BUSS1020 In-semester Examination
Formula Sheet
Range
Range= X largest−Xsmallest
Z-score
Z = X − X¯
S
Population Variance
σ2 =
N∑
i=1
(
Xi −µ
)2
N
Population Standard Deviation
σ=
√√√√√√
N∑
i=1
(
Xi −µ
)2
N
Sample Variance
S2 =
n∑
i=1
(
Xi − X¯
)2
n−1
Sample Standard Deviation
S =
√√√√√ n∑
i=1
(
Xi − X¯
)2
n−1
Coefficient of Variation
CV =
(
S
X¯
)
·100%
Sample Mean
X¯ =
n∑
i=1
Xi
n
= X1+X2+X3+ ...+Xn
n
Sample Covariance
Cov(X ,Y )=
n∑
i=1
(
Xi − X¯
)(
Yi − Y¯
)
n−1
Empirical Rule
If distribution is bell-shaped, then approximately:
• 68% of data is within±1 standard deviation of
the mean.
• 95% of data is within±2 standard deviation of
the mean.
• 99.7% of data is within ±3 standard deviation
of the mean.
Chebyshev’s Rule
Regardless of how the data is distributed, at least:
(
1− 1
k2
)
×100%, k > 1
will fall within k standard deviations of the mean.
Counting Rules
• Multiplication rule: (k1)(k2)...(kn)
• Repetition rule: kn
• Factorial: n!= (n)(n−1)(n−2)...(2)(1)
• Permutations: nPk =
n!
(n−k)!
• Combinations: nCk =
n!
k !(n−k)!
Sample Coefficient of Correlation
r = Cov(X ,Y )
SX SY
General Addition Rule
P (A or B)= P (A)+P (B)−P (A and B)
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BUSS1020 In-semester Examination
Conditional Probability
P (A|B)= P (A and B)
P (B)
Independence
A and B are independent if:
P (A|B)= P (A)
Marginal Probability
P (A)=
k∑
i=1
P (A|Bi )P (Bi )
= P (A|B1)P (B1)+P (A|B2)P (B2)
+P (A|B3)P (B3)+ ...+P (A|Bk)P (Bk)
Bayes Theorem
P (Bi |A)= P (A|Bi )P (Bi )
P (A)
= P (A|Bi )P (Bi )
k∑
i=1
P (A|Bi )P (Bi )
= P (A|Bi )P (Bi )
P (A|B1)P (B1)+ ...+P (A|Bk)P (Bk)
Expected Value
µ= E(X )=
N∑
i=1
xiP (X = xi )
Variance
σ2 = E(X 2)− [E(X )]2
= E [X −E(X )]2
=
N∑
i=1
[xi −E(X )]2P (X = xi )
Covariance
σXY =
N∑
i=1
[xi −E(X )]
[
yi −E(Y )
]
P (xi , yi )
= E(XY )− [E(X )][E(Y )]
Binomial Distribution Formula
P (X = x|n,pi)= nCxpix(1−pi)n−x
= n!
x!(n−x)!pi
x(1−pi)n−x
Mean and Standard deviation:
µ= npi
σ=
√
npi(1−pi)
Poisson Distribution Formula
P (X = x|λ)= e
−λλx
x!
Mean and Standard deviation:
µ=λ
σ=
p
λ
Hypergeometric Distribution Formula
P (X = x|n,N ,A)=
ACx ·N−ACn−x
NCn
Mean and Standard deviation:
µ= nA
N
σ=
√
nA(N − A)
N2
· N −n
N −1
Sum of Two Random Variables: Expected Value
E(X +Y )= E(X )+E(Y )
E(aX +bY )= E(aX )+E(bY )= aE(X )+bE(Y )
Sum of Two Random Variables: Variance
Var (X +Y )=σ2X+Y
=σ2X +σ2Y +2σX ,Y
Var (aX +bY )=σ2aX+bY
=σ2aX +σ2bY +2σaX ,bY
= a2σ2X +b2σ2Y +2abσX ,Y
Sum of Two Random Variables: Standard Deviation
σX+Y =
√
σ2X+Y
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BUSS1020 In-semester Examination
Z-score
Z = X −µ
σ
Assessing Normality
IQR ∼= 1.33S
Uniform Distribution Formula
P (X < x)= x−a
b−a
Mean and Standard deviation:
µ= a+b
2
σ=
√
(b−a)2
12
Exponential Distribution Formula
P (X< x)= 1−e−λx
Mean and Standard deviation:
µ= 1
λ
σ= 1
λ
Standard Error of the Mean
σX¯ =
σp
n
Z-score for Sampling Distribution
Z = X¯ −µσp
n
Standard Error of Population Proportion
σp =
√
pi(1−pi)
n
Z-score for Proportions
Z = p−pi√
pi(1−pi)
n
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Finite population correction factor
= � −
− 1
Standard error of the sample mean for finite populations

√
� = � − − 1
Standard error of the sample proportion for finite populations
= �(1 − ) � − − 1