Measures of Central Tendency
Measures Of Central Tendency
Quantitative Aptitude & Business Statistics 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.
Quantitative aptitude & Business 2 Statistics: Measures Of Central
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 ∑ xi + + + + X = x1 x2 x3 ...... xn = i=1 n n
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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 ∑ fi xi f1x1 + f 2 x2 + f3 x3 +...... + f n xn i=1 X = = n f1 + f2 + f3 + .... + fn ∑ fi i=1
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Arithmetic Mean for Continuous Series
fd X = A + ∑ ×C N
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Calculation of Arithmetic mean in case of Continuous Series From the following data calculate Arithmetic mean
Marks 0- 10- 20- 30- 40- 50- 10 20 30 40 50 60 No. of 10 20 30 50 40 30 Students
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Marks Mid No.of d= X-45 f.d values Students 10 (X) (f) 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 No.of d= X-45 f.d values Students 10 (X) (f) 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 − fd −180*10 x = A + ∑ ×C = 45 + N 180 Mean = = 35 Quantitative aptitude & Business 19 Statistics: Measures Of Central
Calculation Of Arithmetic Mean in case of Less than series
Marks 10 20 30 40 50 60 less than /up to No. of 10 30 60 110 150 180 students
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Solution: Let us first convert Less than series into continuous series as follows
Marks 0-10 10- 20- 30- 40- 50-60 20 30 40 50 No. of 10 20 30 50 40 30 students 180- 150=30
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Calculation Of Arithmetic Mean in case of more than series
Marks 0 10 20 30 40 50 60 more than No. of 180 170 150 120 70 30 0 students
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Solution: Let us first convert More than series into continuous series as follows Marks 0-10 10- 20- 30- 40-50 50- 20 30 40 60 No. of 10 20 30 50 40 30 students
180-170=10 170-150=20 30-0=30
70-30=40
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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- 31- 41- 51- 30 40 50 60 No. of 10 20 30 50 40 30 Students
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Solution
Let us take assumed mean =45.5 Calculation from assumed mean
− fd −180*10 x = A + ∑ ×C = 45 + Mean = N 180 = 35 Quantitative aptitude & Business 25 Statistics: Measures Of Central
Marks Mid No.of d=X-45.5 f.d values Students 10
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 Quantitative aptitude & Business 26 Statistics: Measures Of Central
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 10 60 50 40 20 Students
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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
− fd − 220X10 Mean= x = A + ∑ ×C = 45 + N 180 = 32.778
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Marks Mid No. of d= X-45.5 f.d values Students 10
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 X = 30; X = 15 1 2 Step2: N X + N X = 1 1 2 2 X 12 N1 + N 2
=20
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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 ∑W.X ∑ X w = ∑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. ∑(X − X ) = 0 2.The sum of squared deviations of the items from arithmetic mean is minimum or 2 the least ∑(X − X ) ≤ 0
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3.The formula of Arithmetic mean can be extended to compute the combined average of two or more related series Quantitative aptitude & Business 37 Statistics: Measures Of Central
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. Quantitative aptitude & Business 41 Statistics: Measures Of Central
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 Median
Mean
Mean 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
N N +1 Median (M)=Average of= + 2 2 Item
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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
N +1 Median (M)=Size of = Item 2
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Median for continuous series N − m M = L + 2 × c f 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 Quantitative aptitude & Business 54 Statistics: Measures Of Central
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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N +1 Q = Size th Item 1 4 3(N +1) Q = Size th Item 3 4
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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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j(N +1) O j = Size thItem 8 4(N +1) O4 = Size th Item 8
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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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j(N +1) D = Size th j 10 Item 5(N +1) D = Size th 5 10 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) Quantitative aptitude & Business 66 Statistics: Measures Of Central
j(N +1) P = Size th j 100 Item 50(N +1) p = Size th 50 100 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 From the following data calculate Median
Marks 0-10 10-20 20-30 30-40 40-50 50- 60
No. of 10 20 30 50 40 30 Students
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Marks No. of Cumulative Students Frequencies (f) (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 Quantitative aptitude & Business 70 Statistics: Measures Of Central
Calculate size of N/2 N 180 = = 90 2 2
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180 − 60 M = 30 + 2 ×10 50
M = 30 + 6 = 36
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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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Median By Graph Y
3N/4 Less than Cumulative N/2 curve Frequency N/4 More than Cumulative Curve
X 0 Q1 M Q3 CI Quantitative aptitude & Business 75 Statistics: Measures Of Central
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 Mode = 9 No Mode
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
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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
f1 − f0 Z = L + ×c 2 f1 − f0 − f2
Where 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 From the following data calculate Mode
Marks 0- 10- 20- 30- 40- 50- 10 20 30 40 50 60 No. of 10 20 30 50 40 30 Students
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Marks No. of Students (f) 0-10 10 10-20 20
20-30 30 f0 30-40 50 f1 40-50 40 f2 50-60 30 N=180 Quantitative aptitude & Business 82 Statistics: Measures Of Central
f1 − f0 Z = L + ×c 2 f1 − f0 − f2 50 − 60 Z = 30 + ×10 2×50 − 30 − 40 = 30 + 6.667 = 36.667 Quantitative aptitude & Business 83 Statistics: Measures Of Central
Y Calculation Mode Graphically 50
40
30
20
10
60 10 20 40 50 x 0 30 Z Quantitative aptitude & Business 84 Statistics: Measures Of Central
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
n xG = x1x2 xi xn 1/ n n = ∏ xi i=1
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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 log X ∑ G.M = Anti log n
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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 f .log X ∑ G.M = Anti log N Quantitative aptitude & Business 95 Statistics: Measures Of Central
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
Quantitative aptitude &∑ Businesslog X=8.2107101 Statistics: Measures Of Central
log X ∑ G.M = Anti log n 8.2107 = Al 6 = Anti log(1.3685) = 23.36
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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 logx the end of the decade 1 10 110 2.0414 2 20 120 2.0792 3 30 130 2.1139
∑log X=6.2345
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log X ∑ G.M = Antilog n 6.2345 = Al = Antilog(2.0782) =119.8 Average Rate of increase in Population is 19.8%
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Weighted Geometric Mean
w.log X = ∑ G.M Antilog ∑ w
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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, N H.M = 1 1 1 1 + + + ...... + X 1 X 2 X 3 X n
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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 1 2 3 4 5 6 No Mark 5 15 25 35 45 55 s
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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 Quantitative aptitude & Business 114 Statistics: Measures Of Central
n xH = n 1 ∑i=1 xi 6 = 0.3576 =15.9744
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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 4 H.M = 1 1 1 1 H.M=9.93 + + + 6 8 12 36
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Weighted Harmonic Mean
∑ wi HM = w ∑( i ) X i
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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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∑Wi.Xi 13 + 23 + 33 + ..... + n 3 Weighted AM = ∑Wi 12 + 22 + 32 + ..... + n 2 n 2 (n +1) 2 4 = n(n +1)(2n +1) 6 3n(n +1) = 2(2n +1)
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2 2 2 23 ∑ wi 1 + 2 + 3 + ..... + n HM = w 1+ 2 + 3 + ..... + n ∑( i ) X n(n +1)(2n +1) i 6 = n(n +1) 2 2n +1 = 3 Quantitative aptitude & Business 120 Statistics: Measures Of Central
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
2. In a asymmetrical distribution ____ (a) AM = GM = HM (b) AM
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 Quantitative aptitude & Business 135 Statistics: Measures Of Central
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
Quantitative aptitude & Business 146 Statistics: Measures Of Central
13.The sum of individual observations is Zero When taken from (a) Mean (b) Mode (C) Median (d) All the above
Quantitative aptitude & Business 147 Statistics: Measures Of Central
13.The sum of individual observations is Zero When taken from (a) Mean (b) Mode (C) Median (d) All the above
Quantitative aptitude & Business 148 Statistics: Measures Of Central
14.The sum of absolute deviations from median is (a) Minimum (b) Zero (C) Maximum (d) A negative figure
Quantitative aptitude & Business 149 Statistics: Measures Of Central
14.The sum of absolute deviations from median is (a) Minimum (b) Zero (C) Maximum (d) A negative figure
Quantitative aptitude & Business 150 Statistics: Measures Of Central
15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
Quantitative aptitude & Business 151 Statistics: Measures Of Central
15.The mean of first natural numbers (a)n/2 (b)n-1/2 (c)(n+1)/2 (d) none
Quantitative aptitude & Business 152 Statistics: Measures Of Central
16.The calculation of Speed and velocity (a) G.M (b) A.M (c) H.M (d) none is used
Quantitative aptitude & Business 153 Statistics: Measures Of Central
16.The calculation of Speed and velocity (a)G.M (b)A.M (c)H.M (d)none is used
Quantitative aptitude & Business 154 Statistics: Measures Of Central
17. The class having maximum frequency is called A) Modal class B) Median class C) Mean Class D) None of these
Quantitative aptitude & Business 155 Statistics: Measures Of Central
17. The class having maximum frequency is called A) Modal class B) Median class C) Mean Class D) None of these
Quantitative aptitude & Business 156 Statistics: Measures Of Central
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
Quantitative aptitude & Business 157 Statistics: Measures Of Central
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
Quantitative aptitude & Business 158 Statistics: Measures Of Central
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)
Quantitative aptitude & Business 159 Statistics: Measures Of Central
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)
Quantitative aptitude & Business 160 Statistics: Measures Of Central
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
Quantitative aptitude & Business 161 Statistics: Measures Of Central
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
Quantitative aptitude & Business 162 Statistics: Measures Of Central
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
Quantitative aptitude & Business 163 Statistics: Measures Of Central
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
Quantitative aptitude & Business 164 Statistics: Measures Of Central
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
Quantitative aptitude & Business 165 Statistics: Measures Of Central
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
Quantitative aptitude & Business 166 Statistics: Measures Of Central
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
Quantitative aptitude & Business 167 Statistics: Measures Of Central
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
Quantitative aptitude & Business 168 Statistics: Measures Of Central
24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these
Quantitative aptitude & Business 169 Statistics: Measures Of Central
24. Ogive is used to obtain. A) Mean B) Mode C) Quartiles D) All of these
Quantitative aptitude & Business 170 Statistics: Measures Of Central
25. The presence of extreme observations does not affect A) A.M. B) Median C) Mode D) Any of these
Quantitative aptitude & Business 171 Statistics: Measures Of Central
25. The presence of extreme observations does not affect A) A.M. B) Median C) Mode D) Any of these
Quantitative aptitude & Business 172 Statistics: Measures Of Central
THE END
Measures Of Central Tendency