Download STAT Formulas 05082005

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
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 -
Related documents