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 =

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= pqˆˆwith 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

xE−<µ <+ 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

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 (xx−−− )(µµ ) σ / n z = 12 1 2 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 (pp12− ) (xx−− )(µ − µ ) t = 12 12 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 ()()ppˆˆ−−− pp z = 12 12 − 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 /   

s2 = 1 22≥ Hypothesis Test Statistic: F2 where ss12 s2 numerator df . .= n−= 1 and denominator df . . n− 1 12

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 n − 2 SSBET MSBET = where d . f .BET = k − 1 df..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)

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  

          

                                                                                                                                                                                                                                                                                                                                                                                                    

               

   

          

                                                                                                                                                                                                                                                                                                                                                                                                                          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

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