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Transcript of Measure of central tendency
Measures Of Central Tendency
Quantitative Aptitude & Business Statistics
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Statistics in Plural Sense as Statistical data.
Statistics in Plural Sense refers to numerical data of any phenomena placed in relation to each other.
For example ,numerical data relating to population ,production, price level, national income, crimes, literacy ,unemployment ,houses etc.,
Statistical in Singular Scene as Statistical method.
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According to Prof.Horace Secrist:
“By Statistics we mean aggregate of facts affected to marked extend by multiplicity of causes numerically expressed, enumerated or estimated according to reasonable standard of accuracy ,collected in a systematic manner for a pre determined purpose and placed in relation to each other .”
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Measures of Central Tendency
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Def:Measures of Central Tendency A single expression
representing the whole group,is selected which may convey a fairly adequate idea about the whole group.
This single expression is
known as average.
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Averages are central part of distribution and, therefore ,they are also called measures of central tendency.
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Types of Measures central tendency:
There are five types ,namely 1.Arithmetic Mean (A.M) 2.Median 3.Mode 4.Geometric Mean (G.M) 5.Harmonic Mean (H.M)
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Features of a good average 1.It should be rigidly defined 2.It should be easy to
understand and easy to calculate
3.It should be based on all the observations of the data
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4.It should be easily subjected to further mathematical calculations
5.It should be least affected by fluctuations of sampling
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Arithmetic Mean (A.M) The most commonly used measure of central tendency. When people ask about the “average" of a group of scores, they usually are referring to the mean.
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The arithmetic mean is simply dividing the sum of variables by the total number of observations.
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Arithmetic Mean for raw data is given by
n
x
nX
n
ii
xxxx n
∑=++++ == 1......321
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Find mean for the data 17,16,21,18,13,16,12 and 11
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Arithmetic Mean for Discrete Series
∑
∑
=
=++++ =++++
= n
ii
n
iii
n
xfxfxfxf
f
xf
ffffX nn
1
1
321
......
....332211
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Arithmetic Mean for Continuous Series
CN
fdAX ×+= ∑
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Calculation of Arithmetic mean in case of Continuous Series
Marks 0-10
10-20
20-30
30-40
40-50
50-60
No. of Students
10 20 30 50 40 30
From the following data calculate Arithmetic mean
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Marks Mid values
(X)
No.of Students
(f)
d= X-45 10
f.d
0-10 5 10 -4 -40 10-20 15 20 -3 -60 20-30 25 30 -2 -60 30-40 35 50 -1 -50
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Marks Mid values
(X)
No.of Students
(f)
d= X-45 10
f.d
40-50 45 40 0 0 50-60 55 30 1 30
N=180 ∑fd=-180
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Solution
Let us take assumed mean =45
Calculation from assumed mean
Mean =
35180
10*18045x
=
−+=×+=
− ∑ CN
fdA
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Calculation Of Arithmetic Mean in case of Less than series
Marks less than /up to
10 20 30 40 50 60
No. of students
10 30 60 110 150 180
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Solution: Let us first convert Less than series into continuous series as follows
Marks 0-10 10-20
20-30
30-40
40-50
50-60
No. of students
10 20 30 50 40 30 180-150=30
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Calculation Of Arithmetic Mean in case of more than series
Marks more than
0 10 20 30 40 50 60
No. of students
180 170 150
120 70 30 0
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Solution: Let us first convert More than series into continuous series as follows
Marks 0-10 10-20
20-30
30-40
40-50 50-60
No. of students
10 20 30 50 40 30
180-170=10 170-150=20
70-30=40
30-0=30
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Calculation of Arithmetic Mean in case of Inclusive series
From the following data ,calculate Arithmetic Mean
Marks 1-10 11-20 21-
30 31-40
41-50
51-60
No. of Students
10
20 30 50 40 30
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Solution
Let us take assumed mean =45.5
Calculation from assumed mean
Mean = 35
18010*18045x
=
−+=×+=
− ∑ CN
fdA
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Marks Mid values
No.of Students
d=X-45.5 10
f.d
0.5-10.5 5.5 10 -4 -40 10.5-20.5 15.5 20 -3 -60 20.5-30.5 25.5 30 -2 -60 30.5-40.5 35.5 50 -1 -50 40.5-50.5 45.5 40 0 0 50.5-60.5 55.5 30 1 30
N=180 ∑fd= -180
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Calculation of Arithmetic Mean in case of continuous exclusive series when class intervals are unequal
From the following data ,calculate Arithmetic Mean
Marks 0-10 10-30 30-40 40-50 50-60
No. of Students
10
60 50 40 20
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Since class intervals are unequal, frequencies have been adjusted to make the class intervals equal on the assumption that they are equally distributed throughout the class
Let us take assumed mean =45
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Calculation of Deviations from assumed mean
Mean=
778.32180
1022045x
=
−+=×+=
− ∑ XCN
fdA
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Marks Mid values
No. of Students
d= X-45.5 10
f.d
0-10 5 10 -4 -40 10-20 15 30 -3 -90 20-30 25 30 -2 -60 30-40 35 50 -1 -50 40-50 45 40 0 0 50-60 55 20 1 30
N=180 ∑fd=-220
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Combined Arithmetic Mean (A.M)
An average daily wages of 10 workers in a factory ‘A’ is Rs.30 and an average daily wages of 20 workers in a factory B’ is Rs.15.Find the average daily wages of all the workers of both the factories.
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Solution
Step 1;N1=10 N2=20
Step2:
=20
15;30 21 == XX
21
221112 NN
XNXNX++
=
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Weighted Arithmetic Mean
The term ‘ weight’ stands for the relative importance of the different items of the series. Weighted Arithmetic Mean refers to the Arithmetic Mean calculated after assigning weights to different values of variable. It is suitable where the relative importance of different items of variable is not same
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Weighted Arithmetic Mean is specially useful in problems relating to
1)Construction of Index numbers. 2)Standardised birth and death rates
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Weighted Arithmetic Mean is given by
∑∑∑ =
WXW
X w
.
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Mathematical Properties of Arithmetic Mean
1.The Sum of the deviations of the items from arithmetic mean is always Zero. i.e.
2.The sum of squared deviations of the items from arithmetic mean is minimum or the least
( ) 0=−∑ XX
( ) 02≤−∑ XX
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3.The formula of Arithmetic
mean can be extended to
compute the combined
average of two or more
related series
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4.If each of the values of a
variable ‘X’ is increased or decreased by some constant C, the arithmetic mean also increased or decreased by C .
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Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k .
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When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity
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Merits Of Arithmetic Mean
1.Its easy to understand and easy to calculate.
2.It is based on all the items of the samples.
3.It is rigidly defined by a mathematical formula so that the same answer is derived by every one who computes it.
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4.It is capable for further algebraic treatment so that its utility is enhanced
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6.The formula of arithmetic mean can be extended to compute the combined average of two or more related series.
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7.It has sampling stability .It is least affected by sampling fluctuations
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Limitations of Arithmetic Mean
1.Affected by extreme values i.e . Very small or very big values in the data unduly affect the value of mean because it is based on all the items of the series.
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2.Mean is not useful for studying the qualitative phenomenon.
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Median
The middle score of the distribution when all the scores have been ranked.
If there are an even number of scores, the median is the average of the two middle scores.
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In an ordered array, the median is the “middle” number 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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Potential Problem with Means
Mean
Mean
Median
Median
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Median
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
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Median for raw data
When given observation are even First arrange the items in ascending
order then
Median (M)=Average of Item
21
2+
+=NN
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Find the Median for the raw data
25,55,5,45,15 and 35 Solution ;Arrange the items 5,15,25,35,45,55,here N=6 Median =Average of 3rd and 4th
item=30
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Median for raw data
When given observation are odd First arrange the items in ascending
order then
Median (M)=Size of Item
21+
=N
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Median for continuous series
cf
mN
LM ×
−+= 2
Where M= Median; L=Lower limit of the Median Class,m=Cumulative frequency above median class f=Frequency of the median class N=Sum of frequencies
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Quartiles
The values of variate that divides the series or the series or the distribution into four equal parts are known as Quartiles .
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The first Quartile (Q1),known as a lower Quartile is the value of variate below which 25% of the observations.
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The Second Quartile known as middle Quartile(Q2)known as middle Quartile or median ,the value of variates below which 50% of the observations
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The Third Quartile known as Upper Quartile(Q3)known as middle Quartile or median ,the value of variates below which 75 % of the observations.
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thNSizeQ4
11
+= Item
thNSizeQ4
)1(33
+= Item
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Octiles
The values of variate that divides the series or the distribution into eight equal parts are known as Octiles .
Each octile contains 12.5% of the total number of observations .
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Since seven points are required to divide the data into 8 equal parts ,we have 7 octiles.
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thNjSizeOj 8)1( +
= Item
thNSizeO8
)1(44
+= Item
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Deciles
The values of variate that divides the series or the distribution into Ten equal parts are known as Deciles .
Each Decile contains 10% of the total number of observations .
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Since 9 points are required to divide the data into 10 equal parts ,we have 9 deciles(D1 to D9)
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thNjSizeDj 10)1( +
= Item
thNSizeD10
)1(55
+= Item
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Percentiles The values of variate that divides
the series or the distribution into hundred equal parts are known as Percentiles .
Each percentile contains 10% of the total number of observations .
Since 99 points are required to divide the data into 10 equal parts ,we have 99 deciles(p1 to p99)
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thNjSizePj 100)1( +
= Item
thNSizep100
)1(5050
+= Item
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Relation Ship Between Partition Values 1.Q1=O2=P25 value of variate which
exactly 25% of the total number of observations
2.Q2=D5=P50,value of variate which exactly 50% of the total number of observations.
3. Q3=O6=P75,value of variate which exactly 75% of the total number of observations
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Calculation of Median in case of Continuous Series
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of Students
10 20 30 50 40 30
From the following data calculate Median
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Marks No. of Students
(f)
Cumulative Frequencies
(c.f.) 0-10 10 10 10-20 20 30 20-30 30 60 30-40 50 110 40-50 40 150 50-60 30 180
N=180
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Calculate size of N/2
902
1802
==N
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1050
602
180
30 ×
−+=M
36630 =+=M
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Merits of Median 1.Median is not affected by
extreme values . 2.It is more suitable average
for dealing with qualitative data ie.where ranks are given.
3.It can be determined by graphically.
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Limitations of Median 1.It is not based all the items of
the series . 2.It is not capable of algebraic
treatment .Its formula can not be extended to calculate combined median of two or more related groups.
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0 X
Y
M
Less than Cumulative curve
More than Cumulative Curve
Median By Graph
Q3 Q1 CI
Frequency N/2
3N/4
N/4
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Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or
categorical data There may be no mode or several
modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
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Mode The most frequent score in the
distribution. A distribution where a single
score is most frequent has one mode and is called unimodal.
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A distribution that consists of only one of each score has n modes.
When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie.
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Calculate the mode from the following data of marks obtained by 10 students.
20,30,31,32,25,25,30,31,30,32
Mode (Z)=30
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Mode for Continuous Series
cfff
ffLZ ×
−−
−+=
201
01
2Where Z= Mode ;L=Lower limit of the Mode Class f0 =frequency of the pre modal class f1=frequency of the modal class f2=frequency of the post modal class C=Class interval of Modal Class
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Calculation of Mode :Continuous Series
Marks 0-10
10-20
20-30
30-40
40-50
50-60
No. of Students
10 20 30 50 40 30
From the following data calculate Mode
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Marks No. of Students
(f) 0-10 10 10-20 20 20-30 30 30-40 50 f1 40-50 40 50-60 30
N=180
f0
f2
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667.36667.630
104030502
605030
2 201
01
=+=
×
−−×−
+=
×
−−
−+=
Z
cfff
ffLZ
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x 0
Y
Z 10 20 30 40 50 60
10
20
30
40
50
Calculation Mode Graphically
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Relationship between Mean, Median and Mode
The distance between Mean and Median is about one third of distance between the mean and the mode.
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Karl Pearson has expressed the relationship as follows. Mean –Mode=(Mean-Median)/3 Mean-Median=3(Mean-Mode)
Mode =3Median-2Mean Mean=(3Median-Mode)/2
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Example
For a moderately skewed distribution of marks in statistics for a group of 200 students ,the mean mark and median mark were found to be 55.60 and 52.40.what is the modal mark?
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Solution
Since in this case mean=55.60and median =52.40 applying ,we get
Mode=3median -2Mean =3(52.40)-2(55.60) Mode =46
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Example
If Y=2+1.50X and mode of X is 15 ,What is mode of Y
Solution Y m=2+1.50*15=24.50
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Merits of Mode
1.Mode is the only suitable average e.g. ,modal size of garments, shoes.,etc
2.It is not affected by extreme values.
3.Its value can be determined graphically.
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Limitations of Mode
1.In case of bimodal /multi modal series ,mode cannot be determined.
2.It is not capable for further algebraic treatment, combined mode of two or more series cannot be determined.
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3.It is not based on all the items of the series
4.Its value is significantly affected by the size of the class intervals
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Geometric mean
nn
ii
nniG
x
xxxxx/1
1
21
=
=
∏=
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Take the logarithms of each item of variable and obtain their total i.e ∑ log X
Calculate G M as follows
= ∑
nX
AntiMGlog
log.
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Computation of G.M -Discrete Series Take the logarithms of each item of
variable and multiply with the respective frequencies obtain their total
i.e ∑ f .log X Calculate G M as follows
= ∑
NXf
AntiMGlog.
log.
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Merits of Geometric Mean
1.It is based on all items of the series .
2 It is rigidly defined 3.It is capable for algebraic
treatment.
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4.It is useful for averaging ratios and percentages rates are increase or decrease
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Limitations of Geometric Mean
1.Its difficult to understand and calculate.
2.It cannot be computed when there are both negative and positive values in a series
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3.It is biased for small values as it gives more weight to small values .
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Calculation of G.M :Individual Series
From the following data calculate Geometric Mean Roll No 1 2 3 4 5 6
Marks 5 15 25 35 45 55
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Computation of G.M :Individual Series
X log X 5 0.6990 15 1.1761 25 1.3979 35 1.5441 45 1.6532 55 1.7404
∑log X=8.2107
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36.23)3685.1log(
62107.8
loglog.
==
=
= ∑
Anti
Al
nX
AntiMG
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Find the average rate of increase population which in the first decade has increased by 10% ,in the second decade by 20% and third by 30%
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Decade % rise Population at the end of the decade
logx
1 2 3
10 20 30
110 120 130
2.0414 2.0792 2.1139
∑log X=6.2345
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8.119)0782.2log(
2345.6
loglog.
==
=
= ∑
Anti
Al
nX
AntiMG
Average Rate of increase in Population is 19.8%
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Weighted Geometric Mean
=
∑∑
wXw
AntiMGlog.
log.
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Harmonic Mean (H.M)
Harmonic Mean of various items of a series is the reciprocal of the arithmetic mean of their reciprocal .Symbolically,
nXXXX
NMH1.......111
.
321
++++=
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Where X1,X2,X3…….X n refer to the value of various series.
N= total no. of series
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Merits of Harmonic Mean
1.It is based on all items of the series .
2 It is rigidly defined 3.It is capable for algebraic
treatment.
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4.It is useful for averaging measuring the time ,Speed etc
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Limitations of Harmonic Mean
1.Its difficult to understand and calculate.
2.It cannot be computed when one or more items are zero
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3.It gives more weight to smallest values . Hence it is not suitable for analyzing economic data .
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Calculation of H.M :Individual Series
From the following data calculate Harmonic Mean Roll No
1 2 3 4 5 6
Marks
5 15 25 35 45 55
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Computation of H.M :Individual Series
X l/x 5 0.2000
15 0.0666 25 0.0400 35 0.0286 45 0.0222 55 0.0182
∑(1/x)=0.3756
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9744.153576.06
11
=
=
=∑ =
n
ii
H
x
nx
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116
Compute AM ,GM and HM for the numbers 6,8,12,36
AM=(6+81+12++36)/4=15.50 GM=(6.8.12.36)1/4=12
H.M=9.93
361
121
81
61
4.+++
=MH
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Weighted Harmonic Mean
∑∑=
)(i
i
i
Xww
HM
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118
Find the weighted AM and HM of first n natural numbers ,the weights being equal to the squares of the Corresponding numbers.
X 1 2 3 …n
W 12 22 32 ..n2
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Weighted ∑∑=
WiXiWi
AM.
)12(2)1(3
++
=nnn
++
+
=
++++
++++
6)12)(1(
4)1(.....321.....321
22
2222
3333
nnn
nnnn
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∑∑=
)(i
i
i
Xww
HM
312
2)1(
6)12)(1(
.....321.....321 23222
+=
+
++
=
++++
++++
n
nn
nnnnn
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121
The AM and GM of two observations are 5 and 4 respectively ,Find the two observations.
Solution : Let the Two numbers are a and b given
( a+b)/2=10 ;a + b=10 GM=4 ab=16 (a-b)2=(a+b)2-4ab=100-64=36
a-b=6 a=8 and b=2
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The relationship between AM ,GM and HM
G2=A.H
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1.The empirical relationship among mean, median and mode is ______
(a) mode=2median–3mean (b) mode=3median-2mean (c) mode=3mean-2median (d) mode=2mean-3median
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1. The empirical relationship among mean, median and mode is ______
(a) mode=2median–3mean (b) mode=3median-2mean (c) mode=3mean-2median (d) mode=2mean-3median
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2. In a asymmetrical distribution ____ (a) AM = GM = HM (b) AM<GM<AM (c) AM<GM>HM (d)
HMGMAM ≠≠
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2. In a asymmetrical distribution ____
(a) AM = GM = HM (b) AM<GM<AM (c) AM<GM>HM (d)
HMGMAM ≠≠
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3. The points of intersection of the “less than and more than” ogive corresponds to ___
(a) mean (b) mode (c) median (d) all of above
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.3. The points of intersection of the “less than and more than” ogive corresponds to ___
(a) mean (b) mode (c) median (d) all of above
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•4. Pooled mean is also called
(a) mean (b) geometric mean (c) grouped mean (d) none of these
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4. Pooled mean is also called
(a) mean (b) geometric mean (c) grouped mean (d) none of these
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5. Relation between mean, median and mode is
(a) mean–mode=2(mean-median) (b) mean–median=3(mean–mode)
(c) mean–median=2(mean–mode
(d) mean–mode=3(mean–median)
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5. Relation between mean, median and mode is
(a) mean–mode=2(mean-median) (b) mean–median=3(mean–mode)
(c) mean–median=2(mean–mode
(d) mean–mode=3(mean–median)
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6. The geometric mean of 9, 81, 729 is _____
(a) 9 (b) 27 (c) 81 (d) none of these
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6. The geometric mean of 9, 81, 729 is _____
(a) 9 (b) 27 (c) 81 (d) none of these
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7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be _____
(a) 4 (b) 5 (c) 6 (d) 7
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7. The mean of the data set of 1000 items is 5. From each item 3 is subtracted and then each number is multiplied by 2. The new mean will be
(a) 4 (b) 5 (c) 6 (d) 7
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8. If each item is reduced by 15, AM is ____
(a) reduced by 15 (b) increased by 15 (c) reduced by 10 (d) none of these
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8. If each item is reduced by 15, AM is ____
(a) reduced by 15 (b) increased by 15 (c) reduced by 10 (d) none of these
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9. In a series of values if one value is 0 ____
(a) both GM and HM are zero (b) both GM and HM are intermediate (c) GM is intermediate and HM is zero (d) GM is zero and HM is intermediate
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9. In a series of values if one value is 0 ____
(a) both GM and HM are zero (b) both GM and HM are intermediate (c) GM is intermediate and HM is zero (d) GM is zero and HM is intermediate
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10.Histogram is useful to determine graphically the value of
(a) Mean (b) Mode (c) Median (d) all of above
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10.Histogram is useful to determine graphically the value of
(a) Mean (b) Mode (c) Median (d) all of above
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11.The positional measure of central Tendency
(a) Arithmetic Mean (b) Geometric Mean (c) Harmonic Mean (d) Median
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11.The positional measure of central Tendency
(a) Arithmetic Mean (b) Geometric Mean (c) Harmonic Mean (d) Median
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12.The average has relevance for (a) Homogeneous population (b) Heterogeneous population (c) Both (d) none
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12.The average has relevance for (a) Homogeneous population (b) Heterogeneous population (c) Both (d) none
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13.The sum of individual observations is Zero When taken from
(a) Mean (b) Mode (C) Median (d) All the above
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13.The sum of individual observations is Zero When taken from
(a) Mean (b) Mode (C) Median (d) All the above
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14.The sum of absolute deviations from median is
(a) Minimum (b) Zero (C) Maximum (d) A negative figure
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14.The sum of absolute deviations from median is
(a) Minimum (b) Zero (C) Maximum (d) A negative figure
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15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
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15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
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16.The calculation of Speed and velocity
(a) G.M (b) A.M (c) H.M (d) none is used
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16.The calculation of Speed and velocity
(a)G.M (b)A.M (c)H.M (d)none is used
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17. The class having maximum frequency is called
A) Modal class B) Median class C) Mean Class D) None of these
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17. The class having maximum frequency is called
A) Modal class B) Median class C) Mean Class D) None of these
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18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is
A) 7 B) 10 C) 15 D) 7 and 15
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18. The mode of the numbers 7, 7, 9, 7, 10, 15, 15, 15, 10 is
A) 7 B) 10 C) 15 D) 7 and 15
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19. Which of the following measures of central tendency is based on only 50% of the central values?
A) Mean B) Mode C) Median D) Both (a) and (b)
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19. Which of the following measures of central tendency is based on only 50% of the central values?
A) Mean B) Mode C) Median D) Both (a) and (b)
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20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?
A) 17 B) 16 C) 15.75 D) 12
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20. What is the value of the first quartile for observations 15, 18, 10, 20, 23, 28, 12, 16?
A) 17 B) 16 C) 15.75 D) 12
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21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is
A) 13 B) 10.70 C) 11 D) 11.50
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21. The third decile for the numbers 15, 10, 20, 25, 18, 11, 9, 12 is
A) 13 B) 10.70 C) 11 D) 11.50
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22. In case of an even number of observations which of the following is median?
A) Any of the two middle-most value.. B) The simple average of these two
middle values C) The weighted average of these two
middle values. D) Any of these
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22. In case of an even number of observations which of the following is median?
A) Any of the two middle-most value.. B) The simple average of these two middle
values C) The weighted average of these two middle
values. D) Any of these
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23. A variable is known to be _______ if it can assume any value from a given interval.
A) Discrete B) Continuous C) Attribute D) Characteristic
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23. A variable is known to be _______ if it can assume any value from a given interval.
A) Discrete B) Continuous C) Attribute D) Characteristic
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24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these
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24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these
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25. The presence of extreme observations does not affect
A) A.M. B) Median C) Mode D) Any of these
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25. The presence of extreme observations does not affect
A) A.M. B) Median C) Mode D) Any of these
THE END
Measures Of Central Tendency