Statistics Formula Cheat Sheet

Frequently Used Statistics Formulas and Tables Chapter 2 Class Width = highest value - lowest value (increase to next integer) number classes Class Midpoint = upper limit + lower limit 2 Chapter 3 Chapter 3 n = sample size N = population size f = frequency Σ = sum w = weight Limits for Unusual Data Below : µ - 2σ Above: µ + 2 σ Empirical Rule About 68%: µ -σ to µ + σ About 95%: µ -2 σ to µ + 2 σ About 99.7%: µ -3 σ to µ + 3σ ∑x n ∑x Population mean: µ = N ∑ (w • x ) Weighted mean: x = ∑w Sample mean: x = Sample coefficient of variation: CV = Population coefficient of variation: CV = ∑( f • x ) Mean for frequency table: x = ∑f highest value + lowest value Midrange = 2 s= Sample variance: s2 Population variance: σ 2 2 n [ ∑( f • x ) ] − [ ∑( f • x) ] n ( n −1) Sample z-score: z = 2 ∑( x − x ) n −1 Population standard deviation: σ = σ  100% µ Sample standard deviation for frequency table: Range = Highest value - Lowest value Sample standard deviation: s = s  100% x 2 ∑ (x − µ ) N x− x s Population z-score: z = x− µ σ Interquartile Range: (IQR) = Q3 − Q1 Modified Box Plot Outliers lower limit: Q1 - 1.5 (IQR) upper limit: Q3 + 1.5 (IQR) 2 Chapter 4 Chapter 5 Probability of the complement of event A P( not A) = 1 - P( A) Discrete Probability Distributions: Mean of a discrete probability distribution: µ = ∑[ x • P( x)] Multiplication rule for independent events 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) P( A and B) = P( A) • P( A, given B) σ = ∑[ x2 • P ( x )] − µ2 Addition rule for mutually exclusive events P( A or B) = P( A) + P( B) Binomial Distributions General addition rule P( A or B) = P( A) + P( B) − P( A and B) q = probability of failure q =1 − p p + q=1 Permutation rule: n Pr = r = number of successes (or x) p = probability of success Binomial probability distribution P (r ) = n C r p rq n− r n! (n − r )! Mean: µ = np n! Combination rule: n C r = r !(n − r )! Standard deviation: σ = npq Poisson Distributions Permutation and Combination on TI 83/84 n n Math Math PRB PRB nPr nCr enter enter r = number of successes (or x ) µ = mean number of successes r (over a given interval) Poisson probability distribution r P r ( =) e− µ µ r r ! e ≈ 2.71828 Note: textbooks and formula sheets interchange “r” and “x” for number of successes µ = mean (over some interval) σ= µ σ2 = µ 2 Chapter 6 Chapter 7 Confidence Interval: Point estimate ± error Normal Distributions Raw score: x = z σ + µ Standard score: z = Point estimate = Upper limit + Lower limit 2 x −µ Error = Upper limit - Lower limit 2 σ Mean of x distribution: µ x = µ Sample Size for Estimating σ Standard deviation of x distribtuion: σ x = n (standard error) Standard score for x : z = means: 2  z σ n =  α /2   E  x −µ σ/ n proportions: 2 z  ˆ ˆ  α / 2  with preliminary estimate for p n = pq  E  Chapter 7 2 z  n = 0.25  α / 2  without preliminary estimate for p  E  One Sample Confidence Interval for proportions (p) : ( np > 5 and nq > 5) variance or standard deviation: *see table 7-2 (last page of formula sheet) pˆ − E < p < pˆ + E where E = zα / 2 pˆ = r n p (1− p ) n Confidence Intervals Level of Confidence for means (µ ) when σ is known: x−E<µ< x+E σ where E = zα / 2 n for means (µ ) when σ is unknown: x−E<µ < x+E s n with d. f . = n −1 where E = tα / 2 for variance (σ 2 ) : ( n − 1) s 2 χ2R <σ2 < ( n − 1) s 2 χL2 with d . f . = n −1 3 z-value ( zα / 2 ) 70% 1.04 75% 1.15 80% 1.28 85% 1.44 90% 1.645 95% 1.96 98% 2.33 99% 2.58 Chapter 8 Chapter 9 One Sample Hypothesis Testing Difference of means μ 1 -μ 2 (independent samples) for p ( np > 5 and nq > 5) : z= Confidence Interval when σ 1 and σ 2 are known (x1 − x2 ) − E < ( µ1 − µ2 ) < (x1 − x2 ) + E pˆ − p pq / n for µ (σ known): z= for µ (σ unknown): for σ 2 : χ 2 = x −µ (n − 1)s σ2 2 σ 12 x−µ with d. f . = n − 1 s/ n n1 E = tα / 2 n2 + σ 22 n2 2 s1 s2 + n1 n2 with d. f . = smaller of n1 −1 and n2 −1 Two Sample Confidence Intervals and Tests of Hypotheses Hypothesis Test when σ 1 and σ 2 are unknown Difference of Proportions (p1 − p2 ) t= Confidence Interval: (pˆ1 − pˆ 2 ) − E < ( p1 − p2 ) < (pˆ1 − pˆ 2 ) + E ( x1 − x2 ) − ( µ1 − µ 2) 2 2 s1 s + 2 n1 n2 with d. f . =smaller of n1 −1 and n2 −1 pˆ1qˆ1 pˆ 2qˆ2 + n1 n2 Matched pairs (dependent samples) Confidence Interval d − E < µd < d + E s where E = tα / 2 d with d.f. = n −1 n pˆ1 = r1 / n1; pˆ 2 = r2 / n2 and qˆ1 = 1 − pˆ1; qˆ2 = 1 − pˆ 2 Hypothesis Test: Hypothesis Test d − µd t= with d. f . = n −1 sd n ( pˆ1 − pˆ 2 ) − ( p1 − p2 ) pq pq + n1 n2 where the pool ed proportion is p p= σ22 ( x1 − x2 ) − E < ( µ1 − µ2 ) < ( x1 − x2 ) + E 2 z= + Confidence Interval when σ 1 and σ 2 are unknown with d . f . = n −1 Chapter 9 where E = zα / 2 n1 Hypothesis Test when σ 1 and σ 2 are known ( x − x ) − (µ1 − µ2 ) z= 1 2 σ/ n t= σ12 where E = zα / 2 where q = 1 − p; pˆ = r / n Two Sample Variances r1 + r2 and q =1 − p n1 + n 2 2 2 Confidence Interval for σ1 and σ2  s2 1  σ12  s12 1   12 • <  < •  s 2 Fright  σ 22  s 22 Fleft      pˆ1 = r1 / n1 ; pˆ 2 = r2 / n2 s12 where s12 ≥ s22 s22 numerator d . f . = n1 −1 and denominator d . f . = n 2 −1 Hypothesis Test Statistic: F = 4 Chapter 10 Chapter 11 Regression and Correlation χ2 = ∑ Linear Correlation Coefficient (r) r= OR r= n (∑ x ) − (∑ x ) Σ ( zx zy ) n −1 2 2 n (∑ y ) − (∑ y ) where E = (row total)(column total) sample size 2 Goodness of fit d. f . = (number of categories) − 1 where zx = z score for x and zy = z score for y Coefficient of Determination: r 2 = Standard Error of Estimate: Chapter 12 explained variation total variation One Way ANOVA ∑( y − yˆ )2 n−2 se = ∑ y2 − b0 ∑ y− b1 ∑ xy or se = n− 2 k = number of groups; N = total sample size 2 SSTOT = ∑ xTOT − Prediction Interval: yˆ − E < y < yˆ + E where E = t α / 2 s e 1+ SSBET = n( x0 − x) 1 + n n (Σ x 2 ) − (Σx ) 2 all groups MS BET = MSW = b1 = r sy  ( ∑ xi ) 2   ni  SSBET where d . f . BET = k − 1 d . f .BET SSW d . f .W where d . f .W = N − k MSBET where d. f . numerator = d. f .BET = k − 1 MS W d . f . denominator = d . f .W = N − k F= sx or b0 = y − b1 x Confidence interval for y-intercept β0 b0 − E < β0 < b0 + E where E = tα / 2 se  SSTOT = SS BET + SSW note that b 0 is the y-intercept and b1 is the slope 1 + n  ( ∑ xi ) 2  ( ∑ xTOT ) 2  − N  ni  2 ∑  ∑ xi − SSW = Least-Squares Line (Regression Line or Line of Best Fit) n ∑ xy − (∑ x )(∑ y ) where b1 = or n(∑ x2 ) − (∑ x)2 and 2 (∑ y )(∑ x ) − (∑ x )(∑ xy ) where b0 = n( ∑ x 2 ) − ( ∑ x) 2 2 ( ∑ xTOT ) N ∑  all groups 2 Sample test statistic for r r t= with d . f . = n − 2 1− r2 n− 2 yˆ =b 0 +b1x 2 Tests of Independence d. f . = ( R −1)( C −1) n ∑ xy − ( ∑ x)(∑ y) 2 ( O − E) E Two - Way ANOVA r = number of rows; c = number of columns MS row factor Row factor F : MS error MS column factor Column factor F : MS error MS interaction Interaction F : MS error x2 (∑ x) 2 ∑x2− n Confidence interval for slope β 1 b1 − E < β1 < b1 + E se where E = tα / 2 • (∑ x) 2 ∑x 2 − n with degrees of freedom for row factor = r − 1 column factor = c − 1 interaction = (r − 1)(c − 1) error = rc (n − 1) 5                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         critical z-values for hypothesis testing α = 0.05 c-level = 0.95 α = 0.10 c-level = 0.90 ≠ 0.05 z = - 1.645 0.05 z=0 z = 1.645 ≠ z = - 1.28 ≠ 0.025 z = - 1.96 0.025 z=0 z = 1.96 < < 0.10 α = 0.01 c-level = 0.99 0.005 0.005 z = - 2.575 z=0 z = 2.575 < 0.01 0.05 z = - 1.645 z=0 z = - 2.33 z=0 > z = 1.28 > > 0.10 z=0 z=0 0.01 0.05 z=0 z = 1.645 Figure 8.4 z=0 z = 2.33                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Greek Alphabet