Statistics Formulas
Formula for the range:
R = highest value – lowest value
Formula for the standard deviation for population data:
σ=
Formula for determining class width:
range
Class width =
# of classes desired
 ( X  ) 2
N
Formula for the standard deviation for sample data:
 ( X 2 )  [(  X ) 2 / n]
n 1
Formula for the size of the class width:
Class width = upper boundary – lower boundary
s=
Formula for the class midpoint:
lower boundary + upper boundary
Xm =
2
Formula for the standard deviation for grouped data:
Formula for relative frequency:
f
Relative frequency =
n
Formula for the percentage of values in each class:
f
%=
∙ 100%
n
Formula for the degrees for each section of a pie graph:
f
Degrees =
∙ 360°
n
Formula for the mean for a population:
ΣX
μ=
N
Formula for the mean for a sample (ungrouped):
ΣX
X = n
Formula for the mean for grouped data:
Σ(f∙Xm)
X =
n
s=
(  f  X m 2 )  [(  f  X m ) 2 / n]
n1
Formula for the coefficient of variation:
Sample:
s
CVar =
∙ 100%°
X
Population:
σ
CVar =
∙ 100%°
μ
Range rule of thumb:
range
s≈
4
Chebyshev’s theorem: The proportion of values from a data
set that will fall within k standard deviations of the mean
will be at least
1
1–
k2
where k is a number greater than 1.
Formula for the z score (standard score);
Sample:
Formula for the weighted mean:
ΣwX
X =
Σw
X– X
s
Population:
X–μ
z=
σ
Formula for the midrange:
lowest value – highest value
MR =
2
Formula for cumulative percentage:
Cumul freq
Cumulative % =
∙ 100%
n
Formula for the variance for population data:
Σ(X – μ)2
σ2 =
N
Formula for the percentile rank of a value X:
# of values below X + 0.5
Percentile =
∙ 100%
Total # of values
Formula for the variance for sample data:
Σ(X2) – [(ΣX)2/n]
s2 =
n-1
Formula for finding a value corresponding to a given
percentile (gives data position):
n∙p
c=
100
Formula for the variance for grouped data:
(Σf∙Xm2) – [(Σf∙Xm)2/n]
s2 =
n-1
z=
Formula for interquartile range:
IQR = Q3 – Q1
Formulas for range to exclude outliers:
Q1 – IQR(1.5) and Q3 + IQR(1.5)
Permutations: (order is important) “How many different
ways…”
The arrangement of n objects in a specific order using r
objects at a time.
n
Probability Formulas
 P( X )  1 and 0  P( X )  1
Classical Probability:
Number of outcomes in E
Total number of outcomes in sample space
or
P(E)
=
n(E)
n(S)
Complementary Events:
Pr 
n!
(n  r )!
n = total # in population; r = # selected
Combinations: (order is not important)
The number of combinations of r objects selected from n
objects.
n
Cr 
n!
(n  r )! r !
Mean of a Probability Distribution:
X = ∑[X ∙ P(X)]
P( E ) = 1 – P(E) or P(E) = 1 – P( E ) or
Variance for a Population Distribution:
P(E) + P( E ) = 1
 2  [( X 2  P( X )]   2
Empirical Probability: Round to 2 or 3 decimal places or
fully reduce fraction.
frequency for the class
f
P(E) =
=
total frequency in the distribution
n
Standard Distribution for a Probability Distribution:
Addition Rules (Or events):
When two events are mutually exclusive, the probability
that A or B will occur is:
P(A or B) = P(A) + P(B)
If A and B are not mutually exclusive, then:
P(A or B) = P(A) + P(B) – P(A and B)
Multiplication Rules (And events):
When two events are independent, the probability of both
occurring is:
P(A and B) = P(A) ∙ P(B)
When two events are dependent, the probability of both
occurring is:
P(A and B) = P(A) ∙ P(B|A)
Conditional Probability—The probability that the second
event B occurs given that the first event A has occurred can
be found by dividing the probability that both events
occurred by the probability that the first event has occurred.
The formula is:
P(A and B)
P(B|A) =
P(A)
Fundamental Counting Rule:
k1 ∙ k2 ∙ k3 ∙ ∙ ∙ kn
Factorial Notation:
5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
4! = 4 ∙ 3 ∙ 2 ∙ 1
Note that 0! = 1

 [ X 2  P( X )]   2 or  
2
Expected Value:
E(X) = ∑[X ∙ P(X)] or
E(X) = X ∙ P(X) of gain + X ∙ P(X) of loss
Binomial probability formula: (round to 3 decimal places)
P( X ) 
n!
 p x  q n X
(n  X )! X !
P(S) = probability of success P(S) = p; P(F) = probability
of failure P(F) = 1-p = q; p = numerical probability of
success; q = numerical probability of failure; n = number of
trials; X = number of successes in n trials.
Mean of a binomial distribution:
μ=n∙p
Variance of a binomial distribution:
2  n p q
Standard deviation of a binomial distribution:
  n  p  q or    2
Formula for the z value (or standard score):
X 
z
or

value - mean
z=
standard deviation
Formula for finding a specific data value:
X  z   
Formula for the mean of the sample means:
X  
Formula for the standard error of the mean:

X 
n
Formula for the z value for the central limit theorem (for a
sample mean when the variable is normally distributed or
when the sample size is 30 or more):
X 
z
/ n
Formula for the z value for the central limit theorem (for
individual data when the variable is normally distributed):
X 
z

Finite population correction factor when large samples are
taken from small population:
N n
N is the population size and n is the sample size.
N 1
Standard area of the mean using correction factor:

N n
X 

N 1
n
Formula for z value using correction factor:
X 
z

N n

N 1
n
 s 
 s 
X  t / 2 
    X  t / 2 
 STAT; TESTS; 8
 n
 n
t /2 found in Table F; use C.I. in top row and degree of
freedom (n – 1) in left column.
Proportion Notation:
p = population proportion; p = sample proportion
X
p 
and q  1  p where X = number of sample units
n
that possess the characteristics of interest and n = sample
size.
Formula for a specific C.I. for a proportion:


pq
pq
when np and nq are ≥ 5.
p  z / 2
 p  p  z / 2
n
n
STAT; TESTS; A
Formula for minimum sample size needed for interval
estimate of a population proportion:
2
z 
   / 2  ; round up to obtain whole number.
n  pq
 E 
If sample proportion unavailable, use 0.5 for p and q .
Formula for the C.I. for a variance:
n  1s 2 2 n  1s 2
 
; d.f. = n – 1
2
2
X right
X left
s2 = variance; s = standard deviation; square number in
formula only when appropriate.
Formula for the C.I. for a standard deviation:
Formulas for the mean and standard deviation for the
binomial distribution:
  n  p and   n  p  q ; n  p  5 , n  q  5
Formula for a Specific Confidence Interval for the Mean
when σ (pop. S.D.) is known or n ≥ 30) and Sample Size:
  
  
X  z / 2 
    X  z / 2 
 STAT; TESTS; 7
 n
 n
z /2 for 90% C.I. 1.65, for 95% 1.96, for 98% 2.33, for
99% 2.58
To find other z /2 : C.I./2, find answer in body of Table E
(closest or higher if halfway), use corresponding z score.
Formula for minimum sample size needed for an interval
estimate of the population mean:
 z  
n    /2
 ; E = maximum error of estimate; always
 E 
round to next whole number
2
Formula for a Specific Confidence Interval for the Mean
when σ is unknown and n < 30:
n  1s 2
2
X right
 
n  1s 2
2
X left
; d.f. = n – 1
2
To find: X right
: Calculate (1 – C.I.)/2. Go to Table G—use
2
with d.f. to find X right
.
2
X left
: 1 – [(1 – C.I.)/2]. Look up in Table G.
(1 – C.I. is α)
Chapter 8—Hypothesis Testing
z Test for a Mean (n ≥ 30 or σ is known):
(observed value) – (expected value)
Test value =
standard error
z
X
STAT; TESTS; 1
/ n
X = sample mean, μ = hypothesized population mean,
σ = population standard deviation, n = sample size
Critical Value for Specific α Values (Table E):
One-tailed: 0.5 –α
Two-tailed: 0.5 –α/2
Find value obtained in Table E; use closest value.
Find z-value corresponding to area.
P-value (≤ α = reject Ho):
One-tailed: Find area corresponding to z score.
0.5 – area
Two-tailed: Find area corresponding to z score.
(0.5 – area)2
t Test for a Mean (σ unknown and n < 30):
t
X
s/ n
P-value Interval (≤ α = reject Ho):
Right-tailed: Look across row with d.f. needed and find two
values that X2 falls between. Look to top and find
corresponding α values.
Left-tailed: Above then subtract both from 1. :Look to top
and find corresponding α values.
Two-tailed: Above for right- or left-tailed then double.
Chapter 9—Difference Between Two Means, Variances,
and Proportions
z Test for Comparing Two Means from Independent
Populations: Large Samples
z
( X1  X 2 )  ( 1  2 )
12
n1
Critical Value (Table F):
One-tailed: Where α for one-tailed and d.f. meet (use
appro.– or + #).
Two-tailed: Where α for two-tailed and d.f. meet (both
signs).
P-value (if α not in interval, reject Ho or P-value < α):
Find the two values that the t score in row with appropriate
d.f. fall between; look up corresponding α’s at top (one- or
two-tailed); put into inequality format (smallest # first).
STAT; TESTS; 3
n2
Confidence Interval for Difference Between Means: Large
Samples (if C.I. does not contain zero, reject Ho).
( X1  X 2 )  z / 2
12
n1

 22
 1  2 STAT; TESTS; 9
n2
12
 ( X1  X 2 )  z / 2
STAT; TESTS; 2
s = sample standard deviation

 22
n1
When n1 ≥ 30 and n2 ≥ 30,
s12

 22
n2
and s22 can be used in place
of  12 and  22 .
F Test for the Difference Between Two Variances
F
s12
s22
STAT; TESTS; E
Larger variance always in the numerator.
Hypothesis: 12   22 , etc.
Two-tailed test: α/2; C.V. on right side
Square standard deviations if used.
Table H—If d.f. not found, use closest smaller value.
z Test for a Proportion:
p  p
z
p 
pq / n
STAT; TESTS; 5
X
(sample proportion), p = population proportion,
n
n = sample size, q = 1 – p
Critical Values and P-value: As in z test for mean.
2
X Test for a Variance or Standard Deviation:
X2 
(n  1) s2
2
n = sample size, s2 = sample variance, σ2 = population
variance; d.f. = n – 1.
Critical Value (Table G):
Right-tailed: Find where d.f. and α meet.
Left-tailed: Find where d.f. and 1 – α meet.
Two-tailed: Find where d.f. meets α/2 and 1 – α/2.
t Test for Difference Between Two Means—Small
Independent Samples
Variances assumed to be unequal:
t
( X1  X 2 )  ( 1  2 )
s12 s22

n1 n2
STAT; TESTS; 4; Pool No
d.f. = small of n1 – 1 or n2 – 1.
Variances assumed to be equal:
t
( X1  X 2 )  ( 1  2 )
(n1  1) s12  (n2  1) s22
n1  n2  2
d.f. = n1  n2  2
1
1

n1 n2
Pool Yes
Confidence Intervals for the Difference of Two Means:
Small Independent Samples
Variances unequal:
( X1  X 2 )  t / 2
s12 s22

 1  2 STAT; TESTS; 0; Pool No
n1 n2
s12 s22

n1 n2
( X1  X 2 )  t / 2
( X1  X 2 )  t / 2
(n2  1) s22

n1  n2  2

1
1

n1 n2
d.f. = n1 + n2 -2
Pool Yes
t Test for Difference Between Two Means: Small
Dependent Samples
t
D  D
STAT; TESTS; 2 (L3 = L1 – L2)
sD / n
(  D)2
n
; D  X1  X2 ; and
n1
 D2 
D
; sD 
D  0 ; D 
n
D2  ( X1  X2 )2
Confidence Interval for the Mean Difference
sD
n
 D  D  t / 2
sD
n
STAT; TESTS; 8 (L3 above)
d.f. = n – 1
z Test for Comparing Two Proportions
z
( p1  p2 )  ( p1  p2 )
1
1
pq ( 
n1 n2
STAT; TESTS; 6
where
X1  X 2
; q  1 p ;
n1  n2
X
X
p1  1 ; p2  2
n1
n2
p
Confidence Interval for Difference Between Two
Proportions
( p1  p2 )  z / 2
p1q1 p2q2

n1
n2
STAT; TESTS; B
 p1  p2  ( p1  p2 )  z / 2
p1q1 p 2q2

n1
n2
Chapter 10—Correlation and Regression
Correlation Coefficient r
r
Regression Line y’ = a + bx
(  y)(  x 2 )  (  x )(  xy )
n(  x 2 )  (  x ) 2
n(  xy )  (  x )(  y )
n(  x 2 )  (  x ) 2
a is the y1 intercept and b is the slope of the line.
Standard Error of the Estimate
sest 
 ( y  y  )2
or
n 2
 y 2  a y  b xy
n 2
Prediction Interval about a Value y’
y   t / 2 sest 1 
1
n( x  X ) 2

n n x 2  (  x )2
 y  y   t / 2 sest 1 
d.f. = n – 1
D  t / 2
n 2
1  r2
d.f. = n – 2
b
(n1  1) s12  (n2  1) s22
1
1


n1  n2  2
n1 n2
 1  2  ( X1  X 2 )  t / 2
tr
a
d.f. = smaller of n1 – 1 or n2 – 1.
Variances equal:
(n1  1) s12
t Test for Correlation Coefficient
n(  xy )  (  x )(  y )
[n(  x )  (  x ) 2 ][n(  y 2 )  (  y ) 2 ]
2
n is number of data pairs
STAT; CALC; 8
d.f. = n – 2
1
n( x  X ) 2

n n x 2  (  x )2