Stat & Prob Formula Sheet

Probability and Statistics Formula Sheet I. Descriptive Statistics Sample Mean (List of Data) ∑x x= n

Sample Mean (Freq. Distrib.) ∑ xf x= ∑f

Sample Variance (List of Data)

Pop. Variance (List of Data)

s2 =

∑ ( x− x ) n −1

2

σ 2=

∑ ( x−µ ) N

Variance (Grouped Data)  ( ∑ f ⋅ xm ) 2  2  ∑ f ⋅ xm −   n   s2 = n −1 Chebyshev’s Theorem 1 at least 1− 2 k

Depth of Median d ( x )=

Midrange

n +1 2

Midrange=

Sample St. Deviation (List of Data)

2

s=

Range

∑ ( x− x ) n −1

H +L 2

R=H −L

Pop. St. Deviation (List of Data)

2

σ=

∑ ( x−µ ) N

2

St. Deviation (Grouped Data)  ( ∑ f ⋅ xm ) 2  2  ∑ f ⋅ xm −   n   s= n −1

Standard Score z-score x −µ z= σ

Percentile of a Piece of Data percentile=

kth Percentile

number of values below + 0.5 ⋅100 total number of values

nk Pk = 100

Interquartile Range IQR =Q3 −Q1

II. Probability Permutation Rule:

The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taken n! r objects at a time. It is written n Pr and the formula is n Pr = ( n − r )!

Combination Rule:

Empirical Probability n ( A) P′( A) = n

n The number of combinations of r objects selected from n objects is denoted by n Cr , or   , and is given by the r  n! formula n Cr = . r !( n − r )!

Theoretical Probability n ( A) P ( A) = n( S )

General Multiplication Rule P ( Aand B ) = P ( A)⋅P ( B| A)

Complement Rule P ( A ) =1− P ( A)

General Addition Rule P ( Aor B ) =P ( A) + P ( B )− P ( Aand B )

Special Multiplication Rule for Independent Events P ( Aand B and ... and D ) = P ( A)⋅P ( B )⋅...⋅ P ( D )

Mean of a Variance of a St. Deviation of a Prob. Distribution Prob. Distribution Prob. Distribution

µ =∑ [ x ⋅ P ( x )] Mean of a Binomial Random Variable µ =np

σ 2 =∑  x 2 ⋅ P ( x )  − µ 2

Special Addition Rule Mutually Exclusive Events

σ = ∑  x 2 ⋅ P ( x )  − µ 2

Variance of a Binomial Random Variable σ 2 =npq

P ( Aor B or ... D ) =P ( A) + P ( B )+...+ P ( D )

Conditional Probability P ( Aand B ) P ( A|B ) = P( B) Binomial Prob. Function n   P ( x ) =  ⋅ p x ⋅ q n−x for x =0,1,2,...  x Standard Deviation of a Binomial Random Variable σ = npq

III. Inferential Statistics z-score (Piece of Data)

z-score (Sample Mean)

x −µ z= σ

z=

x −µx σx

Confidence Interval Estimate for µ , ( σ Unknown) x ± tα ⋅ 2

s n

yˆ =b0 +b1 x

Confidence Iterval Estimate for p

Sample Size Needed for an Interval Estimate of Pop. Mean  zα ⋅σ n = 2  E 

σ x±z ⋅ n α 2

Sample Size Needed for an Interval Estimate of Pop. Proportion

Slope for y-intercept for Line of Best Fit Line of Best Fit ∑ ( x − x )( y − y ) b1 = b0 = y −b1 x 2 ∑ ( x− x )

( n −1) s 2 with df = n −1 σ2

Goodness-of-Fit Chi-Square Test (O − E ) 2 χ 2 =∑ E df = # categories – 1

σ σx= n

Test Statistic – Mean ( σ Unknown) x −µx t* = with df = n −1 s n

Test Statistic Variance or St. Deviation

χ2=

Confidence Interval Estimate for µ , ( σ Known)

ˆˆ pq pˆ ± zα ⋅ 2 n

Test Statistic – Mean ( σ Known) x −µ x z* = σ n Equation for Line of Best Fit

Standard Error of the Mean

Confidence Interval Variance ( n −1) s 2 ( n −1) s2