KEY FORMULAS
Chapter 2


Mean for ungrouped data: µ=∑x/N and 𝑥̅ =∑x/n
Mean for grouped data: µ=∑mf/N and 𝑥̅ =∑mf/n
where m is the midpoint and f is the frequency of
a class

Median for ungrouped data = Value of the (


term in a ranked data set
Range = Largest Value-Smallest Value
Standard deviation for ungrouped data:
𝜎=√
2
∑𝑁
𝑖=1(𝑥𝑖 −µ)

2
∑𝑛
𝑖=1(𝑥𝑖 −𝑥̅ )
2
∑𝐾
𝑖=1 𝑓𝑖 (𝑚𝑖 −µ)
Standard deviation of discrete random variable x:
𝜎 = √∑ 𝑥 2 𝑃(𝑥) − 𝜇2





Hypergeometric probability Formula: P(x)=
(𝑥𝑟 )(𝑁−𝑟
𝑛−𝑥 )
(𝑁
𝑛)
𝑒 −𝜆 𝜆𝑥

Poisson probability Formula: P(x)=

Mean and variance of the Poisson probability
distribution: E(x)= 𝜇 = 𝜆, Var(x)= 𝜇 = 𝜆.
𝑥!
Chapter 5

2
∑𝐾
𝑖=1 𝑓𝑖 (𝑚𝑖 −𝑥̅ )
𝑛−1
𝜎
𝑠
𝜇
𝑥̅
Chebyshev’s Theorem:
For any population with mean µ, Standard
deviation 𝜎 , and k>1, the percent of observations
that lie within the interval [µ±k 𝜎] is
at least [1-(1/𝑘 2 )]%
where k is the number of Standard deviations.
Emprical Rule:
For many large populations the empirical rule
provides an estimate of the approximate
percentage of observations that are contained
within one, two, or three Standard deviations of
the mean:
o Approximately 68% of the observations
are in the interval (𝜇 ± 𝜎) ,
o
Approximately 95% are in the interval
(𝜇 ± 2𝜎),
o Almost all of the observations are in the
interval (𝜇 ± 3𝜎).
Interquartile Range: IQR = 𝑄3 − 𝑄1 where 𝑄3 is
the third quartile and 𝑄1 is the first quartile.
∑(𝑥𝑖 −𝑥̅ )(𝑦𝑖 −𝑦̅)

𝐶𝑜𝑣(𝑥, 𝑦) =

Correlation coefficient;
𝑛−1
𝑟𝑥𝑦 =
Mean and Standard deviation of the uniform
probability distribution:
𝜇=
Coefficient of Variation:
𝐶𝑜𝑣(𝑥, 𝑦)
𝑠𝑥 𝑠𝑦
Chapter 3

(N
)=
n



P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

N!
n!(N−n)!
Bayes′ Theorem: P( Ai ∣ B ) =
P(Ai )P(B∣Ai )
P(A1 )P( B∣∣A1 )+⋯+P(An )P( B∣∣An )
𝑎+𝑏
2
and 𝜎 = √
(𝑏−𝑎)2
12
𝑧 value for an x value:
𝑥−𝜇
𝜎
Normal Distribution Approximation for Binomial
Distribution:
𝑧=

Z=
𝑋−𝑛𝑃
√𝑛𝑃(1−𝑃)

Exponential Distribution
The probability that the time btw arrivals is 𝑡𝑎 or
less is as follows:
𝑃(𝑇 ≤ 𝑡𝑎 ) = (1 − 𝑒 −𝜆𝑡𝑎 )
Chapter 6


Standard deviation (Standard error) of 𝑋̅
𝜎
𝜎𝑋̅ =
√𝑛
If n/N>0.05 Standard deviation (Standard error) of
𝑋̅
𝜎𝑋̅ =

𝜎
√𝑛
𝑁−𝑛
.√
𝑁−1
Standard deviation of 𝑝̂
𝑃(1 − 𝑃)
𝜎𝑝̂ = √
𝑛
Chapter 7
Confidence Interval for;
 population mean with known variance
𝜎
𝑥̅ ±𝑧𝛼/2
√𝑛
 population mean with unknown variance
𝑠
𝑥̅ ± 𝑡𝑛−1,𝛼/2
√𝑛
 population proportion
𝑝̂ ± 𝑧𝛼/2 √
P(A ∩ B) = P(B)P(A ∣ B)
Chapter 4
Mean and Standard deviation of the binomial
distribution: 𝜇 = 𝑛𝑝 𝑎𝑛𝑑 𝜎 = √𝑛𝑝𝑞
CV = 𝑥 100% or 𝑥 100%

Binomial probability Formula: P(x)=(𝑛𝑥)𝑝 𝑥 𝑞𝑛−𝑥 ,
where p+q=1.
and
𝑁
𝑠= √

where 𝜎 and 𝑠 are population
𝑛−1
and sample Standard deviations, respectively.
Standard deviation for grouped data:
𝜎=√

2
Mean of a discrete random variable x: 𝜇 =
∑ 𝑥𝑃(𝑥)
)th
and
𝑁
𝑠= √
𝑛+1


𝑝̂ (1 − 𝑝̂ )
𝑛
Confidence interval for the population variance
(𝑛 − 1)𝑠 2
(𝑛 − 1)𝑠 2
< 𝜎2 < 2
2
ℵ𝑛−1,𝛼/2
ℵ𝑛−1,1−𝛼/2
Chapter 8
Confidence Interval for;


the difference between means
𝑠𝑑
𝑑̅ ± 𝑡𝑛−1,𝛼/2
√𝑛
Independent samples with known population
variances
(𝑥̅ − 𝑦̅) ± 𝑧𝛼/2 √

𝜎𝑥2
𝑛𝑥

Tests of the variance of a normal distribution;
(𝑛 − 1)𝑠 2
ℵ2 =
𝜎02
Chapter 10
Hypothesis Testing for Two Sample

Tests of the Difference Between Two Population
Means: Dependent Samples;
𝑑̅ − 𝐷0
𝑡(𝒏−𝟏,𝜶) =
𝑠𝑑 /√𝑛

Tests of the Difference Between Two Population
Means: Independent Samples;
1. Known Variances
𝑥̅ − 𝑦̅ − 𝐷0
𝑧𝛼 =
𝜎2 𝜎2
√ 𝑥+ 𝑦
𝑛𝑥 𝑛𝑦
𝜎𝑦2
+
𝑛𝑦
Independent samples with unknown population
variances and the variances are assumed to be
equal
𝑠𝑝2 𝑠𝑝2
(𝑥̅ − 𝑦̅) ± 𝑡𝑛𝑥 +𝑛𝑦−2,𝛼/2 √ +
𝑛𝑥 𝑛𝑦
𝑠𝑝2 =

[(𝑛𝑥 − 1) 𝑠𝑥2 + (𝑛𝑦 − 1) 𝑠𝑦2 ]
𝑛𝑥 + 𝑛𝑦 − 2
2.
Independent samples with unknown population
variances and the variances are not assumed to be
equal
𝑠𝑥2 𝑠𝑦2
(𝑥̅ − 𝑦̅) ± 𝑡(𝜗,𝛼/2) √ +
𝑛𝑥 𝑛𝑦
[
𝜗=

Unknown Population Variances: Assumed to
be Equal
𝑥̅ − 𝑦̅ − 𝐷0
𝑡(𝑛𝑥 +𝑛𝑦−2,𝛼) =
𝑠2 𝑠2
√ 𝑝+ 𝑝
𝑛𝑥 𝑛𝑦
[(𝑛𝑥 − 1) 𝑠𝑥2 + (𝑛𝑦 − 1) 𝑠𝑦2 ]
𝑛𝑥 + 𝑛𝑦 − 2
Population Variances Unknown and Not
Equal;
𝑠 2 𝑠𝑦2
[ 𝑥 + ]2
𝑛𝑥 𝑛𝑦
𝜗=
2
2
𝑠𝑦2
𝑠𝑥2
(
)
( )
𝑛𝑦
𝑛𝑥
+
𝑛𝑥 − 1 𝑛𝑦 − 1
𝑠𝑝2 =
3.
𝑠𝑥2 𝑠𝑦2 2
+ ]
𝑛𝑥 𝑛𝑦
2
2
𝑠𝑦2
𝑠2
( )
( 𝑥)
𝑛𝑦
𝑛𝑥
+
𝑛𝑥 − 1 𝑛𝑦 − 1
the difference between two population
proportion
𝑝̂𝑥 (1 − 𝑝̂𝑥 ) 𝑝̂𝑦 (1 − 𝑝̂𝑦 )
(𝑝̂𝑥 − 𝑝̂𝑦 ) ± 𝑧𝛼/2 √
+
𝑛𝑥
𝑛𝑦
𝑡(𝜗,𝜶) =
√
Sample Size determination


for the mean of a normally distributed population
with known population variance
2
zα/2
σ2
n=
ME2
for population proportion
(zα/2 )2 0.25
n=
ME2
Chapter 9
Hypothesis Testing for One Sample

Tests of the mean of a normal distribution:
population variance known;
̅̅̅
(𝑥 − 𝜇0 )
𝑧𝛼 =
𝜎/√𝑛

Tests of the mean of a normal distribution:
population variance unknown;
̅̅̅ − 𝜇0 )
(𝑥
𝑡(𝑛−1,𝛼) =
𝑠/√𝑛

Tests of the population proportion;
𝑧𝛼 =
𝑝̂ − 𝑃
√𝑃(1 − 𝑃)/𝑛
𝑥̅ − 𝑦̅ − 𝐷0

𝑠𝑥2 𝑠𝑦2
+
𝑛𝑥 𝑛𝑦
Tests of the Difference Between Two Population
Proportions;
𝑛𝑥 𝑝
̂𝑥 + 𝑛𝑦 𝑝
̂𝑦
𝑝
̂0 =
𝑛𝑥 + 𝑛𝑦
𝑧𝛼 =
(𝑝
̂𝑥 + 𝑝
̂)
𝑦
𝑝
̂(1 − 𝑝
̂) 𝑝
̂(1 − 𝑝
̂)
√ 0 𝑛 0 + 0 𝑛 0
𝑥
𝑦