Frequently Used Statistics Formulas and Tables Chapter 2 highest value - lowest value Class Width = (increase to next integer) number classes upper limit + lower limit Class Midpoint = 2 Chapter 3 Chapter 3 n = sample size Limits for Unusual Data N = population size µσ Below : - 2 f = frequency Above: µσ+ 2 Σ=sum w= weight Empirical Rule About 68%: µσ- to µ+ σ ∑ x About 95%: µσ-2 to µ+ 2 σ Sample mean: x = n About 99.7%: µσ-3 to µ+ 3 σ ∑ x Population mean: µ = N s Sample coefficient of variation: CV = 100% ∑•()wx x Weighted mean: x = ∑ σ w Population coefficient of variation: CV = 100% ∑•()fx µ Mean for frequency table: x = ∑ f highest value + lowest value Sample standard deviation for frequency table: Midrange = 2 n [ ∑•( fx22 ) ] −∑• [ ( fx ) ] s = nn (− 1) Range = Highest value - Lowest value xx− Sample z-score: z = ∑−()xx2 s = Sample standard deviation: s n −1 x − µ Population z-score: z = ∑−()x µ 2 σ Population standard deviation: σ = N Interquartile Range: (IQR) =QQ31 − Sample variance: s2 Modified Box Plot Outliers Population variance: σ 2 lower limit: Q1 - 1.5 (IQR) upper limit: Q3 + 1.5 (IQR) Chapter 4 Chapter 5 Probability of the complement of event A Discrete Probability Distributions: P( not A) = 1 - P( A ) Mean of a discrete probability distribution: Multiplication rule for independent events µ =∑•[x Px ( )] P( A and B) = P( A ) • P( B ) Standard deviation of a probability distribution: General multiplication rules P( A and B) = P( A ) • P( B , given A) σµ=∑•[x22 Px ( )] − P ( A and B) = P( A ) • P( A , given B) Addition rule for mutually exclusive events Binomial Distributions PAorB( ) = PA( ) + PB( ) r = number of successes (or x) p = probability of success General addition rule q = probability of failure P( A or B) = P( A ) + P( B )− P ( A and B) =−+ q1 p pq = 1 Binomial probability distribution n! r nr− Permutation rule: P = Pr()= nr Cpq nr (nr− )! Mean: µ = np n! Combination rule: C = Standard deviation: σ = npq nr rn!(− r )! Poisson Distributions Permutation and Combination on TI 83/84 rx= number of successes (or ) µ = mean number of successes n Math PRB nPr enter r (over a given interval) Poisson probability distribution n Math PRB nCr enter r e−µ µ r Pr()= r! e ≈ 2.71828 Note: textbooks and formula sheets interchange “r” and “x” µ = mean (over some interval) for number of successes σµ= σµ2 = 2 Chapter 6 Chapter 7 Normal Distributions Confidence Interval: Point estimate ± error Raw score: xz=σµ + Point estimate = Upper limit + Lower limit 2 x − µ Standard score: z = σ Error = Upper limit - Lower limit 2 Mean of x distribution: µµx = Sample Size for Estimating σ means: Standard deviation of x distribtuion: σ x = 2 zα /2σ n n = (standard error) E x − µ Standard score for xz: = proportions: σ 2 / n zα /2 n= pqˆˆwith preliminary estimate for p E Chapter 7 2 zα /2 np= 0.25 without preliminary estimate for One Sample Confidence Interval E >> for proportions (p ) : ( np 5 and nq 5) variance or standard deviation: *see table 7-2 (last page of formula sheet) pEˆˆ−<<+ p pE pp(1− ) Confidence Intervals where Ez= α /2 n r Level of Confidence z-value ( zα /2 ) pˆ = n 70% 1.04 for means (µσ ) when is known: 75% 1.15 xE−<µ <+ xE σ 80% 1.28 where Ez= α /2 n 85% 1.44 for means (µσ ) when is unknown: 90% 1.645 −<µ <+ xE xE 95% 1.96 s where Et= α /2 n 98% 2.33 with df. .= n − 1 99% 2.58 22 22(ns−− 1) ( ns 1) for variance (σσ ) : < < χχ22 RL with df. .= n − 1 3 Chapter 8 Chapter 9 One Sample Hypothesis Testing Difference of means μ -μ (independent samples) 1 2 Confidence Interval when σσ12 and are known ppˆ − −−<−µµ <−+ for p ( np> 5 and nq >=5) : z ()(xx12 E 1 2 )() xx 12 E pq/ n 22 σσ12 where Ez=α /2 + where q=−=1 pp ;ˆ r / n nn12 x − µ Hypothesis Test when σσ and are known for µσ ( known): z = 12 σ (xx12−−− )(µµ 1 2 ) / n z = σσ22 x − µ 12+ for µσ ( unknown): t= with df. .= n − 1 nn sn/ 12 2 22(ns− 1) Confidence Interval when σσ and are unknown for σχ: = with df. .= n − 1 12 σ 2 ()(xx12−−<− Eµµ 1 2 )() <−+ xx 12 E 22 ss12 Et=α /2 + Chapter 9 nn12 with df. = smaller of n−−1 and n 1 Two Sample Confidence Intervals 12 and Tests of Hypotheses Hypothesis Test when σσ12 and are unknown Difference of Proportions (pp− ) 12 (xx12−− )(µ12− µ ) t = 22 ss12 Confidence Interval: + nn12 ()()()ppˆˆ12−−<−<−+ E pp 12 ppˆˆ 12 E with df. .= smaller of n12 −−1 and n 1 pqˆˆ11 pq ˆˆ 2 2 where Ez=α /2 + Matched pairs (dependent samples) nn12 Confidence Interval dE−+ < µ < dE pˆ1= r 1/ np 1 ; ˆ 2 = r 2 / n 2 and q ˆ 1 =−=−1 pq ˆˆ 12 ; 1 p ˆ 2 d sd where Et= α /2 with d.f. = n−1 n Hypothesis Test: Hypothesis Test ()()ppˆˆ12−−− pp 12 z = − µ d d pq pq t= with df. .= n − 1 + sd nn 12 n where the pooled proportion is p Two Sample Variances rr12+ σσ22 p= and qp=1 − Confidence Interval for 12 and 2 22 nn12+ ss11 σ 1• <<11 • 2 22 ss22FFright σ 2 left pˆˆ1= rnp 112/ ; = rn 22/ 2 s1 22 = ≥ Hypothesis Test Statistic: F2 where ss12 s2 numerator df. .= n−= 1 and denominator df. n− 1 12 4 Chapter 10 Chapter 11 ()OE− 2 (row total)(column total) χ 2 =∑= where E Regression and Correlation E sample size Linear Correlation Coefficient (r) Tests of Independence df. .=−− ( R 1)( C 1) n∑ xy −∑( x )( ∑ y ) r = nx(∑2 )( −∑ x ) 22 ny ( ∑ )( −∑ y ) 2 OR Goodness of fit df. .= (number of categories) −1 Σ()zz rz= xy where = z score for x and z= z score for y n −1 xy Chapter 12 explained variation Coefficient of Determination: r 2 = total variation One Way ANOVA ∑−()yyˆ 2 Standard Error of Estimate: s = kN= number of groups; = total sample size e n − 2 2 2 ∑y − b01 ∑− y b ∑ xy 2 ()∑ xTOT or s = SS=∑− xTOT e n − 2 TOT N Prediction Interval: yEˆˆ−<<+ y yE ()∑∑xx22 ( ) = i − TOT SSBET ∑ all groups nNi 2 1 nx()0 − x where Et=α s1 ++ /2 e n nx(Σ22 )( −Σ x ) ()∑ x 2 = ∑−2 i SSWi∑ x all groups ni Sample test statistic for r r t = with df..= n − 2 SSTOT= SS BET + SS W 1− r 2 SS n − 2 MS = BET where d. f .= k − 1 BET df.. BET BET Least-Squares Line (Regression Line or Line of Best Fit) SSW yˆ = b01 + bx note that b0 is the y-intercept and b1 is the slope MSW = where df. .W = N − k d..f W n∑ xy −∑( x )( ∑ y ) sy = = MSBET where b1122 or br = = − nx(∑ )( −∑ x ) s F where df. numerator = df. .BET k 1 x MS and W df. denominator = df. = N − k (∑y )( ∑ x2 ) −∑ ( x )( ∑ xy ) W where b = or b= y − bx 0nx(∑22 )( −∑ x ) 01 β Confidence interval for y-intercept 0 Two - Way ANOVA bE0−<β 00 < bE + 1 x 2 rc= number of rows; = number of columns where E = tsα + /2 e n ()∑ x 2 MS row factor ∑−x2 Row factor F : n MS error MS column factor Column factor F : Confidence interval for slope β MS error 1 MS interaction bE1−<β 11 <+ bE Interaction F : MS error se where E = tα /2 • ()∑ x 2 ∑−x2 with degrees of freedom for n row factor = r −1 column factor = c −1 interaction = (rc−− 1)( 1) error = rc( n − 1) 5 critical z-values for hypothesis testing α = 0.10 α = 0.05 α = 0.01 c-level = 0.90 c-level = 0.95 c-level = 0.99 ≠ ≠ ≠ 0.05 0.05 0.025 0.025 0.005 0.005 z = - 1.645 z = 0 z = 1.645 z = - 1.96 z = 0 z = 1.96 z = - 2.575 z = 0 z = 2.575 < < < 0.01 0.10 0.05 z = - 2.33 z = 0 z = - 1.28 z = 0 z = - 1.645 z = 0 > > > 0.01 0.10 0.05 z = 0 z = 2.33 z = 0 z = 1.28 z = 0 z = 1.645 Figure 8.4 . . . . Greek Alphabet http://www.keyway.ca/htm2002/greekal.htm .