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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