Sample Standard Deviation
Chapter 3 Descriptive Statistics
Population Mean
𝜇=
∑𝑥 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑁
=
𝑁
𝑁
Sample Mean
𝑥=
∑𝑥 𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛
=
𝑛
𝑛
∑(𝑥 − 𝑥)2
𝑠=√
𝑛−1
Computational Formulas for Population Variance
and Standard Deviation
𝜎2 =
(∑ 𝑥 )2
𝑁
𝑁
∑ 𝑥2 −
Interquartile Range
Q3 – Q1
𝜎 = √𝜎 2
Sum of Deviations from the Arithmetic Mean is
Always Zero
∑(𝑥 − 𝜇) = 0
Computational Formulas for Sample Variance and
Standard Deviation
𝑠2 =
(∑ 𝑥 )2
𝑛
𝑛−1
∑ 𝑥2 −
Mean Absolute Deviation
𝑠 = √𝑠 2
∑|𝑥 − 𝜇|
𝑀𝐴𝐷 =
𝑁
z Score
𝑧=
Population Variance
𝜎2 =
∑(𝑥 − 𝜇)2
𝑁
Coefficient of Variation
𝑐𝑣 =
Population Standard Deviation
∑(𝑥 − 𝜇)2
𝜎=√
𝑁
Empirical Rule*
Distance from the Mean
𝜇 ± 1𝜎
𝜇 ± 2𝜎
𝜇 ± 3𝜎
1
𝑘2
proportion of the values.
Assumption: k > 1
Sample Variance
𝑠2 =
∑(𝑥 − 𝑥)2
𝑛−1
∑𝑓 𝑀 ∑𝑓 𝑀
=
∑𝑓
𝑁
𝑓1 𝑀1 + 𝑓2 𝑀2 + ⋯ + 𝑓𝑖 𝑀𝑖
=
𝑓1 + 𝑓2+ ⋯ + 𝑓𝑖
𝜇𝑔𝑟𝑜𝑢𝑝𝑒𝑑 =
Values within the Distance
68%
95%
99.7%
Chebyshev’s Theorem
Within k standard deviations of the mean, 𝜇 ± 𝑘𝜎, lie at
1−
𝜎
(100)
𝜇
Mean of Grouped Data
*Based on the assumption that the data are
approximately normally distributed.
least
𝑥− 𝜇
𝜎
where
i = the number of classes
f = class frequency
M = class midpoint
N = total frequencies (total number of data values)
Medium of Grouped Data
mediangrouped
where
𝑁
− 𝐹
= 𝑙 + (2
)𝑤
𝑓
l = lower endpoint of the class containing the median
w = width of the class containing the median
f = frequency of the class containing the median
F = cumulative frequency of classes preceding the
class containing the median
N = total frequencies (total number of data values)
Formulas for Population Variance and Standard
Deviation of Grouped Data
Original Formula
Computational Version
𝜎2 =
∑ 𝑓 (𝑀 − 𝜇)
𝑁
2
𝜎2 =
(∑ 𝑓𝑀 )2
𝑁
𝑁
∑ 𝑓𝑀2 −
𝜎 = √𝜎 2
where
f = frequency
M = class midpoint
N = ∑𝑓, or total of the frequencies of the population
𝜇 = grouped mean for the population
Formulas for Sample Variance and Standard
Deviation of Grouped Data
Original Formula
Computational Version
2
𝑠2 =
∑ 𝑓 (𝑀 − 𝑥̅ )
𝑛−1
𝑠2 =
(∑ 𝑓𝑀 )2
𝑛
𝑛−1
∑ 𝑓𝑀2 −
𝑠 = √𝑠 2
where
f = frequency
M = class midpoint
N = ∑𝑓, or total of the frequencies of the population
𝑥̅ = grouped mean for the sample
Coefficient of Skewness
𝑠𝑘 =
3(𝜇 − 𝑀𝑑)
𝜎
where
𝑠𝑘 = coefficient of skewness
𝑀𝑑 = median
Chapter 4 Probability
Classical Method of Assigning Probabilities
𝑛𝑒
𝑃(𝐸) =
𝑁
where
N = total possible number of outcomes of an
experiment
𝑛𝑒 = the number of outcomes in which the event
occurs out of N outcomes
Range of Possible Probabilities
0 ≤ 𝑃(𝐸) ≤ 1
Probability by Relative Frequency of Occurrence
Number of Times an Event Occurred
Total Number of Opportunities for the Event to Occur
Mutually Exclusive Events X and Y
𝑃(𝑋 ∩ 𝑌) = 0
Independent Events X and Y
𝑃(𝑋|𝑌) = 𝑃(𝑋) and 𝑃 (𝑌|𝑋) = 𝑃(𝑌)
Probability of the Complement of A
𝑃(𝐴′ ) = 1 − 𝑃(𝐴)
The mn Counting Rule
For an operation that can be done m ways and a second
operation that can be done n ways, the two operations
can then occur, in order, in mn ways. This rule can be
extended to cases with three or more operations.
General Law of Addition
𝑃(𝑋 ∪ 𝑌) = 𝑃(𝑋) + 𝑃(𝑌) − 𝑃(𝑋 ∩ 𝑌)
where X, Y, are events and (𝑋 ∩ 𝑌) is the intersection of
X and Y.
Special Law of Addition
If X, Y are mutually exclusive,
𝑃(𝑋 ∪ 𝑌) = 𝑃(𝑋) + 𝑃(𝑌)
General Law of Multiplication
𝑃(𝑋 ∩ 𝑌) = 𝑃(𝑋) ∙ 𝑃(𝑌|𝑋) = 𝑃(𝑌) ∙ 𝑃(𝑋|𝑌)
Special Law of Multiplication
If X, Y are independent,
𝑃(𝑋 ∩ 𝑌) = 𝑃(𝑋) ∙ 𝑃(𝑌)
Law of Conditional Probability
𝑃(𝑋 | 𝑌) =
𝑃 (𝑋 ∩ 𝑌)
𝑃(𝑋) ∙ 𝑃(𝑌 | 𝑋)
=
𝑃(𝑌)
𝑃(𝑌)
Independent Events X, Y
If X and Y are independent events, the following must be
true:
𝑃(𝑋|𝑌) = 𝑃(𝑋) and 𝑃(𝑌|𝑋) = 𝑃(𝑌)
Bayes’ Rule
𝑃(𝑋𝑖 | 𝑌) =
𝑃(𝑋𝑖 ) ∙ 𝑃(𝑌|𝑋𝑖 ) ⋯
𝑃(𝑋1 ) ∙ 𝑃(𝑌 | 𝑋1 ) + 𝑃(𝑋2 ) ∙ 𝑃(𝑌 | 𝑋2 ) + ⋯ + 𝑃(𝑋𝑛 ) ∙ 𝑃(𝑌 | 𝑋𝑛 )
Chapter 5 Discrete Distributions
Hypergeometric Formula
Mean or Expected Value of a Discrete Distribution
𝜇 = 𝐸(𝑥) = ∑[𝑥 ∙ 𝑃(𝑥)]
where
E(x) = long-run average
x = an outcome
P(x) = probability of that outcome
Variance of a Discrete Distribution
𝜎 2 = ∑[(𝑥 − 𝜇)2 ∙ 𝑃(𝑥)]
where
x = an outcome
P(x) = probability of a given outcome
𝜇= mean
Standard Deviation of a Discrete Distribution
𝜎 = √∑[(𝑥 − 𝜇)2 ∙ 𝑃(𝑥)]
Assumptions of the Binomial Distribution
- The experiment involves n identical trials.
- Each trial has only two possible outcomes denoted as
success or failure.
- Each trial is independent of the previous trials.
- The terms p and q remain constant throughout the
experiment, where the term p is the probability of
getting a success on any one trial and the term q = 1 –
p is the probability of getting a failure on any one
trial.
Binomial Formula
𝑛𝐶𝑥 . 𝑝
𝑥
. 𝑞 𝑛−𝑥 =
𝑛!
. 𝑝 𝑥 . 𝑞 𝑛−𝑥
𝑥! (𝑛 − 𝑥)!
where
n = the number of trials (or the number being
sampled)
x = the number of successes desired
p = the probability of getting a success in one trial
𝑞 = 1 – p = the probability of getting a failure in one
trial
Mean and Standard Deviation of a Binomial
Distribution
𝜇= 𝑛 ∙ 𝑝
𝜎 = √𝑛 ∙ 𝑝 ∙ 𝑞
Poisson Formula
𝑃(𝑥) =
where
𝜆𝑥 𝑒 −𝜆
𝑥!
x = 0, 1, 2, 3, …
λ = long-run average
e = 2.718281…
𝑃(𝑥) =
𝑛𝐶𝑥 . 𝑁−𝐴𝐶𝑛−𝑥
𝑁𝐶𝑛
where
N = size of the population
n = sample size
A = number of successes in the population
x = number of successes in the sample; sampling is
done without replacement
Chapter 6 Continuous Distributions
Probability Density Function of a Uniform
Distribution
1
𝑓(𝑥) = {𝑏 − 1 for a ≤ x ≤ b
0
for all other values
Mean and Standard Deviation of a Uniform
Distribution
𝑎+𝑏
2
𝜇=
𝑏−𝑎
𝜎=
√12
Probabilities in a Uniform Distribution
𝑃(𝑥) =
where
𝑥2 − 𝑥1
𝑏−𝑎
𝑎 ≤ 𝑥1 ≤ 𝑥2 ≤ 𝑏
z formula
𝑧=
𝑥− 𝜇
,𝜎 ≠ 0
𝜎
Exponential Probability Density Function
𝑓(𝑥) = λ𝑒−λ𝑥
where
x≥0
λ>0
and e = 2.271828…
Probabilities of the Right Tail of the Exponential
Distribution
𝑃(𝑥 ≥ 𝑥0 ) = 𝑒−λ𝑥0
where
𝑥0 ≥ 0
Chapter 7 Sampling and Sampling
Distributions
Determining the Value of k
𝑘=
where
𝑁
𝑛
n = sample size
N = population size
k = size of interval for selection
Central Limit Theorem
If samples of size n are drawn randomly from a
population that has a mean of 𝜇 and a standard
deviation of 𝜎, the sample means, 𝑥, are approximately
normally distributed for sufficiently large samples (n ≥
30*) regardless of the shape of the population
distribution. If the population is normally distributed,
the sample means are normally distributed for any
sample size.
From mathematical expectation, it can be shown that
the mean of the sample means is the population mean:
𝜇𝑥̅ = 𝜇
and the standard deviation of the sample means (called
the standard error of the mean) is the standard
deviation of the population divided by the square root
of the sample size:
𝜎
𝜎𝑥̅ =
√𝑛
z Formula for Sample Means
𝑧=
𝑥̅ − 𝜇
𝜎
√𝑛
z Formula for Sample Means of a Finite Population
𝑧=
𝑥̅ − 𝜇
𝜎 √𝑁 − 𝑛
√𝑛 𝑁 − 1
Sample Proportion
𝑝̂ =
where
𝑥
𝑛
x = number of items in a sample that have the
characteristic
n = number of items in the sample
z Formula for Sample Proportions for n ∙ 𝒑 > 𝟓 and n
∙𝒒>𝟓
𝑧=
𝑝̂ − 𝑝
𝑝 ∙𝑞
𝑛
√
where
𝑝̂ = sample proportion
n = sample size
p = population proportion
q=1–p
Chapter 8 Statistical Inference: Estimation for
Single Populations
100(1 – a)% Confidence Interval to Estimate 𝛍: 𝝈
Known (8.1)
𝑥̅ − 𝑧∝⁄2
𝜎
√𝑛
≤ 𝜇 ≤ 𝑥̅ + 𝑧∝⁄2
√𝑛
𝑛=
the area under the normal curve outside the
confidence interval area
 = the area in one end (tail) of the distribution
outside the confidence interval
Confidence Interval to Estimate 𝛍 Using the Finite
Correction Factor (8.2)
𝜎 𝑁−𝑛
𝜎 𝑁−𝑛
√
√
≤ 𝜇 ≤ 𝑥̅ + 𝑧∝⁄2
√𝑛 𝑁 − 1
√𝑛 𝑁 − 1
Confidence Interval to Estimate 𝛍: Population
Standard Deviation Unknown and the Population
Normally Distributed (8.3)
𝑥̅ − 𝑡∝⁄2,𝑛−1
𝑠
√𝑛
≤ 𝜇 ≤ 𝑥̅ + 𝑡∝⁄2,𝑛−1
𝑠
√𝑛
df = 𝑛 − 1
Confidence Interval to Estimate p (8.4)
𝑝̂ . 𝑞̂
𝑝̂ . 𝑞̂
𝑝̂ − 𝑧∝⁄2 √
≤ 𝑝 ≤ 𝑝̂ + 𝑧∝⁄2 √
𝑛
𝑛
where
𝑝̂ = sample proportion
𝑞̂ = sample size
p = population proportion
n = sample size
𝝌𝟐 Formula for Single Variance (8.5)
(𝑛 − 1)𝑠 2
𝜒2 =
𝜎2
𝑑𝑓 = 𝑛 − 1
Confidence Interval to Estimate the Population
Variance (8.6)
(𝑛 − 1)𝑠 2
(𝑛 − 1)𝑠 2
2
≤
𝜎
≤
2
𝜒𝛼2⁄2
𝜒1−𝛼
⁄2
𝑛=
𝑧∝2⁄2 𝜎 2
𝑧∝⁄2 𝜎 2
=(
)
2
𝐸
𝐸
Sample Size When Estimating p (8.8)
𝜎
where
=
𝑥̅ − 𝑧∝⁄2
Sample Size When Estimating 𝝁 (8.7)
where
𝑧∝2⁄2 𝑝. 𝑞
𝐸2
p = population proportion
q= 1 – p
E = error of estimation
n = sample size
Chapter 9 Statistical Inference: Hypothesis
Testing for Single Populations
z Test for a Single Mean (9.1)
𝑥̅ − 𝜇
𝜎
√𝑛
𝑧=
Formula to Test Hypotheses about 𝝁 with a Finite
Population (9.2)
𝑧=
𝑥̅ − 𝜇
𝜎 √𝑁 − 𝑛
√𝑛 𝑁 − 1
t Test for 𝝁 (9.3)
𝑡=
𝑥̅ − 𝜇
𝑠
√𝑛
𝑑𝑓 = 𝑛 − 1
z Test of a Population Proportion (9.4)
𝑝̂ − 𝑝
𝑧=
𝑝 ∙𝑞
𝑛
√
where
𝑝̂ = sample proportion
p = population proportion
q=1–p
Formula for Testing Hypotheses about a Population
Variance (9.5)
𝜒2 =
(𝑛 − 1)𝑠 2
𝜎2
𝑑𝑓 = 𝑛 − 1
Chapter 10 Statistical Inference: About Two
Populations
z Formula for the Difference in Two Sample Means
(Independent Samples and Population Variances
Known) (10.1)
𝑧=
(𝑥̅1 − 𝑥̅2 ) − (𝜇1 − 𝜇2 )
𝜎2
𝜎2
√ 1 + 2
𝑛1
𝑛2
where
𝜇1 = mean of population 1
𝜇2 = mean of population 2
𝑛1 = size of sample 1
𝑛2 = size of sample 2
df = n1 +n1 − 2
t Formula to Test the Difference in two Dependent
Populations (10.6)
t=
̅ −D
d
sd
√n
df = n − 1
Confidence Interval to Estimate 𝝁𝟏 − 𝝁𝟐 (10.2)
σ2 σ2
σ2 σ2
(x̅1 − x̅2 ) − zα⁄2 √ 1 + 2 ≤ μ1 − μ2 ≤ (x̅1 − x̅2 ) + zα⁄2 √ 1 + 2
n1 n2
n1 n2
t Formula to Test the Difference in Means Assuming
𝝈𝟐𝟏 and 𝝈𝟐𝟐 are Equal (10.3)
t=
s 2 (n1 − 1) + s22 (n2 − 1) 1
1
(x̅1 − x̅2 ) + t√ 1
√ +
n1 + n2 − 2
n1 n2
(x̅1 − x̅2 ) − (μ1 − μ2 )
1
1
sp √n + n
1
2
df = n1 +n1 − 2
where
n = number of pairs
d = sample difference in pairs
D = mean population difference
sd = standard deviation of sample difference
d = mean sample difference
Formulas for d and sd (10.7 and 10.8)
̅=
d
∑d
n
(∑ d)2
̅ )2 √∑ d2 −
∑(d
−
d
n
sd = √
=
n−1
n−1
Confidence Interval Formula to Estimate the
Difference in Related Populations, D (10.9)
where
s12 (n1 − 1) + s22 (n2 − 1)
sp = √
n1 + n2 − 2
̅ −t
d
t Formula to Test the Difference in Means (10.4)
df = n − 1
t=
(x̅1 − x̅2 ) − (μ1 − μ2 )
s2
s2
√ 1 + 2
n1
n2
2
df =
s2
s2
(n1 + n2 )
1
2
2
s12
(n )
1
n1 − 1 +
2
s22
(n )
2
n2 − 1
Confidence Interval to Estimate μ1 − μ2 Assuming
the Population Variances are Unknown and Equal
(10.5)
s 2 (n1 − 1) + s22 (n2 − 1) 1
1
(x̅1 − x̅2 ) − t√ 1
√ + ≤ μ1 − μ2 ≤
n1 + n2 − 2
n1 n2
sd
√n
̅ +t
≤D≤ d
sd
√n
z Formula for the Difference in Two Population
Proportions (10.10)
z=
(p̂1 − p̂2 ) − (p1 − p2 )
p .q
p .q
√ 1n 1 + 2n 2
1
2
where
p̂1 = proportion from sample 1
𝑝̂2 = proportion from sample 2
𝑛1 = size of sample 1
𝑛2 = size of sample 2
𝑝1 = proportion from population 1
𝑝2 = proportion from population 2
𝑞1 = 1 − 𝑝1
𝑞2 =1 − 𝑝2
z Formula to Test the Difference in Population
Proportions (10.11)
z=
(p̂1 − p̂2 ) − (p1 − p2 )
1
1
√(p
̅ . q̅) (n + n )
1
2
where
p
̅=
x1 + x2 (n1 p̂1 + n2 p̂2 )
=
n1 + n2
n1 + n2
and
q̅ = 1 − p
̅
Confidence Interval to Estimate p1 - p2 (10.12)
p̂1 . q̂1 p̂2 . q̂2
(p̂1 − p̂2 ) − z√
+
≤ p1 − p2 ≤
n1
n2
p̂1 . q̂1 p̂2 . q̂2
(p̂1 − p̂2 ) + z√
+
n1
n2
F Test for Two Population Variances (10.13)
F=
s12
s22
dfnumerator = v1 = n1 − 1
dfdenominator = v2 = n2 − 1
Formula for Determining the Critical Value for the
Lower-Tail F (10.14)
F1−∝,v1 ,v2 =
1
F∝,v1 ,v2
Chapter 11 Analysis of Variance and Design of
Experiments
𝐶
𝑛
SSE = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅𝑗 − 𝑥̅𝑖
𝑖=1 𝑗=1
− 𝑥̅ )
Formulas for computing a one-way ANOVA
𝑐
2
SST = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅ )
SSC = ∑ 𝑛𝑗 (𝑥̅𝑗 − 𝑥̅ )
SSE
𝑁−𝑛−𝐶+1
F𝑏𝑙𝑜𝑐𝑘𝑠 =
2
𝑖=1 𝑗=1
𝑛𝑗 𝑐
𝑅
𝑖=1 𝑗=1
df𝐶 = 𝐶 − 1
𝑖=1
𝐶
SSC = 𝑛𝑅 ∑(𝑥̅𝑗 − 𝑥̅ )
df𝐸 = 𝑁 − 𝐶
df𝐼 = (𝑅 − 1)(𝐶 − 1)
2
df𝐸 = 𝑅𝐶(𝑛 − 1)
𝑗=1
df 𝑇 = 𝑁 − 1
𝐶
𝑅
SSI = 𝑛 ∑ ∑(𝑥̅𝑖𝑗 − 𝑥̅𝑖 − 𝑥̅𝑗 − 𝑥̅ )
SSC
df𝐶
𝑖=1 𝑗=1
𝑅
𝐶
SSE
MSE =
df𝐸
𝑖=1 𝑗=1 𝑘=1
𝑅
𝐶
𝑛
SST = ∑ ∑ ∑(𝑥𝑖𝑗𝑘 − 𝑥̅ )
MSC
F=
MSE
2
df 𝑇 = 𝑁 − 1
MSR =
SSR
𝑅−1
MSC =
SSC
𝐶−1
𝑛
SSE = ∑ ∑ ∑(𝑥𝑖𝑗𝑘 − 𝑥̅ 𝑖𝑗 )
2
2
MSI =
SSI
(𝑅 − 1)(𝐶 − 1)
𝑖=1 𝑗=1 𝑘=1
MSE =
Tukey’s HSD test
MSE
HSD = 𝑞∝,𝐶,𝑁−𝐶 √
𝑛
Tukey-Kramer formula
MSE 1
1
q∝,𝐶,𝑁−𝐶 √
( + )
2 𝑛𝑟 𝑛 𝑠
Formulas for computing a randomized block design
2
𝑗=1
𝑛
SSR = 𝐶 ∑(𝑥̅𝑖 − 𝑥̅ )2
MSE
df𝑅 = 𝑅 − 1
SSR = 𝑛𝐶 ∑(𝑥̅𝑖 − 𝑥̅ )2
df𝐶 = 𝐶 − 1
𝐶
MSR
Formulas for computing a two-way ANOVA
2
SST = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅ )
SSC = 𝑛 ∑(𝑥̅𝑗 − 𝑥̅ )
MSC
MSE
F𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠 =
SSE = ∑ ∑(𝑥𝑖𝑗 − 𝑥̅𝑗 )
MSC =
MSE =
2
𝑖=1 𝑗=1
𝑗=1
𝑛𝑗 𝑐
SSR
𝑛−1
2
𝐶
𝑛
MSR =
df𝐶 = 𝐶 − 1
df𝑅 = 𝑅 − 1
df𝐸 = (𝐶 − 1)(𝑛 − 1)
= 𝑁−𝑛−𝐶+1
𝑖=1
MSC =
SSC
𝐶−1
SSE
(𝑛
𝑅𝐶 − 1)
F𝑅 =
MSR
MSE
F𝐶 =
MSC
MSE
F𝐼 =
MSI
MSE
Chapter 12 Correlation and Simple Regression
Analysis
(12.1) Pearson product-moment correlation
coefficient
r=
∑(x − x̅)(y − y̅)
√∑(x − x̅)2 ∑(y − y̅)2
∑x∑y
∑ xy −
n
=
2
2
(∑
x)
2 − (∑ y) ]
√[∑ x 2 −
y
]
[∑
n
n
Equation of the simple regression line
ŷ = β0 + β1 x
Standard error of the estimate
se = √
SSE
n−2
(12.5) Coefficient of determination
r2 = 1 −
SSE
=1−
SSyy
SSE
(∑ y)2
∑ y2 −
n
Computational formula for r2
r2 =
b12 SSxx
SSyy
Sum of squares
SSxx = ∑ x 2 −
SSyy = ∑ y 2 −
SSxy = ∑ xy −
(∑ x)2
n
(∑ y)2
n
(∑ x)(∑ y)
n
(12.2) Slope of the regression line
b1 =
=
∑(x − x̅)(y − y̅) ∑ xy − nx̅ y̅
=
∑(x − x̅)2
∑ x 2 − nx̅ 2
(∑ x)(∑ y)
n
(∑ x)2
2
∑x −
n
t=
b1 − β1
sb
sb =
se
√SSxx
(12.6) Confidence interval to estimate E(yx) for a
given value of x
1 (xo − x̅)2
ŷ ± t ∝⁄2,n−2 se √ +
n
SSxx
∑ xy −
(12.3) Alternative formula for slope
b1 =
t test of slope
SSxy
SSxx
(12.4) y intercept of the regression line
b0 = y̅ − b1 x̅ =
∑y
(∑ x)
− b1
n
n
Sum of squares of error
SSE = ∑(y − ŷ)2
= ∑ y 2 − b0 ∑ y − b1 ∑ xy
(12.7) Prediction interval to estimate y for a given
value of x
ŷ ± t ∝⁄2,n−2 se √1 +
1 (xo − x̅)2
+
n
SSxx
Chapter 13 Multiple Regression Analysis
The F value
F=
MSreg SSreg⁄dfreg
SSR⁄k
=
=
MSerr SSerr⁄dferr SSE⁄(N − k − 1)
Sum of squares of error
SSE = ∑(y − ŷ)2
Standard error of the estimate
se = √
SSE
n−k−1
Coefficient of multiple determination
R2 −
SSR
SSE
=1−
SSyy
SSyy
Adjusted R2
Adjusted R2 = 1 −
SSE⁄(n − k − 1)
SSyy ⁄(n − 1)
Chapter 14 Building Multiple Regression
Models
(14.1) Variance inflation factor
VIF =
1
1 − R2i
Chapter 15 Time-Series forecasting and Index
Numbers
Individual forecast error
et = X t − Ft
Mean absolute deviation
MAD =
∑ |𝑒𝑖 |
Number of Forecasts
Mean square error
∑ e2i
MSE =
Number of Forecasts
Exponential smoothing
Ft+1 =∝. X t − (1−∝). Ft
Durbin-Watson test
D=
∑nt=2(et − et−1 )2
∑nt=1 e2t
Chapter 16 Analysis of Categorical Data
χ2 goodness-of-fit test (16.1)
(𝑓𝑜 − 𝑓𝑒 )2
𝑓𝑒
df = k − 1 − c
𝜒2 = ∑
χ2 test of independence (16.2)
(𝑓𝑜 − 𝑓𝑒 )2
𝑓𝑒
df = (r − 1)(c − 1)
𝜒2 = ∑ ∑
Chapter 17 Nonparametric Statistics
Large-sample runs test
𝜇𝑅 =
𝑐
𝑇𝑗2
12
𝐾=
(∑ ) − 3(𝑛 + 1)
𝑛(𝑛 + 1)
𝑛𝑗
2𝑛1 𝑛2
+1
𝑛1 +𝑛2
𝑗=1
2𝑛1 𝑛2 (2𝑛1 𝑛2 − 𝑛1 − 𝑛2 )
𝜎𝑅 = √
(𝑛1 +𝑛2 )2 (𝑛1 +𝑛2 − 1)
𝑅 − 𝜇𝑅
𝑧=
=
𝜎𝑅
Kruskal-Wallis test (17.3)
2𝑛 𝑛
𝑅 − ( 1 2 + 1)
𝑛1 +𝑛2
2𝑛 𝑛 (2𝑛1 𝑛2 − 𝑛1 − 𝑛2 )
√ 1 2
(𝑛1 +𝑛2 )2 (𝑛1 +𝑛2 − 1)
Friedman test (17.4)
𝑐
𝜒𝑟2
12
=
∑ 𝑅𝑗2 − 3𝑏(𝑐 + 1)
𝑏𝑐(𝑐 + 1)
𝑗=1
Spearman’s rank correlation (17.5)
𝑟𝑠 = 1 −
Mann-Whitney U test (small sample)
𝑈1 = 𝑛1 𝑛2 +
𝑛1 (𝑛1 + 1)
− 𝑊1
2
𝑈2 = 𝑛1 𝑛2 +
𝑛2 (𝑛2 + 1)
− 𝑊2
2
𝑈 ′ = 𝑛1 𝑛2 − 𝑈
Mann-Whitney U test (large sample) (17.1)
𝜇𝑈 =
𝑛1 𝑛2
2
𝜎𝑈 = √
𝑧=
𝑛1 𝑛2 (𝑛1 + 𝑛2 + 1)
12
𝑈 − 𝜇𝑈
𝜎𝑈
Wilcoxon matched-pairs signed rank test (17.2)
𝜇𝑇 =
(𝑛)(𝑛 + 1)
4
(𝑛)(𝑛 + 1)(2𝑛 + 1)
𝜎𝑇 = √
24
𝑧=
𝑇 − 𝜇𝑇
𝜎𝑇
6 ∑ 𝑑2
𝑛(𝑛2 − 1)
Chapter 18 Statistical Quality Control
̅ 𝐂𝐡𝐚𝐫𝐭𝐬
𝒙
Centreline:
∑ x̅
k
x̅ =
UCL:
̅
x̅ + A2 R
LCL:
̅
x̅ − A2 R
OR
UCL:
x̅ + A3 s̅
LCL:
x̅ − A3 s̅
R charts
Centreline:
∑R
k
̅=
R
UCL:
̅
D4 R
LCL:
̅
D3 R
p charts
Centreline:
UCL:
LCL:
p=
∑ p̂
k
p + 3√
p. q
n
p − 3√
p. q
n
c charts
Centreline:
c̅ =
c1 + c2 + c3 … + ci
i
UCL:
c̅ + 3√c̅
LCL:
c̅ − 3√c̅
Chapter 19 Decision Analysis
Bayes’ Rule
𝑃(𝑋𝑖 | 𝑌) =
𝑃(𝑋𝑖 ) ∙ 𝑃(𝑌|𝑋𝑖 )
𝑃(𝑋1 ) ∙ 𝑃(𝑌 | 𝑋1 ) + 𝑃(𝑋2 ) ∙ 𝑃(𝑌 | 𝑋2 ) + ⋯ + 𝑃(𝑋𝑛 ) ∙ 𝑃(𝑌 | 𝑋𝑛 )