Revision Of All Topics
Quantitative Aptitude & Business Statistics
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.
Quantitative Aptituide & Business 2 Statistics: Revision Of All Toics Averages are central part of distribution and, therefore ,they are also called measures of central tendency.
Quantitative Aptituide & Business 3 Statistics: Revision Of All Toics 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)
Quantitative Aptituide & Business 4 Statistics: Revision Of All Toics
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.
Quantitative Aptituide & Business 5 Statistics: Revision Of All Toics • The arithmetic mean is simply dividing the sum of variables by the total number of observations.
Quantitative Aptituide & Business 6 Statistics: Revision Of All Toics Arithmetic Mean for Ungrouped data is given by n ∑ xi + + + + X = x1 x2 x3 ...... xn = i=1 n n
Quantitative Aptituide & Business 7 Statistics: Revision Of All Toics Arithmetic Mean for Discrete Series
n ∑ fi xi f1x1 + f 2 x2 + f 3 x3 +...... + f n xn i=1 X = = n f1 + f2 + f3 + .... + fn ∑ fi i=1
Quantitative Aptituide & Business 8 Statistics: Revision Of All Toics Arithmetic Mean for Continuous Series
fd X = A + ∑ ×C N
Quantitative Aptituide & Business 9 Statistics: Revision Of All Toics 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
Quantitative Aptituide & Business 10 Statistics: Revision Of All Toics • Weighted Arithmetic Mean is specially useful in problems relating to • 1)Construction of Index numbers. • 2)Standardised birth and death rates
Quantitative Aptituide & Business 11 Statistics: Revision Of All Toics • Weighted Arithmetic Mean is given by W.X = ∑ ∑W
Quantitative Aptituide & Business 12 Statistics: Revision Of All Toics 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 the least 2 ∑(X − X ) ≤ 0 Quantitative Aptituide & Business 13 Statistics: Revision Of All Toics • 3.The formula of Arithmetic mean can be extended to compute the combined average of two or more related series
Quantitative Aptituide & Business 14 Statistics: Revision Of All Toics
• 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 .
Quantitative Aptituide & Business 15 Statistics: Revision Of All Toics • Similarly When the value of the variable ‘X’ are multiplied by constant say k,arithmetic mean also multiplied the same quantity k .
Quantitative Aptituide & Business 16 Statistics: Revision Of All Toics • When the values of variable are divided by a constant say ‘d’ ,the arithmetic mean also divided by same quantity
Quantitative Aptituide & Business 17 Statistics: Revision Of All Toics 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
Quantitative Aptituide & Business 18 Statistics: Revision Of All Toics Median for raw data
• When given observation are even • First arrange the items in ascending order then
N N + 1 • Median (M)=Average of + Item 2 2
Quantitative Aptituide & Business 19 Statistics: Revision Of All Toics Median for raw data
• When given observation are odd • First arrange the items in ascending order then
• Median (M)=Size of N + 1 Item = 2
Quantitative Aptituide & Business 20 Statistics: Revision Of All Toics 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 frequenciesQuantitative Aptituide & Business 21 Statistics: Revision Of All Toics 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. Quantitative Aptituide & Business 22 Statistics: Revision Of All Toics 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. Quantitative Aptituide & Business 23 Statistics: Revision Of All Toics 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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 No Mode Quantitative Aptituide & Business 24 Statistics: Revision Of All Toics • 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. Quantitative Aptituide & Business 25 Statistics: Revision Of All Toics 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 Quantitative Aptituide & Business 26 Statistics: Revision Of All Toics
Relationship between Mean, Median and Mode • The distance between Mean and Median is about one third of distance between the mean and the mode.
Quantitative Aptituide & Business 27 Statistics: Revision Of All Toics 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
Quantitative Aptituide & Business 28 Statistics: Revision Of All Toics 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.
Quantitative Aptituide & Business 29 Statistics: Revision Of All Toics Geometric mean
n xG = x1x2 xi xn 1/ n n = ∏ xi i=1
Quantitative Aptituide & Business 30 Statistics: Revision Of All Toics 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 Aptituide & Business 31 Statistics: Revision Of All Toics Weighted Geometric Mean
w.log X = ∑ G.M Anti log ∑ w
Quantitative Aptituide & Business 32 Statistics: Revision Of All Toics • It is useful for averaging ratios and percentages rates are increase or decrease
Quantitative Aptituide & Business 33 Statistics: Revision Of All Toics 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
Quantitative Aptituide & Business 34 Statistics: Revision Of All Toics • 4.It is useful for averaging measuring the time ,Speed etc
Quantitative Aptituide & Business 35 Statistics: Revision Of All Toics Quartiles
• N + 1 Q = Size th Item 1 4 3(N +1) Q = Size th Item 3 4
Quantitative Aptituide & Business 36 Statistics: Revision Of All Toics Octiles
• j(N +1) O j = Size thItem 8 4(N +1) O = Size th Item 4 8
Quantitative Aptituide & Business 37 Statistics: Revision Of All Toics Deciles
• j(N +1) D = Size th j 10 Item 5(N +1) D = Size th 5 10 Item
Quantitative Aptituide & Business 38 Statistics: Revision Of All Toics 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
Quantitative Aptituide & Business 39 Statistics: Revision Of All Toics
Measures of Dispersion Why Study Dispersion?
• An average, such as the mean or the median only locates the centre of the data. • An average does not tell us anything about the spread of the data.
Quantitative Aptituide & Business 41 Statistics: Revision Of All Toics Properties of Good Measure of Dispersion • Simple to understand and easy to calculate • Rigidly defined • Based on all items • A meanable to algebraic treatment • Sampling stability • Not unduly affected by Extreme items.
Quantitative Aptituide & Business 42 Statistics: Revision Of All Toics Absolute Measure of Dispersion
Based on selected Based on all items items
1.Range 1.Mean Deviation 2.Inter Quartile Range 2.Standard Deviation
Quantitative Aptituide & Business 43 Statistics: Revision Of All Toics
Relative measures of Dispersion
Based on Based on Selected all items items
1.Coefficient of 1.Coefficient of MD Range 2.Coefficient of SD & Coefficient of Variation 2.CoefficientQuantitative of Aptituide & Business 44 QD Statistics: Revision Of All Toics The Range
• The simplest measure of dispersion is the range. • For ungrouped data, the range is the difference between the highest and lowest values in a set of data.
Quantitative Aptituide & Business 45 Statistics: Revision Of All Toics • RANGE = Highest Value - Lowest Value
Quantitative Aptituide & Business 46 Statistics: Revision Of All Toics Coefficient of Range
• Coefficient of Range =
L − S L + S
Quantitative Aptituide & Business 47 Statistics: Revision Of All Toics Interquartile Range
• The interquartile range is used to overcome the problem of outlying observations. • The interquartile range measures the range of the middle (50%) values only
Quantitative Aptituide & Business 48 Statistics: Revision Of All Toics • Inter quartile range = Q3 – Q1 • It is sometimes referred to as the quartile deviation or the semi-inter quartile range.
Quantitative Aptituide & Business 49 Statistics: Revision Of All Toics Lower Quartile Deviation
N − c.f 4 Q1 = L + × C f
Quantitative Aptituide & Business 50 Statistics: Revision Of All Toics Upper Quartile Deviation
N 3. − c. f Q = L + 4 ×C 3 f
Quantitative Aptituide & Business 51 Statistics: Revision Of All Toics • Inter Quartile Range=Q3-Q1
• Coefficient of Quartile Deviation
Q − Q = 3 1 Q3 + Q1
Quantitative Aptituide & Business 52 Statistics: Revision Of All Toics Mean Deviation
• The mean deviation takes into consideration all of the values • Mean Deviation: The arithmetic mean of the absolute values of the deviations from the arithmetic mean Quantitative Aptituide & Business 53 Statistics: Revision Of All Toics ∑ x − x MD = n Where: X = the value of each observation X = the arithmetic mean of the values n = the number of observations || = the absolute value (the signs of Quantitative Aptituide & Business 54 the deviationsStatistics: are Revision disregarded) Of All Toics Frequency Distribution Mean Deviation • If the data are in the form of a frequency distribution, the mean deviation can be calculated using the following formula:
_ f | x − x | MD = ∑ ∑ f Where f = the frequency of an observation x n = Σf = the sum of the frequencies Quantitative Aptituide & Business 55 Statistics: Revision Of All Toics Standard Deviation
• Standard deviation is the most commonly used measure of dispersion • Similar to the mean deviation, the standard deviation takes into account the value of every observation
Quantitative Aptituide & Business 56 Statistics: Revision Of All Toics 2 _ ∑ x − x σ = N
Quantitative Aptituide & Business 57 Statistics: Revision Of All Toics 2 f.x2 f.x σ = ∑ − ∑ N N
2 f.x 2 = ∑ − (X) N
Quantitative Aptituide & Business 58 Statistics: Revision Of All Toics In a normal distribution there is fixed relationship 2 QD = σ 3 4 MD = σ 5
Thus SD is never less than QD and MD
Quantitative Aptituide & Business 59 Statistics: Revision Of All Toics Properties of Standard Deviation
• Independent of change of origin • Not independent of change of Scale. • Fixed Relationship among measures of Dispersion.
Quantitative Aptituide & Business 60 Statistics: Revision Of All Toics Bell - Shaped Curve showing the relationsh ip between σ and µ.
68.27%
95.45%
99.73%
µ−3σ µ−2σ µ−1σ µ µ+1σ µ+2σ µ+ 3σ Quantitative Aptituide & Business 61 Statistics: Revision Of All Toics • Minimum sum of Squares; The Sum of Squares of Deviations of items in the series from their arithmetic mean is minimum. • Standard Deviation of n natural numbers N 2 −1 = 12 Quantitative Aptituide & Business 62 Statistics: Revision Of All Toics • Combined standard deviation
2 2 2 2 N 1σ1 + N 2σ2 + N 1d 1 + N 2d 2 σ12 = N 1 + N 2
Quantitative Aptituide & Business 63 Statistics: Revision Of All Toics • Where σ 12 =Combined standard Deviation of two groups σ • 1 =Standard Deviation of first group • N1=No. of items of First group • N2=No. of items of Second group • σ 2 = Standard deviation of Second group
Quantitative Aptituide & Business 64 Statistics: Revision Of All Toics
d 1 = X1 − X12 d 2 = X2 − X12
Where X 12 is the combined mean of two groups
Quantitative Aptituide & Business 65 Statistics: Revision Of All Toics Limitations of Standard Deviation • It can’t be used for comparing the variability of two or more series of observations given in different units. A coefficient of Standard deviation is to be calculated for this purpose. • It is difficult to compute and Quantitative Aptituide & Business 66 comparedStatistics: Revision Of All Toics Variance
• Variance is the arithmetic mean of the squares of deviations of all the items of the distributions from arithmetic mean .In other words, variance is the square of the Standard deviation= 2 σ
• Variance= σ = var iance
Quantitative Aptituide & Business 67 Statistics: Revision Of All Toics Interpretation of Variance
• Smaller the variance ,greater the uniformity in population. • Larger the variance ,greater the variability
Quantitative Aptituide & Business 68 Statistics: Revision Of All Toics The Coefficient of Variation
• The coefficient of variation is a measure of relative variability It is used to measure the changes that have taken place in a population over time
Quantitative Aptituide & Business 69 Statistics: Revision Of All Toics • Formula: σ Where: CV = ×100 X
X = mean σ = standard deviation
Quantitative Aptituide & Business 70 Statistics: Revision Of All Toics Correlation
• Correlation is the relationship that exists between two or more variables. • If two variables are related to each other in such a way that change increases a corresponding change in other, then variables are said to be correlated.
Quantitative Aptituide & Business 71 Statistics: Revision Of All Toics Methods of studying correlation
Method of studying Correlation
Graphic Algebraic
1.Karl Pearson Scatter Diagram 2.Rank method
Method Quantitative Aptituide &3.Concurrent Business Deviation72 Statistics: Revision Of All Toics Scatter Diagram Method
• Scatter diagrams are used to demonstrate correlation between two quantitative variables.
Quantitative Aptituide & Business 73 Statistics: Revision Of All Toics Scatter Plots of Data with Various Correlation Coefficients Y Y Y
X X X r = -1 r = -Ve r = 0 Y Y
X X Quantitative Aptituide & Business 74 r = +VeStatistics: Revision Of All Toicsr = 1 The value of r lies between - 1 and +1 • If r=0 There exists no relationship between the variables • If +0.75 ≤r ≤ +1 There exists high positive relationship between the variables . • If -0.75 ≥ r ≥ -1 There exists high negative relationship between the variables
Quantitative Aptituide & Business 75 Statistics: Revision Of All Toics
• If +0.5 ≤r ≤ 0.75 There exists Moderate positive relationship between the variables . • If -0.50 ≥ r >-0.75 There exists moderate negative relationship between the variables. • If r > -0.50 There exists low negative relationship between the variables • If r <0.5 There exists low positive relationship between the variables .
Quantitative Aptituide & Business 76 Statistics: Revision Of All Toics
• If +0.5 ≤r ≤ 0.75 There exists Moderate positive relationship between the variables . • If -0.50 ≥ r >-0.75 There exists moderate negative relationship between the variables. • If r > -0.50 There exists low negative relationship between the variables • If r <0.5 There exists low positive relationship between the variables .
Quantitative Aptituide & Business 77 Statistics: Revision Of All Toics
Covariance
• Definition : Given a n pairs of
observations (X1,Y1),(X2,Y2) .,,,,,, (Xn,Yn) relating to two variables X and Y ,the Covariance of X and Y is usually represented by Cov(X,Y)
(X − X ).(Y −Y ) Cov(X ,Y ) = ∑ N xy = ∑ Quantitative Aptituide & Business 78 Statistics:N Revision Of All Toics Properties of Co-Variance
• Independent of Choice of origin • not Independent of Choice of Scale. • Co-variance lies between negative infinity to positive infinity. • In other words co-variance may be positive or negative or Zero.
Quantitative Aptituide & Business 79 Statistics: Revision Of All Toics Coefficient of Correlation
• Measures the strength of the linear relationship between two quantitative variables
n
( Xii−− XYY)( ) ∑ r = i=1 nn 22 ∑∑( Xii−− X) ( YY) ii=11=
Quantitative Aptituide & Business 80 Statistics: Revision Of All Toics Properties of KralPear son’s Coefficient of Correlation • Independent of choice of origin • Independent of Choice Scale • Independent of units of Measurement
Quantitative Aptituide & Business 81 Statistics: Revision Of All Toics Assumptions of Karl Pearson’s Coefficient of Correlation • Linear relationship between variables. • Cause and effect relationship. • Normality.
Quantitative Aptituide & Business 82 Statistics: Revision Of All Toics • The correlation coefficient lies between -1 and +1 • The coefficient of correlation is the geometric mean of two regression coefficients.
Quantitative Aptituide & Business 83 Statistics: Revision Of All Toics • The correlation coefficient lies between -1 and +1 • The coefficient of correlation is the geometric mean of two regression coefficients.
Quantitative Aptituide & Business 84 Statistics: Revision Of All Toics Correlation for Bivariate analysis
(∑ f .dx )(∑ f .d y ) ∑ fdx.d y − r = N 2 2 2 (∑ f .dx) 2 (∑ f .dx) ∑ f .dx − ∑ f .d y − N N
Quantitative Aptituide & Business 85 Statistics: Revision Of All Toics Standard error
• Standard error of co efficient of correlation is used foe ascertaining the probable error of coefficient of correlation • Where r=Coefficient of correlation • N= No. of Pairs of observations 1− r 2 SE = N Quantitative Aptituide & Business 86 Statistics: Revision Of All Toics Probable Error
• The Probable error of coefficient of correlation is an amount which if added to and subtracted from value of r gives the upper and lower limits with in which coefficients of correlation in the population can be expected to lie. It is 0.6745 times of standard error.
Quantitative Aptituide & Business 87 Statistics: Revision Of All Toics Uses of Probable Error
• PE is used to for determining reliability of the value of r in so far as it depends on the condition of random sampling.
Quantitative Aptituide & Business 88 Statistics: Revision Of All Toics Case Interpretation 1.If |r |< 6 PE The value of r is not at all significant. There is no evidence of correlation. 2. 1.If |r | >6 PE The value of r is significant. There is evidence of correlation
Quantitative Aptituide & Business 89 Statistics: Revision Of All Toics Spearman’s Rank Correlation
Spearman’s Rank Correlation uses ranks than actual observations and make no assumptions about the population from which actual observations are drawn. 6∑d 2 r = 1− n(n 2 −1)
Quantitative Aptituide & Business 90 Statistics: Revision Of All Toics Spearman’s Rank Correlation for repeated ranks • Where m=the no of times ranks are repeated • n=No of observations • r= Correlation Coefficient
3 2 m − m 6∑ D + + ..... 12 r = 1− n(n 2 −1)
Quantitative Aptituide & Business 91 Statistics: Revision Of All Toics Features of Spearman’s Rank Correlation • Spearman’s Correlation coefficient is based on ranks rather than actual observations . • Spearman’s Correlation coefficient is distribution –free and non-parametric because no strict assumptions are made about the form of population from which sample observation are drawn. Quantitative Aptituide & Business 92 Statistics: Revision Of All Toics
Merits of Spearman’s Rank Correlation • Simple to understand and easy to apply • Suitable for Qualitative Data • Suitable for abnormal data. • Only method for ranks • Appliacble even for actual data.
Quantitative Aptituide & Business 93 Statistics: Revision Of All Toics Limitations of Spearman’s Rank Correlation • Unsuitable data • Tedious calculations • Approximation
Quantitative Aptituide & Business 94 Statistics: Revision Of All Toics When is used Spearman’s Rank Correlation method • The distribution is not normal • The behavior of distribution is not known • only qualitative data are given
Quantitative Aptituide & Business 95 Statistics: Revision Of All Toics Meaning of Concurrent Deviation Method • Concurrent Deviation Method is based on the direction of change in the two paired variables .The coefficient of Concurrent Deviation between two series of direction of change is called coefficient of Concurrent Deviation .
Quantitative Aptituide & Business 96 Statistics: Revision Of All Toics • rc=Coefficient of Concurrent deviation • C= no of positive signs after multiplying the change direction of change of X- series and Y-Series • n=no. of pairs of observations computed 2c − n rc = ± ±
Quantitative Aptituide & Business n 97 Statistics: Revision Of All Toics Limitations of Concurrent Deviation Method • This method does not differentiate between small and big changes . • Approximation
Quantitative Aptituide & Business 98 Statistics: Revision Of All Toics Merits of Concurrent Deviation
• Simple to understand and easy to calculate. • Suitable for large N
Quantitative Aptituide & Business 99 Statistics: Revision Of All Toics Regression • Regression is the measure of average relationship between two or more variables in terms of original units of the data.
Quantitative Aptituide & Business 100 Statistics: Revision Of All Toics Regression lines
• Regression line X on Y X = a + bY
• Where X= Dependent Variable Y =Independent variable a=intercept and b= slope
Quantitative Aptituide & Business 101 Statistics: Revision Of All Toics • Another way of regression line X on Y X − X = bxy (Y −Y )
σ X − X = r x (Y −Y )
σ y
Quantitative Aptituide & Business 102 Statistics: Revision Of All Toics
Calculate bxy
X Y XY− ∑ ∑ ∑ N bxy = 2 ( Y) Y2 − ∑ ∑ N a = X − bY
Quantitative Aptituide & Business 103 Statistics: Revision Of All Toics Regression coefficients
• There are two regression coefficients byx and bxy • The regression coefficient Y on X is
σy byx = r. σx
The regression coefficient X on Y is σx bxy = r. σy
Quantitative Aptituide & Business 104 Statistics: Revision Of All Toics Regression coefficients
The regression coefficient X on Y is
σx bxy = r. σy
Quantitative Aptituide & Business 105 Statistics: Revision Of All Toics • Regression line Y on X Y = a + bX
• Where Y= Dependent Variable • X =Independent variable • a=intercept and b= slope
Quantitative Aptituide & Business 106 Statistics: Revision Of All Toics • Another way of regression line Y on X Y −Y = byx(X − X )
σ Y −Y = r y X − X ( ) σ x
Quantitative Aptituide & Business 107 Statistics: Revision Of All Toics Calculate byx
∑X∑Y XY− ∑ N byx = 2 ( X) X2 − ∑ ∑ N a = Y − bX
Quantitative Aptituide & Business 108 Statistics: Revision Of All Toics Properties of Linear Regression • Two Regression Equations. • Product of regression coefficient. • Signs of Regression Coefficient and correlation coefficient. • Intersection of means.
• Slopes . Quantitative Aptituide & Business 109 Statistics: Revision Of All Toics • Angle between Regression lines Value of r Angle between Regression Lines a) If r=0 Regression lines are perpendicular to each other. b) If r=+1 or -1 Regression lines are coincide to become
Quantitativeidentical Aptituide & Business . 110 Statistics: Revision Of All Toics Properties of regression coefficients 1.Same Sign. 2.Both cannot greater than one . 3.Independent of origin but not of scale . 4.Arithmetic mean of regression coefficients are greater than Correlation coefficient. 5.r,bxy and byx have same sign. Quantitative Aptituide & Business 111 6 .Correlation Statistics:coefficient Revision Of All Toics is the
Independent of origin but not of scale. • This property states that if the original pairs of variables is (x,y) and if they are changed to the pair (u,v), where x=a + p
u and y=cx − a+q v q u = and buv = × bxy and or p p y − c q v = bvu = × byx q p
Quantitative Aptituide & Business 112 Statistics: Revision Of All Toics Measure of Variation: The Sum of Squares
SST = SSR + SSE Total = Unexplained Sample Explained + Variability Variability Variability
Quantitative Aptituide & Business 113 Statistics: Revision Of All Toics Measure of Variation: The Sum of Squares • SST = Total Sum of Squares
– Measures the variation of the Yi values around their mean Y • SSR = Regression Sum of Squares – Explained variation attributable to the relationship between X and Y • SSE = Error Sum of Squares – Variation attributable to factors other than the relationship Quantitative Aptituide & Business 114 between XStatistics: and Revision Y Of All Toics Coefficient of determination(r2)
• The coefficient of determination is the square of the coefficient of correlation. It is equal to r2. • The maximum value of r2 is unity and in the case of all the variation in Y is explained by the variation in X ,it is Explained var inace defined as = T otalVaria nce • Coefficient of determination( r2 )
Quantitative Aptituide & Business 115 Statistics: Revision Of All Toics
Coefficient of non- determination(k2) • Coefficient of non-determination(k2)=1- r2 Un exp lained var inace = TotalVaria nce
Quantitative Aptituide & Business 116 Statistics: Revision Of All Toics
Population or Universe refers to the aggregate of statistical information on a particular character of all the members covered by an investigation/enquiry. For example, constitute population.
Quantitative Aptituide & Business 117 Statistics: Revision Of All Toics SAMPLE
• Sample refers to the part of aggregate statistical information (i.e. Population) which is actually selected in the course of an investigation/enquiry to ascertain the characteristics of the population.
Quantitative Aptituide & Business 118 Statistics: Revision Of All Toics SAMPLE SIZE
• Sample size refers to the number of members of the population included in the sample. • Usually, the sample size is denoted by 'n'
Quantitative Aptituide & Business 119 Statistics: Revision Of All Toics METHODS OF SAMPLING
1.Deliberate, Purposive or Judgment Sampling. 2. Block or Cluster Sampling 3. Area Sampling 4. Quota Sampling 5. Random (or Probability) Sampling 6. Systematic Sampling
Quantitative Aptituide & Business 120 Statistics: Revision Of All Toics METHODS OF SAMPLING
• 7. Stratified Sampling • 8. Multi Stage Sampling
Quantitative Aptituide & Business 121 Statistics: Revision Of All Toics Standard Error Meaning • Standard Error of a given statistic is the standard deviation of sampling distribution of that statistic. In other words, standard error of a given 'statistic is the standard deviation of all possible values of that statistic in repeated sample of a fixed size from given populatiot.It is a measurer of the divergence between‘ the‘ statistic and parameter values. This divergence varies with the sample size (n). Quantitative Aptituide & Business 122 Statistics: Revision Of All Toics HOW TO COMPUTE STANDARD ERROR OF THE MEAN Statistic Standard Error
sample mean SRSWR σ X n
Quantitative Aptituide & Business 123 Statistics: Revision Of All Toics HOW TO COMPUTE STANDARD ERROR OF THE MEAN
Statistic Standard Error sample mean SRSWOR σ − X N n n N − 1
Quantitative Aptituide & Business 124 Statistics: Revision Of All Toics HOW TO COMPUTE STANDARD ERROR OF THE PROPORTION Statistic Standard Error Observed sample Proportion ‘P’ PQ SRS WR n
Quantitative Aptituide & Business 125 Statistics: Revision Of All Toics HOW TO COMPUTE STANDARD ERROR OF THE PROPORTION Statistic Standard Error
Observed sample PQ N − n Proportion ‘P’ . n N −1 SRS WOR
Quantitative Aptituide & Business 126 Statistics: Revision Of All Toics 1.Problem of Estimation
• This problem arises when no information is available about the parameters of the population from which the sample is drawn. Statistics obtained from the sample are used to estimate the unknown parameter of the population from which the sample is drawn. Quantitative Aptituide & Business 127 Statistics: Revision Of All Toics MEANING OF ESTIMATION
• In the context of statistics, an estimation is a statistical technique of estimating unknown population parameters from the corresponding sample statistic. Two ways - A population parameter ,can be estimated in two ways: • 1. Point Estimation, and • 2. Interval Estimation
Quantitative Aptituide & Business 128 Statistics: Revision Of All Toics POINT ESTIMATION
• It provides a single value of a statistic that is used to estimate an unknown population parameter. • Estimator -The statistic which is used to obtain a point estimate is called estimator. • Criteria for a Good Estimator - According to R A: Fisher, the criteria for good estimator are: • (a) Unbiased ness, (b) Consistency, (c) Efficiency and (d) Sufficiency Quantitative Aptituide & Business 129 Statistics: Revision Of All Toics INTERVAL ESTIMATION
• Provides an interval of finite width centered ,if the point estimate of the parameter, within which unknown parameter is expected to lie with a specified probability. Such an interval is called a confidence interval for population parameter.
Quantitative Aptituide & Business 130 Statistics: Revision Of All Toics • Confidence Limits - The lower and upper limits of the confidence interval are called confidence limits. • Confidence Coefficient – The probability with which the confidence interval will include the true value of the parameter.
Quantitative Aptituide & Business 131 Statistics: Revision Of All Toics Calculation of Confidence 'Limits
• The calculation of confidence limits is based on the appropriate statistic. If the population is normal or the sample size (n) is large (say more than 30), percentage of area Under the standard normal curve maybe used to find confidence limits) corresponding to any specified percentage of confidence.
Quantitative Aptituide & Business 132 Statistics: Revision Of All Toics Why hypothesis is used?
• Hypothesis Testing is a process of making a decision on whether to accept or reject an assumption about the population parameter on the basis of sample information at a given level of significance.
Quantitative Aptituide & Business 133 Statistics: Revision Of All Toics NULL HYPOTHESIS (Ho)
• Null hypothesis is the assumption which we wish to test and whose validity is tested for possible rejection on the basis of sample information. • It asserts that there is no significant difference between the sample statistic
Quantitative Aptituide & Business 134 Statistics: Revision Of All Toics NULL HYPOTHESIS (Ho)
• Acceptance - The acceptance of null hypothesis implies that we have no evidence to believe otherwise and indicates that the difference is not significant and is due to sampling fluctuations. • Rejection - The rejection of null hypothesis implies that it is false and indicates that the difference is significant.
Quantitative Aptituide & Business 135 Statistics: Revision Of All Toics ALTERNATIVE HYPOTHESIS(H1) • Alternative hypothesis is the hypothesis which differs from the null hypothesis. It is not tested • Rejection of one implies the acceptance of the other.
• Symbol - It is denoted by H1 • Acceptance - Its depends on the rejection of the null hypothesis. Rejection - Its rejection depends on the acceptance of the null hypothesis.
Quantitative Aptituide & Business 136 Statistics: Revision Of All Toics Example for ALTERNATIVE HYPOTHESIS(H1) • If population mean is 50, an alternative hypothesis may be anyone of the following three:
H1 : µ ≠50,H1 : µ > 50,H1: µ < 50 • Ho and HI are mutually exclusive statements in the sense that both cannot hold good simultaneously. • Rejection of one implies the acceptance of the other. Quantitative Aptituide & Business 137 Statistics: Revision Of All Toics LEVEL OF SIGNIFICANCE(α )
• Level of significance is the maximum probability of rejecting the null hypothesis when it is true. • Symbol - It is usually expressed as % and is denoted by symbol α (called 'Alpha ') • Usefulness - It is used as a guide in decision- making. It is used to indicate the upper limit of the size of the critical rejection.
Quantitative Aptituide & Business 138 Statistics: Revision Of All Toics Example of LEVEL OF SIGNIFICANCE(α ) • At 5% level of significance implies that there are about 5 chances in 100 of rejecting the Ho when it is true or in other words, we are about 95% confident that we will make a correct decision.
Quantitative Aptituide & Business 139 Statistics: Revision Of All Toics Usefulness
• It is used as a guide in decision making regarding acceptance or rejection of Ho' If the value of the test statistic falls in the critical region, the null hypothesis is rejected. If the value of the test statistic does not fall in the critical region, the null hypothesis is accepted.
Quantitative Aptituide & Business 140 Statistics: Revision Of All Toics TEST STATISTIC
Test-Statistic Used for i) Z-test For test of Hypothesis involving large sample i.e. n> 30 ii) t-test Hypothesis involving small sample i.e. n≤ 30 and if SD is unknown.
Quantitative Aptituide & Business 141 Statistics: Revision Of All Toics TEST STATISTIC
Test-Statistic Used for iii) Chi-square test Test For testing the discrepancy between observed frequencies and expected frequencies without any reference to population parameter iv) F-test Testing the sample variances. Quantitative Aptituide & Business 142 Statistics: Revision Of All Toics Two-Tailed Tests level of significance(α ) Two –tailed tests Level of significance α
Critical Acceptance Region Region (1-α) Critical Region
µ
Lower Critical Upper Critical Value Value Quantitative Aptituide & Business 143 Statistics: Revision Of All Toics • by one tail (or side) under the curve when the null hypothesis is tested against 'one sided alternative' right or left. The tests of hypothesis which are based on the critical region represented by one tail (on right side or on left side)
Quantitative Aptituide & Business 144 Statistics: Revision Of All Toics One-Tail Test Two-Tail One-Tail Test (left- tail) Test (right- tail) Acceptance Region
Quantitative Aptituide & Business 145 Statistics: Revision Of All Toics Critical Values of test- statistic
Level of Significance Types of tests 1% 5% 1.Two Tailed ± 2.58 ± 1.96 2.One tailed a. Right tailed + 2.33 + 1.645 b. Left tailed - 2.33 - 1.645
Quantitative Aptituide & Business 146 Statistics: Revision Of All Toics Type I and Type II Errors
• The decision to accept or reject null hypothesis Ho is made on the basis of the information supplied by the sample data. There is always a chance of committing an error. There are two possible types of error in the test of a hypothesis.
Quantitative Aptituide & Business 147 Statistics: Revision Of All Toics Type I and Type II Errors
• Type I Error - This is the error committed by the test in rejecting a true null hypothesis. The probability of committing Type I Error is denoted by α , the level of significance. . • Type II Error - This is the error committed by the test in accepting a false null hypothesis. The probability of committing Type II Error is denoted by β.
Quantitative Aptituide & Business 148 Statistics: Revision Of All Toics POWER OF THE TEST
• Power of the Test is the probability of rejecting a false null hypothesis. It can be calculated as follows: • Power of the Test = I-Probability of Type II Error.=1-β
Quantitative Aptituide & Business 149 Statistics: Revision Of All Toics Decision
Statement Reject H0 AcceptH0 H0 is True Wrong Correct (Type I Error)
H0 is False Correct Wrong (Type II Error)
Quantitative Aptituide & Business 150 Statistics: Revision Of All Toics Confidence Limits • The confidence limits of population parameter are calculated • 95% Confidence Limits 99% Confidence Limits are • Sample Statistic± 1.96 S. E. Sample Statistic ± 2.58 S. E.
Quantitative Aptituide & Business 151 Statistics: Revision Of All Toics TEST OF SIGNIFICANCE OF LARGE SAMPLES • A sample is regarded as large only if its size exceeds 30. • The following assumptions are made while dealing with problems relating to large samples: • The random sampling distribution of a statistic is approximately normal and Values given by the samples are approximately close to the population value. Quantitative Aptituide & Business 152 Statistics: Revision Of All Toics
Testing the significance of Mean of Random Sample If Population SD is known
X − µ Z = S.E.X X − µ = W here σ n X = SampleM ean µ = Population M ean σ = Population SD n = SampleSize Quantitative Aptituide & Business 153 Statistics: Revision Of All Toics Test for difference between the Standard deviations of two samples S − S Z = 1 2 σ2 σ2 1 + 2 2n1 2n 2
Ifσ1and σ2 areunknown Use
S1andS 2
Quantitative Aptituide & Business 154 Statistics: Revision Of All Toics Test for number of Success
p − P Z = npq
Quantitative Aptituide & Business 155 Statistics: Revision Of All Toics Test for Proportion of Success p − P Z = PQ n
Quantitative Aptituide & Business 156 Statistics: Revision Of All Toics Test for difference between the two Proportions P − P Z = 1 2 S.E Where P1 −P2 P − P = 1 2 1 1 PQ + n1 n2 n p + n p Pˆ = 1 1 2 2 n1 + n 2 Quantitative Aptituide & Business 157 Statistics: Revision Of All Toics
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