APSC 254 Equation sheet
Note: If there are typographical errors on this equation sheet, it is the student's responsibility to know and
apply the correct equation
Sample mean (sample average):
Sample variance:
𝑠2 =
𝑛
𝑛−1
𝑠 = √𝑠 2
𝜎2=
(𝑥1 −𝑥̅ )2 +(𝑥2 −𝑥̅ )2 +⋯+(𝑥𝑛 −𝑥̅ )2
𝑛
Population standard deviation:
Interquartile range:
𝑥1 +𝑥2 +⋯+𝑥𝑛
(𝑥1 −𝑥̅ )2 +(𝑥2 −𝑥̅ )2 +⋯+(𝑥𝑛 −𝑥̅ )2
Sample standard deviation:
Population variance:
𝑥̅ =
𝜎 = √𝜎 2
𝐼𝑄𝑅 = 𝑄3 − 𝑄1
Max upper whisker reach:
𝑄3 + 1.5 × 𝐼𝑄𝑅
Max lower whisker reach:
𝑄1 − 1.5 × 𝐼𝑄𝑅
Addition rule (A and B are disjoint events):
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
General addition rule (A and B are disjoint or not):
Complement rule:
𝑃(𝐴𝐶 ) = 1 − 𝑃(𝐴)
Conditional probability:
Multiplication rule:
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴|𝐵) =
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐵)
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵|𝐴) = 𝑃(𝐵) × 𝑃(𝐴|𝐵)
Multiplication rule (independent events):
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
Z-score:
𝑍=
𝑋−𝜇
𝜎
𝑛
𝑛
𝑛!
𝑃(𝑋 = 𝑘) = ( ) 𝑝𝑘 (1 − 𝑝)𝑛−𝑘 , ( ) = 𝑘!(𝑛−𝑘)!
𝑘
𝑘
Binomial distribution general formula:
Binomial distribution mean and SD:
𝜇 = 𝑛𝑝
Geometric distribution general formula:
Geometric distribution mean and SD:
𝜎 = √𝑛𝑝(1 − 𝑝)
𝑃(𝑋 = 𝑘) = (1 − 𝑝)𝑘−1 𝑝
1
1−𝑝
𝜇 = 𝑝, 𝜎 = √ 𝑝 2
𝑝𝑜𝑖𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 ± 1.96 × 𝑆𝐸
95% confidence interval of containing the mean:
𝑆𝐸 =
Standard error:
𝜎
√𝑛
𝑝𝑜𝑖𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 ± 2.58 ×
99% confidence interval of containing the mean:
Geometric distribution general formula:
Geometric distribution mean and SD:
Hypothesis testing:
𝑅=
1
𝑛−1
Line-regression line:
Sensitivity (ratio):
1
1−𝑝
𝜇 = 𝑝, 𝜎 = √ 𝑝 2
see the 4-step procedure in our checklist
Linear-regression line:
Value of R:
𝑃(𝑋 = 𝑘) = (1 − 𝑝)𝑘−1 𝑝
𝑦̂−𝑦̅
∑𝑛𝑖=1
𝑠𝑦
=𝑅
𝑥−𝑥̅
𝑠𝑥
𝑥𝑖 −𝑥̅ 𝑦𝑖 −𝑦̅
𝑠𝑥
𝑠𝑦
𝑦̂ = 𝛽0 + 𝛽1 𝑥,
𝑠𝑦
𝛽0 = 𝑦̅ − 𝑠 𝑅𝑥̅ ,
𝑥
𝑠𝑐𝑎𝑙𝑒 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝑚𝑒𝑎𝑠𝑢𝑟𝑎𝑛𝑑 𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑛𝑔 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
Variance confidence interval:
[
(𝑛−1)𝑠2 (𝑛−1)𝑠2
Χ2 𝛼/2
, Χ2
1−𝛼/2
]
𝑠𝑦
𝛽1 = 𝑠 𝑅
𝑥
𝜎
√𝑛