Important Statistics Formulas

7/30/2019 Important Statistics Formulas

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Diunduh oleh: mBah Darmo

Letter NameSound value

Ancient Modern alpha [a][a] [a]

beta [b] [v]

gamma [] []

delta [d] []

epsilon [e] [e]

zeta [zd] (or [dz][7])[z]

eta [] [i]

theta [t] []

iota [i][i] [i]

kappa [k] [k]

lambda [l] [l]

mu [m] [m]

Letter NameSound value

Ancient Modern

nu [n] [n]

xi [ks] [ks]

omicron [o] [o]

pi [p] [p]

rho [r] [r]

sigma [s] [s]

tau [t] [t]

upsilon [y][y] [i]

phi [p] [f]

chi [kh ] [x]

psi [ps] [ps]

omega [] [o]
http://en.wikipedia.org/wiki/Greek_alphabet#cite_note-woodard_2008_15-4http://en.wikipedia.org/wiki/Greek_alphabet#cite_note-holton_1998_31-5http://en.wikipedia.org/wiki/Alphahttp://en.wikipedia.org/wiki/Alphahttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Betahttp://en.wikipedia.org/wiki/Betahttp://en.wikipedia.org/wiki/Voiced_bilabial_stophttp://en.wikipedia.org/wiki/Voiced_bilabial_stophttp://en.wikipedia.org/wiki/Voiced_bilabial_stophttp://en.wikipedia.org/wiki/Voiced_labiodental_fricativehttp://en.wikipedia.org/wiki/Voiced_labiodental_fricativehttp://en.wikipedia.org/wiki/Voiced_labiodental_fricativehttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Voiced_velar_stophttp://en.wikipedia.org/wiki/Voiced_velar_stophttp://en.wikipedia.org/wiki/Voiced_velar_stophttp://en.wikipedia.org/wiki/Voiced_velar_fricativehttp://en.wikipedia.org/wiki/Voiced_velar_fricativehttp://en.wikipedia.org/wiki/Voiced_velar_fricativehttp://en.wikipedia.org/wiki/Delta_%28letter%29http://en.wikipedia.org/wiki/Voiced_alveolar_stophttp://en.wikipedia.org/wiki/Voiced_alveolar_stophttp://en.wikipedia.org/wiki/Voiced_alveolar_stophttp://en.wikipedia.org/wiki/Voiced_dental_fricativehttp://en.wikipedia.org/wiki/Voiced_dental_fricativehttp://en.wikipedia.org/wiki/Voiced_dental_fricativehttp://en.wikipedia.org/wiki/Epsilonhttp://en.wikipedia.org/wiki/Epsilonhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Zetahttp://en.wikipedia.org/wiki/Greek_alphabet#cite_note-hinge-6http://en.wikipedia.org/wiki/Greek_alphabet#cite_note-hinge-6http://en.wikipedia.org/wiki/Greek_alphabet#cite_note-hinge-6http://en.wikipedia.org/wiki/Voiced_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiced_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiced_alveolar_sibilanthttp://en.wikipedia.org/wiki/Etahttp://en.wikipedia.org/wiki/Etahttp://en.wikipedia.org/wiki/Open-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Voiceless_dental_fricativehttp://en.wikipedia.org/wiki/Voiceless_dental_fricativehttp://en.wikipedia.org/wiki/Voiceless_dental_fricativehttp://en.wikipedia.org/wiki/Iotahttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Kappahttp://en.wikipedia.org/wiki/Kappahttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Lambdahttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Mu_%28letter%29http://en.wikipedia.org/wiki/Mu_%28letter%29http://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Nu_%28letter%29http://en.wikipedia.org/wiki/Nu_%28letter%29http://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Xi_%28letter%29http://en.wikipedia.org/wiki/Omicronhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Pi_%28letter%29http://en.wikipedia.org/wiki/Pi_%28letter%29http://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Rhohttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Sigmahttp://en.wikipedia.org/wiki/Sigmahttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Upsilonhttp://en.wikipedia.org/wiki/Upsilonhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Phihttp://en.wikipedia.org/wiki/Phihttp://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Voiceless_labiodental_fricativehttp://en.wikipedia.org/wiki/Voiceless_labiodental_fricativehttp://en.wikipedia.org/wiki/Voiceless_labiodental_fricativehttp://en.wikipedia.org/wiki/Chi_%28letter%29http://en.wikipedia.org/wiki/Chi_%28letter%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Voiceless_velar_fricativehttp://en.wikipedia.org/wiki/Voiceless_velar_fricativehttp://en.wikipedia.org/wiki/Voiceless_velar_fricativehttp://en.wikipedia.org/wiki/Psi_%28letter%29http://en.wikipedia.org/wiki/Psi_%28letter%29http://en.wikipedia.org/wiki/Omegahttp://en.wikipedia.org/wiki/Omegahttp://en.wikipedia.org/wiki/Open-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Omegahttp://en.wikipedia.org/wiki/Psi_%28letter%29http://en.wikipedia.org/wiki/Voiceless_velar_fricativehttp://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Chi_%28letter%29http://en.wikipedia.org/wiki/Voiceless_labiodental_fricativehttp://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Phihttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Close_front_rounded_vowelhttp://en.wikipedia.org/wiki/Upsilonhttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Voiceless_alveolar_stophttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Voiceless_alveolar_sibilanthttp://en.wikipedia.org/wiki/Sigmahttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Alveolar_trillhttp://en.wikipedia.org/wiki/Rhohttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Voiceless_bilabial_stophttp://en.wikipedia.org/wiki/Pi_%28letter%29http://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_back_rounded_vowelhttp://en.wikipedia.org/wiki/Omicronhttp://en.wikipedia.org/wiki/Xi_%28letter%29http://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Alveolar_nasalhttp://en.wikipedia.org/wiki/Nu_%28letter%29http://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Bilabial_nasalhttp://en.wikipedia.org/wiki/Mu_%28letter%29http://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Alveolar_lateral_approximanthttp://en.wikipedia.org/wiki/Lambdahttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Voiceless_velar_stophttp://en.wikipedia.org/wiki/Kappahttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Iotahttp://en.wikipedia.org/wiki/Voiceless_dental_fricativehttp://en.wikipedia.org/wiki/Aspiration_%28phonetics%29http://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Close_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Etahttp://en.wikipedia.org/wiki/Voiced_alveolar_sibilanthttp://en.wikipedia.org/wiki/Greek_alphabet#cite_note-hinge-6http://en.wikipedia.org/wiki/Zetahttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Close-mid_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Epsilonhttp://en.wikipedia.org/wiki/Voiced_dental_fricativehttp://en.wikipedia.org/wiki/Voiced_alveolar_stophttp://en.wikipedia.org/wiki/Delta_%28letter%29http://en.wikipedia.org/wiki/Voiced_velar_fricativehttp://en.wikipedia.org/wiki/Voiced_velar_stophttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Voiced_labiodental_fricativehttp://en.wikipedia.org/wiki/Voiced_bilabial_stophttp://en.wikipedia.org/wiki/Betahttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Open_front_unrounded_vowelhttp://en.wikipedia.org/wiki/Alphahttp://en.wikipedia.org/wiki/Greek_alphabet#cite_note-holton_1998_31-5http://en.wikipedia.org/wiki/Greek_alphabet#cite_note-woodard_2008_15-4


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Important Statistics Formulas

This web page presents statistics formulas described in the Stat Trek tutorials. Each formula

links to a web page that explains how to use the formula.

Parameters

Population mean = = ( X i ) / N

Population standard deviation = = sqrt [ ( Xi - )2 / N ]

Population variance = 2= ( Xi - )2 / N

Variance of population proportion = P2 = PQ / n

Standardized score = Z = (X - ) /

Population correlation coefficient = = [ 1 / N ] * { [ (Xi - X) / x ] * [ (Yi - Y) / y ]}

Statistics

Unless otherwise noted, these formulas assumesimple random sampling.

Sample mean = x = ( x i ) / n

Sample standard deviation = s = sqrt [ ( x i - x )2 / ( n - 1 ) ]

Sample variance = s2= ( xi - x )2 / ( n - 1 )

Variance of sample proportion = sp2 = pq / (n - 1)

Pooled sample proportion = p = (p1 * n1 + p2 * n2) / (n1 + n2)
http://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Position.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Position.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/Help/Glossary.aspx?Target=Simple%20random%20samplinghttp://stattrek.com/Help/Glossary.aspx?Target=Simple%20random%20samplinghttp://stattrek.com/Help/Glossary.aspx?Target=Simple%20random%20samplinghttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Difference-Proportion.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stathttp://stattrek.com/Help/Glossary.aspx?Target=Simple%20random%20samplinghttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Correlation.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Position.aspx?Tutorial=stathttp://stattrek.com/Lesson3/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Variability.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-1/Central-Tendency.aspx?Tutorial=stat


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Pooled sample standard deviation = sp = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2)]

Sample correlation coefficient = r = [ 1 / (n - 1) ] * { [ (xi - x) / sx ] * [ (yi - y) / sy ] }

Correlation

Pearson product-moment correlation = r = (xy) / sqrt [ ( x2) * ( y2 ) ]

Linear correlation (sample data) = r = [ 1 / (n - 1) ] * { [ (xi - x) / sx ] * [ (yi - y) / sy ] }

Linear correlation (population data) = = [ 1 / N ] * { [ (Xi - X) / x ] * [ (Yi - Y) / y] }

Simple Linear Regression

Simple linear regression line: = b0 + b1x

Regression coefficient = b1= [ (xi - x) (yi - y) ] / [ (xi - x)2]

Regression slope intercept = b0 = y - b1 * x

Regression coefficient = b1 = r * (sy / sx)

Standard error of regression slope = sb1= sqrt [ (yi - i)2 / (n - 2) ] / sqrt [ (xi - x)2 ]

Counting

n factorial: n! = n * (n-1) * (n - 2) * . . . * 3 * 2 * 1. By convention, 0! = 1.

Permutations ofn things, taken rat a time: nPr= n! / (n - r)!

Combinations ofn things, taken rat a time: nCr= n! / r!(n - r)! = nPr/ r!
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4/9

Probability

Rule of addition: P(A B) = P(A) + P(B) - P(A B)

Rule of multiplication: P(A B) = P(A) P(B|A)

Rule of subtraction: P(A') = 1 - P(A)Random Variables

In the following formulas,Xand Yare random variables, and a and b are constants.

Expected value of X = E(X) = x= [ xi * P(xi) ] Variance of X = Var(X) = 2= [ xi - E(x) ]2 * P(xi) = [ xi - x ]2 * P(xi)

Normal random variable = z-score = z = (X - )/

Chi-square statistic = 2 = [ ( n - 1 ) * s2] / 2

f statistic =f= [s12/12 ] / [s22/22 ]

Expected value of sum of random variables = E(X + Y) = E(X) + E(Y)

Expected value of difference between random variables = E(X - Y) = E(X) - E(Y)

Variance of the sum ofindependentrandom variables = Var(X + Y) = Var(X) + Var(Y)

Variance of the difference between independentrandom variables = Var(X - Y) = Var(X)+ Var(Y)

Sampling Distributions

Mean of sampling distribution of the mean = x=
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5/9

Mean of sampling distribution of theproportion = p = P

Standard deviation of proportion = p = sqrt[ P * (1 - P)/n ] = sqrt( PQ / n )

Standard deviation of the mean = x= /sqrt(n)

Standard deviation of difference of sample means = d= sqrt[ (12 / n1) + (22 / n2) ]

Standard deviation of difference of sample proportions = d = sqrt{ [P1(1 - P1) / n1] +[P2(1 - P2) / n2] }

Standard Error

Standard error of proportion = SEp = sp = sqrt[ p * (1 - p)/n ] = sqrt( pq / n )

Standard error of difference for proportions = SEp = sp = sqrt{ p * ( 1 - p ) * [ (1/n1) +(1/n2) ] }

Standard error of the mean = SEx = sx = s/sqrt(n)

Standard error of difference of sample means = SEd = sd = sqrt[ (s12 / n1) + (s22 / n2) ]

Standard error of difference of paired sample means = SEd = sd= { sqrt [ ((di - d)2 / (n -1) ] } / sqrt(n)

Pooled sample standard error = spooled = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ]

Standard error of difference of sample proportions = sd = sqrt{ [p1(1 - p1) / n1] + [p2(1 -p2) / n2] }

Discrete Probability Distributions

Binomial formula: P(X = x) = b(x; n, P) = nCx * Px * (1 - P)n - x = nCx * Px * Qn - x
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6/9

Mean of binomial distribution = x = n * P

Variance of binomial distribution = x2 = n *P* ( 1 -P)

Negative Binomial formula: P(X = x) = b*(x; r, P) = x-1Cr-1 * Pr* (1 - P)x - r

Mean of negative binomial distribution = x = rQ / P

Variance of negative binomial distribution = x2 = r * Q / P2

Geometric formula: P(X = x) = g(x;P) = P * Qx - 1

Mean of geometric distribution = x = Q / P

Variance of geometric distribution = x2 = Q / P2

Hypergeometric formula: P(X = x) = h(x;N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]

Mean of hypergeometric distribution = x = n * k / N

Variance of hypergeometric distribution = x2 = n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1) ]

Poisson formula: P(x; ) = (e-) (x) / x!

Mean of Poisson distribution = x=

Variance of Poisson distribution = x2=

Multinomial formula: P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n

1 * p2n

2 * . . . * pkn

k)

Linear Transformations

For the following formulas, assume that Y is alinear transformationof the random variable X,defined by the equation: Y = aX + b.
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7/9

Mean of a linear transformation = E(Y) = Y = aX + b.

Variance of a linear transformation = Var(Y) = a2 * Var(X).

Standardized score = z = (x - x) / x.

t-score = t = (x - x) / [ s/sqrt(n) ].Estimation

Confidence interval: Sample statistic + Critical value * Standard error of statistic

Margin of error = (Critical value) * (Standard deviation of statistic) Margin of error = (Critical value) * (Standard error of statistic)

Hypothesis Testing

Standardized test statistic = (Statistic - Parameter) / (Standard deviation of statistic)

One-sample z-test for proportions: z-score = z = (p - P0) / sqrt( p * q / n ) Two-sample z-test for proportions: z-score = z = z = [ (p1 - p2) - d ] / SE

One-sample t-test for means: t-score = t = (x - ) / SE

Two-sample t-test for means: t-score = t = [ (x1 - x2) - d ] / SE

Matched-sample t-test for means: t-score = t = [ (x1 - x2) - D ] / SE = (d - D) / SE

Chi-square test statistic = 2= [ (Observed - Expected)2 / Expected ]

Degrees of Freedom

The correct formula for degrees of freedom (DF) depends on the situation (the nature of the teststatistic, the number of samples, underlying assumptions, etc.).
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8/9

One-sample t-test: DF = n - 1

Two-sample t-test: DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ]}

Two-sample t-test, pooled standard error: DF = n1 + n2 - 2

Simple linear regression, test slope: DF = n - 2

Chi-square goodness of fit test: DF = k - 1

Chi-square test for homogeneity: DF = (r - 1) * (c - 1)

Chi-square test for independence: DF = (r - 1) * (c - 1)Sample Size

Below, the first two formulas find the smallest sample sizes required to achieve a fixed margin oferror, using simple random sampling. The third formula assigns sample to strata, based on a

proportionate design. The fourth formula, Neyman allocation, uses stratified sampling to

minimize variance, given a fixed sample size. And the last formula, optimum allocation, uses

stratified sampling to minimize variance, given a fixed budget.

Mean (simple random sampling): n = { z2* 2 * [ N / (N - 1) ] } / { ME2 + [ z2* 2 / (N -1) ] }

Proportion (simple random sampling): n = [ ( z2 * p * q ) + ME2 ] / [ ME2 + z2 * p * q / N]

Proportionate stratified sampling: nh = ( Nh / N ) * n

Neyman allocation (stratified sampling): nh = n * ( Nh* h) / [ ( Ni* i ) ]
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6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSize.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Independence.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Homogeneity.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Goodness-of-Fit.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Test-Slope.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Difference-Means.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Difference-Means.aspx?Tutorial=stathttp://stattrek.com/AP-Statistics-4/Difference-Means.aspx?Tutorial=stathttp://stattrek.com/Lesson5/Mean.aspx?Tutorial=stat


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Optimum allocation (stratified sampling):nh = n * [ ( Nh* h ) / sqrt( ch) ] / [ ( Ni* i ) / sqrt( ci ) ]
http://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stathttp://stattrek.com/Lesson6/SampleSizeStrata.aspx?Tutorial=stat

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Important Statistics Formulas