Stat Formulas Jen.docx

Statistics Formulas Sample Mean

x

x

i

n

Population Mean

x  

xi

N

Interquartile Range

IQR  Q3  Q1 Sample Variance

Sample Standard Deviation

 x  x  

2

s2

2

i

sx 

n 1

Population Variance

  2

 x  x  i

n 1

Population Standard Deviation

2   x    i x

 x   

2

x 

N

i

x

N

Coefficient of Variation

 Standard Deviation  CV   100 % Mean   Z-Score

Zi 

xi  x x  or Z i  i s 

Sample Covariance

s xy 

 x  x y i

i

y



n 1

Population Covariance

 xy  

xi   x yi   y  N

Pearson Product Moment Correlation Coefficient (Pearson R): Sample Data

rxy 

s xy sx s y

Population Data

 xy 

 xy  x y -1-

Statistics Formulas Weighted Mean

xw 

w x w

i i i

Sample Mean for Grouped Data

xg 

fM i

Population Mean for Grouped Data

g  

i

n

Sample Variance for Grouped Data

s g2 

 f M i

i

 xg

n 1



2

sg 

 

n 1



N

Population Variance for Grouped Data

 g2  

Sample Standard Deviation for Grouped Data

f i M i  xg

fi M i

f i M i   g 

2

N

Population Standard Deviation for Grouped Data

2

g 

-2-



f i M i   g 

2

N

Statistics Formulas Counting Rule for Combinations n

Cr 

n! r!(n  r )!

Counting Rule for Permutations n Pr 

n! (n  r )!

Computing Probability Using the Complement

P( A)  1  P( AC ) Addition Law P( A  B)  P( A)  P( B)  P( A  B) Conditional Probability P( A  B) P( A | B)  or P( B) Multiplication Law P( A  B)  P( B) P( A | B)

P( B | A) 

or

P( A  B) P( A)

P( A  B)  P( A) P( B | A)

Multiplication Law for Independent Events P( A  B)  P( A) P( B) Expected Value of a Discrete Random Variable E ( x)     x f ( x) Variance of a Discrete Random Variable 2 Var ( x)   2   x    f ( x) Binomial Probability Function

f ( x) 

n! p x (1  p ) ( n  x ) x!(n  x)!

Expected Value and Variance for the Binomial Distribution

E ( x)    np

Var ( x)   2  np(1  p)

Poisson Probability Function

 xe f ( x)  x!

where f (x) = the probability of x occurrences in an interval = the expected value or mean number of occurrences in an interval  e = 2.718 -3-

Statistics Formulas Sampling & Sampling Distributions

Standard Deviation of For a Finite Population

x 

x

N n     N 1  n 

(Standard Error) For an Infinite Population

x 



n

Sampling Proportions Standard Deviation of

pˆ

For a Finite Population

N n N 1

 pˆ 

For an Infinite Population

pˆ qˆ n

Interval Estimate of a Population Mean:

xE

Where

Where

 pˆ 

pˆ qˆ n



Known



Unknown

   E  Z   n

Interval Estimate of a Population Mean:

xE

(Standard Error)

 s  E  t  n  

Sample Size for an Interval Estimate of a Population Mean

Z 2 2 n E2 Interval Estimate of a Population Proportion

pˆ  Z

pˆ qˆ n

Sample Size for an Interval Estimate of a Population Proportion

Z 2 p *q * n E2

-4-

Statistics Formulas Sampling Proportions Test Statistic for Hypothesis Tests About a Population Proportion

Z

p  p0 p0 1 p0  n

pˆ

Standard Deviation of For a Finite Population

 pˆ 

N n N 1

pˆ 1  pˆ  n

Interval Estimate of a Population Mean:

xE

Where

Where

For an Infinite Population

 pˆ 



pˆ 1  pˆ  n

Known

   E  Z   n

Interval Estimate of a Population Mean:

xE

(Standard Error)

   x  Z   n



Unknown

 s  x  t x   n

 s  E  t x   n

Interval Estimate of a Population Proportion

pZ



p 1 p n



Interval Estimate of the Difference Between Two Population Proportions

p1  p2  Z







p1 1  p1 p2 1  p2  n1 n2



 Z  n

Sample Size for an Interval Estimate of Population Proportions and Normal Distributions 

Z 2 p *q * n E2

2

2

E2

-5-

2 x

Statistics Formulas Z-Value Formulas for Normal and “Approximately Normal” Distributions – Calculations of Test Statistics

Z

x  x 

t

 Known

x  0  Unknown

Sx n

n

Test Statistic for Hypothesis Tests About Two Independent Proportions

p  p  1 1 p 1  p    n n

Z

1

2



1

2



Standard Error of x1  x2

 x x 1

 12 2

n1



 22 n2

Degrees of Freedom for the t Distribution Using Two Independent Random Samples 2

 s12 s22      n1 n2  d. f .  2 2 2 2    1 s 1 s   1    2  n1  1  n1  n2  1  n2  Test Statistic for Hypothesis Tests About Two Independent Means; Population Standard Deviations Unknown

t

x  x  D 1

2

0

s12 s22  n1 n2

Test Statistic for Hypothesis Tests Involving Matched Samples

t

d  d sd n -6-

Statistics Formulas Mean Difference Involving Matched (or Dependent) Samples

 d 

d

i

n

Standard Deviation Notation Used for Matched Samples

 d  d 

2

sd 

i

n 1

Interval Estimate of Means of Matched Samples

d  t / 2  S dn Test Statistic for Hypothesis Tests About a Population Variance



2

 n  1s 2  

 2 d. f .  n 1

2 0

Test Statistic for Hypothesis Tests About Two Population Variances when  12   22

s12 F 2 s2

df numerator  n1  1

and

df denominator  n2  1

Interval Estimate of the Difference Between Two Population Means: σ1 and σ2 Known and Unknown

x1  x2  Z / 2

 12 n1



 22 n2

(σ known)

x1  x2  t / 2

S12 S 22  n1 n2

(σ unknown)

Pooled Sample Standard Deviation (Assumptions that σ1 and σ2 are equal and populations are

approximately normal) & Test Statistic for Comparison of Independent Samples

Sp 

n1  1S12  n2  1S 22 n1  n2  2

d . f .  n1  n2  2

-7-

t

x  x  D 1

Sp

2

1 1  n1 n2

0

Statistics Formulas Chi-Square Goodness of Fit Test Statistic

  2

 f o  f e 2 fe

; d . f .  (c  1)

Chi-Square Test for Independence Test Statistic

   2

i

 f oij  f eij 2 f eij

; d . f .  (c  1)(r  1)

j

Testing for the Equality of k Population Means—ANOVA Sample Mean for Treatment j nj

xj 

x

ij

i 1

nj

Sample Variance for Treatment j

 x nj

s  2 j

ij

i 1

 xj



2

n j 1

Overall Sample Mean (Grand Mean) k

x

nj

 x j 1 i 1

ij

nT



x k

Mean Square Due to Treatments

SSTR MSTR  k 1 Sum of Squares Due to Treatments k



SSTR   n j x j  x j 1

Mean Square Due to Error

-8-



2

Statistics Formulas

MSE 

SSE nT  k

Sum of Squares Due to Error

SSE   n j  1s 2j k

j 1

Test Statistic for the Equality of k Population Means

F

MSTR MSE

Total Sum of Squares nj

k



SST   xij  x j 1 i 1

Partitioning of Sum of Squares



2

SST  SSTR  SSE

Multiple Comparison Procedures Test Statistic for Fisher’s LSD Procedure

t

xi  x j MSE



1 ni

 n1j

Fisher’s LSD

LSD  t / 2 MSE Completely Randomized Designs Mean Square Due to Treatments

 n x k

MSTR 

j 1

j

j



1 ni

-9-

 n1j

x

k 1

Mean Square Due to Error





2



Statistics Formulas

 n k

MSE 

j 1

j

 1s 2j

nT  k

- 10 -

Statistics Formulas F Test Statistic

MSTR F MSE Randomized Block Designs Total Sum of Squares b













k

SST   xij  x i 1 j 1

Sum of Squares Due to Treatments k

2

SSTR  b x  j  x j 1

2

Sum of Squares Due to Blocks b

SSBL  k  x i  x i 1

Sum of Squares Due to Error

2

SSE  SST  SSTR  SSBL

Factorial Experiments Total Sum of Squares a

b

r



SST   xijk  x i 1 j 1 k 1

Sum of Squares for Factor A









a

SSA  br  x i  x i 1

2

Sum of Squares for Factor B b

SSB  ar  x  j  x j 1

- 11 -

2



2

Statistics Formulas Sum of Squares for Interaction a

b



SSAB  r  x ij  x i   x  j  x i 1 j 1

Sum of Squares for Error



SSE  SST  SSA  SSB  SSAB

Simple Linear Regression Formulas Simple Linear Regression Model

y   0  1 x  

Simple Linear Regression Equation

E  y    0  1 x

Estimated Simple Linear Regression Equation

yˆ  b0  b1 x Least Squares Criterion

min   yi  yˆ i 

2

Slope and y-Intercept for the Estimated Regression Equation

b1

 x  x y  y     x  x  i

i

2

i

b0  y  b1 x Total Sum of Squares



SST   yi  y Sum of Squares Due to Error



2

SSE    yi  yˆ i 

2

Sum of Squares Due to Regression

SSR  



- 12 -

yˆ i  y



2

2

Statistics Formulas Total Sum of Squares for Regression

SST  SSR  SSE

Coefficient of Determination

SSR r  SST 2

Sample Correlation Coefficient

rxy  sign of b1  Coefficient of Determination  sign of b1  r 2 Mean Square Error (Estimate of σ2 )

s 2  MSE 

SSE n2

Standard Error of the Estimate

s 2  MSE  Standard Deviation of b1

b  1

SSE n2



 x  x 

2

i

Estimated Standard Deviation of b1

sb1 

s

 x  x 

2

i

t Test Statistic

t

b1 sb1

Mean Square Regression

MSR 

SSR # independent variables

- 13 -

Statistics Formulas F Test Statistic

MSR F MSE Estimated Standard Deviation of

yˆ p

 x  x 1  n  x  x  2

s yˆ p  s Confidence Interval for

p

2

i

E y p 

yˆ p  t / 2 s yˆ p Estimated Standard Deviation of an Individual Value

 x  x 1 1  n  x  x  2

sind  s

p

2

i

Prediction Interval for yp

yˆ p  t / 2 sind

Residual for Observation i

yi  yˆ p Standard Deviation of the i th Residual

s yi  yˆ i  s 1  hi Standardized Residual for Observation i

Leverage of Observation i

ˆi yi  y s yi  yˆ i





2

xi  x 1 hi  n  xi  x



- 14 -



2

Statistics Formulas Nonparametric Method Formulas Mann-Whitney-Wilcoxon Test (Large Sample)

Mean : T 

1

n  n1  n2  1

2 1

Standard Deviation:  T 

n n  n1  n2  1

1 12 1 2

Kruskall-Wallis Test Statistic

k  Ri2  12 W    3  nT  1   nT  nT  1 i 1 ni  Spearman Rank –Correlation Coefficient

rs  1 

6 di2





n n2  1

- 15 -