Download STAT Formulas 05082005

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
s
2
sx 
n 1
Population Variance
 x   


i
i
n 1
Population Standard Deviation
2
2
 x  x
2
i
 x   
2
x
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
sxy 
Population Covariance
 x  xy  y 
i
 xy  
i
n 1
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
g  
i
n
Sample Variance for Grouped Data
s g2
 f M

i
i
Population Mean for Grouped Data
 xg


2
g
Sample Standard Deviation for Grouped Data
sg 

 f i M i  xg
n 1

N
Population Variance for Grouped Data
2
n 1
fi M i
 f M

i
 g 
2
i
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)
or
P( A | B) 
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
f ( x) 
 xe 
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
(Standard Error)
For an Infinite Population
N n  


N 1  n 
x 

n
Sampling Proportions
Standard Deviation of
p̂
For a Finite Population
 pˆ 
N n
N 1
(Standard Error)
For an Infinite Population
pˆ qˆ
n
Interval Estimate of a Population Mean:
xE
Where
Where

Known

Unknown
  
E  Z

 n
Interval Estimate of a Population Mean:
xE
 pˆ 
pˆ qˆ
n
 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
Sample Mean
x
x
i
n
Sample Variance
Sample Standard Deviation
 x  x 

 x  x
2
s
2
2
i
sx 
n 1
i
n 1
Sampling Proportions
Standard Deviation of
For a Finite Population
N n
N 1
 pˆ 
p̂
(Standard Error)
For an Infinite Population
pˆ 1  pˆ 
n
 pˆ 
Interval Estimate of a Population Mean:
xE
Where
Where
Known
  
E  Z

 n
Interval Estimate of a Population Mean:
xE

pˆ 1  pˆ 
n
  
x  Z

 n

Unknown
 s 
E  t x 
 n
 s 
x  t x 
 n
Interval Estimate of a Population Proportion

p 1 p
n
pZ

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
2

* *
Z pq
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S22
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 2j 
ij  x j
i 1

2
n j 1
Overall Sample Mean (Grand Mean)
k
x
nj
 x
j 1 i 1
nT
ij
x


k
Mean Square Due to Treatments
MSTR 
Sum of Squares Due to Treatments
k
SSTR
k 1

SSTR   n j x j  x
j 1
Mean Square Due to Error
SSE
- 8 -nT  k
MSE 

2
Statistics Formulas
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
j 1
j

1
ni

 n1j
x

k 1
Mean Square Due to Error
 n
k
MSE 
j 1
j
 1s 2j
nT  k
-9-
2

Statistics Formulas
F Test Statistic
F
MSTR
MSE
Randomized Block Designs
Total Sum of Squares
b






k
2
SST   xij  x
i 1 j 1
Sum of Squares Due to Treatments
k
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
- 10 -
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
Simple Linear Regression Equation
y  0  1 x  
E  y    0  1 x
Estimated Simple Linear Regression Equation
yˆ  b0  b1 x
Least Squares Criterion
min
2
ˆ


y

y
 i i
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   yˆi  y
- 11 -

2
2
Statistics Formulas
Total Sum of Squares for Regression
SST  SSR  SSE
Coefficient of Determination
r2 
SSR
SST
Sample Correlation Coefficient
rxy  sign of b1  Coefficien t of Determinat ion
 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
b1
t
sb1
Mean Square Regression
MSR 
SSR
# independen t variable s
- 12 -
Statistics Formulas
F Test Statistic
F
Estimated Standard Deviation of
MSR
MSE
ŷ 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  yˆi  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

- 13 -

2
Statistics Formulas
Nonparametric Method Formulas
Mann-Whitney-Wilcoxon Test (Large Sample)
Mean : T  1 2 n1  n1  n2  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
- 14 -