Download - Statistical Formula
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1 |NSG ACADEMY - 9823782121
Statistics Important Formula Statistical Data
Un-Grouped Data Grouped Data
Individual Data Discrete Frequency Distribution Continuous Frequency Distribution
Mean x = xn Where, n = Total Sample Size x = fxN Where, N = f
x = fxN Where, N = f
x = Class Mid Value Weighted Mean Not Applicable x = xw
x = xw
Where, x = Class Mid Value
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Trimmed MeanStep1: Sort all n values Step2: Calculate % Step3: Discard Step2 value from lower and upper end Step4: Find mean of remaining observation using above formula
Not Applicable Not Applicable
Combined Mean = + + Where,
= = = = Correcting Wrong Value for Mean
= +
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Effect of Change of Origin for Mean = , = = Where, A= Constant Note: Change of origin = Add or Subtract a constant number from each value of , . .
Effect of Change of Scale for Mean = , = =
= , = = + Note: Change of scale = Multiply or Divide a constant number to each value of , . . Median n = Total Sample Size Check If n = Even
N = f
Check If N = Even Median(M) = L + N2 + c. f.F h Where,
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Median(M)= n2 obsevation value + n2 + 1 observation value2 If n = Odd Median (M) = n + 12
Median(M)= N2 obsevation value + N2 + 1 observation value2 If N = Odd Median (M) = N + 12 N = f
L = Lower limit of Median clas c. f.= Cumulative Frequency prior to Median class F = frequency of Median class h = Class Width Mode Mode(M) = The value that occurs maximum number of times Mode(M) = The value with maximum freuency Mode(M) = L + f f2f f f h Where, N = f
L = Lower limit of Modal clas f = Frequency of Modal class f = Frequency of Prior Modal class f = Frequency of Post Modal class h = Class Width Quartiles
Lower Quartiles Q1 n = Total Sample Size N = f
Lower Quartile(Q) = L + N4 + c. f.F h
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Check If n = Even Lower Quartile(Q)= n4 obsevation value + n4 + 1 observation value2 If n = Odd Lower Quartile(Q) = n + 14
Check If N = Even Lower Quartile(Q)= N4 obsevation value + N4 + 1 observation value2 If N = Odd Lower Quartile(Q) = N + 14
Where, N = f
L = Lower limit of lower quartile class c. f.= Cumulative Frequency prior to lower quartile class F = frequency of lower quartile class h = Class Width Median Q2 n = Total Sample Size Check If n = Even Median(M/Q)= n2 obsevation value + n2 + 1 observation value2 If n = Odd Median(M/Q) = n + 12
N = f
Check If N = Even Median(M/Q)= N2 obsevation value + N2 + 1 observation value2 If N = Odd Median(M/Q) = N + 12
Median(M/Q) = L + N2 + c. f.F h Where, N = f
L = Lower limit of Median clas c. f.= Cumulative Frequency prior to Median class F = frequency of Median class h = Class Width
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Upper Quartile Q3 n = Total Sample Size Check If n = Even Upper Quartile(Q)= 3n4 obsevation value + 3n4 + 1 observation value2 If n = Odd Upper Quartile(Q) = 3(n + 1)4
N = f
Check If N = Even Upper Quartile(Q)= 3N4 obsevation value + 3N4 + 1 observation value2 If N = Odd Upper Quartile(Q) = 3(N + 1)4
Upper Quartile(Q) = L + 3N4 + c. f.F h Where, N = f
L = Lower limit of upper quartile clas c. f.= Cumulative Frequency prior to upper quartile class F = frequency of upper quartile class h = Class Width Decile iDecile(D) = L + iN10 + c. f.F h Where, = 1,2,3 . .9 N = f
L = Lower limit of i decile clas c. f.= Cumulative Frequency prior to i decile class F = frequency of i decile class h = Class Width
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Percentile iPercentile(P) = L + iN100 + c. f.F h Where, = 1,2,3 . .99 N = f
L = Lower limit of i percentile clas c. f.= Cumulative Frequency prior to i percentile class F = frequency of i percentile class h = Class Width Range
= = + Where, L = Largest Item S = Smallest Item
Quartile Deviation
. = 2 . = +
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Variance = 1( )
=
= 1 ( )
=
Where, N = f
= 1 ( )
=
Where, N = f
x = Class Mid Value Standard Deviation = 1( )
=
= 1 ( )
=
Where,
= 1 ( )
=
Where,
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N = f
N = f
x = Class Mid Value Combined Standard Deviation = ( + ) + ( + ) + Where,
= ; = ; = + + Combined Variance = ( + ) + ( + )
+ Where, = ; = ; = + +
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Coefficient of Variation .. = || 100
Effect of Change of Origin for Range, Standard Deviation and Variance = , & & . . ,& Where, A= Constant
Correcting Wrong Value for Standard Deviation and Variance
= ( ) + ( )
Empirical Relation Mean Mode 3 ( Mean Median )
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Skewness
Karl Pearsons coefficient of Skewness = = = 3( )
= 3( ) Interpretation: If > 0, the distribution is positively skewed If < 0, the distribution is negatively skewed If = 0, the distribution is symmetric Remark: The 1 + 1 . .1 +1 Bowleys coefficient of Skewness
= ( ) ( )( ) + ( ) = + 2 = + 2
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Interpretation: If > 0, the distribution is positively skewed If < 0, the distribution is negatively skewed If = 0, the distribution is symmetric Remark: The 1 + 1 . .1 +1 Kurtosis
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Moments
Raw Moments =
Where,
= 0,1,2, .. =
=
Where, = 0,1,2, .. =
=
Where, = 0,1,2, .. = =
Central Moments = ( )
Where,
= 0,1,2, .. =
= ( )
Where, = 0,1,2, .. =
= ( )
Where, = 0,1,2, .. = =
Relation between Central and Raw Moments = 0 =
= 3 + 2 , = 4 + 6 3
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Pearsons coefficient of Skewness based on Moments
= = : = 0, = 0, < 0, < 0, > 0, > 0, Pearsons coefficient of Kurtosis based on Moments
= = 3 : = 3, = 0, < 3, < 0, > 3, > 0,
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Covariance (,) = ( )( )
=
= ( )( )
=
(,) = ( )
( ) = 0 (,) = ( )
( ) = 0
Karl Pearsons Coefficient of Correlation (or Product Moment Correlation Coefficient) = (,)
Where,
(,) = ( )( )
=
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= 1( )
=
= 1( )
=
= ( )( )
1 ( ) 1 ( ) =
=
= ( )( )
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() = 1 6( 1) Where,
= = Rank Correlation with Ties
= 1 6( + + )( 1) Where,
( ) = 112
( ) = 112
Regression
Regression Lines X on Y Regression Lines Y on X
Regression Equations
= +
= ( )
= +
= ( )
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Normal Equations
= +
= +
= + = +
Regression Coefficients
= = (,)
= () ()( () =
= = (,)
= () ()()
() =
Coefficient of Correlation
= .
Fitting of second degree curve = + + to a bivariate data. The Normal Equations for estimating ,, are
= + +
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= + + = + +
Fitting of an exponential curve = to a bivariate data. The Normal Equations are = +
= + Where, = ; = ; =
Multiple Regression
Equation of Plane of Regression of X1 on X2 and X3
= + . + . Equation of Plane of Regression of X2 on X1 and X3
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= + . + . Equation of Plane of Regression of X3 on X1 and X2
= + . + . Partial Regression Coefficient of Regression Equation of X1 on X2 and X3
. = 1 . = . = 1 . = Where,
, , = . .
Multiple Regression Equations when the deviations are taken from their means [ = , = , = ]
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Regression Equation of X1 on X2 and X3
+ + = 0
+ + = 0
+ + = 0 Multiple Correlation Coefficient
. = + 21 . = + 21 . = + 21
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0 . 1 , , = 1,2,3 Partial Correlation Coefficient
. = (1 )(1 )
. = (1 )(1 )
. = (1 )(1 ) 0 . 1 , , = 1,2,3
Time Series
m-yearly moving averages for the Time Series T t1 t2 t3 . tn Y y1 y2 y3 . yn m-yearly moving averages are gives by, = . = .()
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For fitting linear trend y=a+bt , by least squares method, the Normal Equations are
= + = +
Where, =
=
= ( )
= 12 ( ) ( ) Exponential Smoothing
=
= + (1)() 0 < < 1
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Methods of Measuring Seasonal Variations
Ratio to Trend Method
100
= 1200 400 Link Relative Method
= 100
= 100