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 + 2σ
Empirical Rule
About 68%: µ -σ to µ + σ
About 95%: µ -2σ to µ + 2σ
About 99.7%: µ -3σ to µ + 3σ
∑x
n
∑x
Population mean: µ =
N
∑( w • x)
Weighted mean: x =
∑w
Sample mean: x =
Sample coefficient of variation: CV =
s
 100%
x
Population coefficient of variation: CV =
∑( f • x)
∑f
highest value + lowest value
Midrange =
2
Mean for frequency table: x =
Sample standard deviation for frequency table:
s=
n [ ∑( f • x 2 ) ] − [ ∑( f • x) ]
n (n − 1)
Range = Highest value - Lowest value
Sample standard deviation: s =
∑( x − x )
n −1
Population standard deviation: σ =
Sample z-score: z =
2
∑( x − µ )
N
2
x−x
s
Population z-score: z =
x−µ
σ
Interquartile Range: (IQR)
= Q3 − Q1
Sample variance: s 2
Population variance: σ 2
σ
 100%
µ
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 = ∑[ x 2 • P( x)] − µ 2
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
r = number of successes (or x)
p = probability of success
Binomial probability distribution
P ( r ) = n Cr p r q n − r
n!
Permutation rule: n Pr =
(n − r )!
Combination rule:
n Cr
=
Mean: µ = np
n!
r !(n − r )!
Standard deviation: σ = npq
Poisson Distributions
Permutation and Combination on TI 83/84
n
Math
PRB
nPr
enter
r = number of successes (or x)
µ = mean number of successes
r
(over a given interval)
Poisson probability distribution
n
Math
PRB
nCr
enter
r
e− µ µ r
P(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 zσ + µ
Standard score: z =
Point estimate = Upper limit + Lower limit
2
x−µ
σ
Error = Upper limit - Lower limit
2
Mean of x distribution: µ x = µ
Sample Size for Estimating
Standard deviation of x distribtuion: σ x =
(standard error)
Standard score for x : z =
σ
n
means:
z σ 
n =  α /2 
 E 
x −µ
2
proportions:
σ/ n
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
(np > 5 and nq > 5)
for proportions (p) :
variance or standard deviation:
*see table 7-2 (last page of formula sheet)
pˆ − E < p < pˆ + E
where E = zα / 2
pˆ =
p(1 − p)
n
Confidence Intervals
Level of Confidence
r
n
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
where E = tα / 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
s
n
with d . f .= n − 1
for variance (σ 2 ) :
z-value ( zα / 2 )
(n − 1) s 2
χ R2
< σ2 <
(n − 1) s 2
χ L2
with d . f .= n − 1
3
Chapter 8
Chapter 9
One Sample Hypothesis Testing
Difference of means μ1 -μ 2 (independent samples)
Confidence Interval when σ 1 and σ 2 are known
(x1 − x2 ) − E < ( µ1 − µ2 ) < (x1 − x2 ) + E
pˆ − p
z=
pq / n
for p (np > 5 and nq > 5) :
r/n
where q =
1 − p; pˆ =
for µ (σ known): z =
x −µ
for µ (σ unknown): t=
for σ 2 : χ 2=
(n − 1) s 2
σ2
s/ n
σ 12
with d . f .= n − 1
n1
σ 22
n2
+
σ 22
n2
Confidence Interval when σ 1 and σ 2 are unknown
with d . f .= n − 1
(x1 − x2 ) − E < ( µ1 − µ2 ) < (x1 − x2 ) + E
s12 s22
+
n1 n2
=
E tα / 2
Chapter 9
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
where E zα / 2
=
n1
+
Hypothesis Test when σ 1 and σ 2 are known
( x − x ) − ( µ1 − µ2 )
z= 1 2
σ/ n
x −µ
σ 12
=
where
E zα / 2
( x1 − x2 ) − ( µ1 − µ2 )
s12 s22
+
n1 n2
with d .=
f . smaller of n1 − 1 and n2 − 1
pˆ1qˆ1 pˆ 2 qˆ2
+
n1
n2
Matched pairs (dependent samples)
Confidence Interval
d − E < µd < d + E
pˆ1 =
r1 / n1 ; pˆ 2 =
r2 / n2 and qˆ1 =
1 − pˆ1 ; qˆ2 =
1 − pˆ 2
s
=
where E tα / 2 d with d.f. = n − 1
n
Hypothesis Test:
z=
Hypothesis Test
d − µd
t=
with d . f .= n − 1
sd
( pˆ1 − pˆ 2 ) − ( p1 − p2 )
pq pq
+
n1
n2
n
where the pooled proportion is p
Two Sample Variances
r +r
p = 1 2 and q = 1 − p
n1 + n2
Confidence Interval for σ 12 and σ 22
 s2
1  σ 12  s12
1 
 12 •
< 2 < 2 •

 s 2 Fright  σ 2  s 2 Fleft 




=
pˆ1 r1=
/ n1 ; pˆ 2 r2 / n2
s12
where s12 ≥ s22
s22
numerator d . f . =
n1 − 1 and denominator d . f . =
n2 − 1
Hypothesis Test=
Statistic: F
4
Chapter 10
Chapter 11
Regression and Correlation
where E =
∑
χ2 =
(O − E ) 2
E
Linear Correlation Coefficient (r)
OR
r
Tests of Independence d . f . =( R − 1)(C − 1)
n ∑ xy − (∑ x)(∑ y )
r=
n( ∑ x 2 ) − ( ∑ x ) 2 n( ∑ y 2 ) − ( ∑ y ) 2
Σ( z x z y )
=
where z x
n −1
(row total)(column total)
sample size
z=
score for x and z y
Goodness of fit d . f . (number of categories) − 1
=
z score for y
Chapter 12
Coefficient of Determination: r 2 =
Standard Error of Estimate:
or s e =
explained variation
total variation
∑( y − yˆ ) 2
n−2
se =
One Way ANOVA
=
k
∑ y 2 − b0 ∑ y − b1 ∑ xy
n−2
2
SSTOT =
∑ xTOT
−
Prediction Interval: yˆ − E < y < yˆ + E
where
=
E tα / 2 se 1 +
number
=
of groups; N total sample size
 (∑ xi ) 2
∑  n
i
all groups 
=
SS BET
n( x0 − x ) 2
1
+
n n ( Σx 2 ) − ( Σx ) 2
SS=
W
Sample test statistic for r
r
with d . f .= n − 2
t=
1− r2
n−2
where b1
MS BET =
MSW =
note that b0 is the y-intercept and b1 is the slope
sy
n ∑ xy − (∑ x)(∑ y )
=
or b1 r
2
2
sx
n( ∑ x ) − ( ∑ x )
(∑ y )(∑ x 2 ) − (∑ x)(∑ xy )
n( ∑ x 2 ) − ( ∑ x ) 2
1
+
n
Confidence interval for slope β1
b1 − E < β1 < b1 + E
se
where E = tα / 2 •
(∑ x) 2
∑ x2 −
n
SSW
where d . f .W= N − k
d . f .W
MS BET
where d . f . numerator = d . f .BET = k − 1
MSW
d . f . denominator = d . f .W= N − k
or b0= y − b1 x
x2
(∑ x) 2
∑ x2 −
n
SS BET
where d . f .BET = k − 1
d . f .BET
F=
Confidence interval for y-intercept β 0
b0 − E < β 0 < b0 + E
where E = tα / 2 se



SS
=
SS BET + SSW
TOT
and
where b0=
 (∑ xTOT ) 2
 −
N


(∑ xi ) 2
2
∑
x
−

∑  i
ni
all groups 
Least-Squares Line (Regression Line or Line of Best Fit)
ˆ b0 + b1 x
y=
(∑ xTOT ) 2
N
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
with degrees of freedom for
row factor = r − 1
column factor = c − 1
interaction = ( r − 1)(c − 1)
error = rc(n − 1)
5
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 
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

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    

 
 
 




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




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 
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
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

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
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
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



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






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


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
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


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
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
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

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
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


 




























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
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








 

























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
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


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



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
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

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
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

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


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
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
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

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
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


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
























  

 






















 



















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
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



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
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











































































 














































 








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
0.05
z = - 1.645
z=0
z = 1.28
z = - 2.575
z=0
0.01
z=0
>
0.01
0.05
z=0
z = 1.645
Figure 8.4
z = 2.575
<
>
0.10
z=0
0.005
z = - 2.33
z=0
>
0.005
z=0
z = 2.33
 
  


















































































































































































































































































































































 


 

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

 


  
  
    
  

 
  
 
Greek Alphabet
http://www.keyway.ca/htm2002/greekal.htm