-F1 Mode = L1 + ------------------ * c 2Fm– F1- F2 Where •Fm is mode freq. • F1 is freq. just lower mode class. •F2 is freq. after mode class. • L1 is lower limit of mode class •C is class difference.
  • 33. C I FREQU. (F) 20 - 30 3 30 - 40 20 40 - 50 27 50 - 60 15 60 - 70 9 Q. Find the Mode for group data Age group 20-30 30-40 40-50 50-60 60-70 No. of persons 3 20 27 15 9
  • 34. Q. Find the Mode for group data Age group 20-30 30-40 40-50 50-60 60-70 No. of persons 3 20 27 15 9 C I FREQU. (F) 20 - 30 3 30 - 40 20 40 - 50 27 50 - 60 15 60 - 70 9 L1 Fm F1 F2
  • 35. FFmm -- FF11 Mode = LMode = L11 + ------------------ * c+ ------------------ * c 2F2Fmm – F– F11- F- F22 2727 –– 20 7020 70 Mode = 40 + ------------------ * 10 = 40 + -----Mode = 40 + ------------------ * 10 = 40 + ----- 2*272*27 – 20- 15 19– 20- 15 19 Mode = 43.68Mode = 43.68
  • 36. Calculate the mode for the following frequency distribution:- IQ Range Frequency 90-100 11 100-110 27 110-120 36 120-130 38 130-140 43 140-150 28 150-160 16 160-170 1
  • 37. Modal class by inspection is 130-140 fm= 43 f1= 38 f2= 28 C=10 l = 130 c fff ff l m m       −− − + )(2 )( 21 1 =130.6579
  • 38. Relationship between Mean, Median and Mode Mode = 3 Median – 2 Mean
  • 39. Summary of Central Tendency Measures MeasureMeasure DescriptionDescription MeanMean Balance PointBalance Point MedianMedian Middle ValueMiddle Value When OrderedWhen Ordered ModeMode Most FrequentMost Frequent
  • 40. Ex. Calculate Mean, Median, Mode. Age Group No. of Patients 25-30 4 30-35 3 35-40 2 40-45 3 45-50 4 50-55 8 55-60 6
  • 41. Age Group No. of Patients (F) X F*X C.F 25-30 4 27.5 110 4 30-35 3 32.5 97.5 7 35-40 2 37.5 75 9 40-45 3 42.5 127.5 12 45-50 4 47.5 190 16 50-55 8 52.5 420 24 55-60 6 57.5 345 30 30 1365 MEAN =1365/30 = 45.5 MEDIAN =45+ (15-12)*5/4 = 48.75 MODE=50 + [(8-4)*5/(2*8-4-6)]=53.34
  • 42. Ex. The following table gives the frequency distribution of marks obtained by 2300 medical students of Gujarat in MCQ of PSM exam. Find Mean, Median and Mode. Marks No. of students 11-20 141 21-30 221 31-40 439 41-50 529 51-60 495 61-70 322 71-80 153
  • 43. Geometric Mean • Geometric mean is defined as the positive root of the product of observations. Symbolically, • It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. • Find geometric mean of rate of growth: 34, 27, 45, 55, 22, 34 n nxxxxG /1 321 )( =
  • 44. Geometric mean of Group data • If the “n” non-zero and positive variate-values occur times, respectively, then the geometric mean of the set of observations is defined by: [ ] Nn i f i Nf n ff in xxxxG 1 1 1 21 21       == ∏=  ∑= = n i ifN 1 Where nxxx ,........,, 21 nfff ,.......,, 21
  • 45. Geometric Mean (Revised Eqn.) )( 321 nxxxxG =         = ∑= n i ixLog N AntiLogG 1 1         = ∑= n i ii xLogf N AntiLogG 1 1 )( 321 321 n fff xxxxG = Ungroup Data Group Data
  • 47. What is the geometric mean of 4,8.3,9 and 17?
  • 48. First, multiply the numbers together and then take the 5th root (because there are 5 numbers) G = (4*8*3*9*17)(1/5) G = 6.81
  • 49. Harmonic Mean • Harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. • The harmonic mean is a very specific type of average. • It’s generally used when dealing with averages of units, like speed or other rates and ratios.
  • 50. Harmonic Mean Group Data • The harmonic mean H of the positive real numbers x1,x2, ..., xn is defined to be ∑= = n i i i x f n H 1 ∑= = n i ix n H 1 1 Ungroup Data Group Data
  • 51. What is the harmonic mean of 1,5,8,10? Here, N=4 H = 4 / (1/1) +(1/5) + (1/8) + (1/10) H = 4/ 1.425 H = 2.80 ∑= = n i ix n H 1 1
  • 52. Rahul drives a car at 20 mph for the first hour and 30 mph for the second. What’s his average speed?
  • 53. We need the harmonic mean: = 2/(1/20 + 1/30) = 2(0.05 + 0.033) = 2 / 0.083 = 24.09624 mph.
  • 54. Weighted Mean • The Weighted mean of the positive real numbers x1,x2, ..., xn with their weight w1,w2, ..., wn is defined to be ∑ ∑ = = = n i i n i ii w xw x 1 1 Σ = the sum of (in other words…add them up!). w = the weights. x = the value.
  • 55. A weighted mean is a kind of average. Instead of each data point contributing equally to the final mean, some data points contribute more “weight” than others. If all the weights are equal, then the weighted mean equals the arithmetic mean (the regular “average” you’re used to). Weighted means are very common in statistics, especially when studying populations.
  • 56. Steps: 1.Multiply the numbers in your data set by the weights. 2.Add the numbers in Step 1 up. Set this number aside for a moment. 3.Add up all of the weights. 4.Divide the numbers you found in Step 2 by the number you found in Step 3.
  • 57. You take three 100-point exams in your statistics class and score 80, 80 and 95. The last exam is much easier than the first two, so your professor has given it less weight. The weights for the three exams are: •Exam 1: 40 % of your grade. (Note: 40% as a decimal is .4.) •Exam 2: 40 % of your grade. •Exam 3: 20 % of your grade
  • 58. 1.Multiply the numbers in your data set by the weights: .4(80) = 32 .4(80) = 32 .2(95) = 19 2. Add the numbers up. 32 + 32 + 19 = 83. 3. (0.4+0.4+0.2) = 1 4. 83/1= 83
  • 59. The arithmetic mean is best used when the sum of the values is significant. For example, your grade in your statistics class. If you were to get 85 on the first test, 95 on the second test, and 90 on the third test, your average grade would be 90. Why don't we use the geometric mean here? What about the harmonic mean?  
  • 60. What if you got a 0 on your first test and 100 on the other two? The arithmetic mean would give you a grade of 66.6. The geometric mean would give you a grade of 0!!! The harmonic mean can't even be applied at all because 1/0 is undefined.