Many students find that the obligatory Statistics course comes as a shock. The set textbook is difficult, the curriculum is vast, and secondary-school maths feels infinitely far away. "Statistics" offers friendly instruction on the core areas of these subjects. The focus is overview. And the numerous examples give the reader a "recipe" for solving all the common types of exercise.

Many students find that the obligatory Statistics course comes as a shock. The set textbook is difficult, the curriculum is vast, and secondary-school maths feels infinitely far away.

“Statistics” offers friendly instruction on the core areas of these subjects. The focus is overview. And the numerous examples give the reader a “recipe” for solving all the common types of exercise. You can download this book free of charge.

1 Preface

2 Basic concepts of probability theory
2.1 Probability space, probability function, sample space, event
2.2 Conditional probability
2.3 Independent events
2.4 The Inclusion-Exclusion Formula
2.5 Binomial coefficients
2.6 Multinomial coefficients

3 Random variables
3.1 Random variables, definition
3.2 The distribution function
3.3 Discrete random variables, point probabilities
3.4 Continuous random variables, density function
3.5 Continuous random variables, distribution function
3.6 Independent random variables
3.7 Random vector, simultaneous density and distribution function

4 Expected value and variance
4.1 Expected value of random variables
4.2 Variance and standard deviation of random variables
4.3 Example (computation of expected value, variance and standard deviation)
4.4 Estimation of expected value μ and standard deviation σ by eye
4.5 Addition and multiplication formulas for expected value and variance
4.6 Covariance and correlation coefficient

5 The Law of Large Numbers
5.1 Chebyshev’s Inequality
5.2 The Law of Large Numbers
5.3 The Central Limit Theorem
5.4 Example (distribution functions converge to Φ)

6 Descriptive statistics
6.1 Median and quartiles
6.2 Mean value
6.3 Empirical variance and empirical standard deviation
6.4 Empirical covariance and empirical correlation coefficient

7 Statistical hypothesis testing
7.1 Null hypothesis and alternative hypothesis
7.2 Significance probability and significance level
7.3 Errors of type I and II
7.4 Example

8 The binomial distribution Bin(n, p)
8.1 Parameters
8.2 Description
8.3 Point probabilities
8.4 Expected value and variance
8.5 Significance probabilities for tests in the binomial distribution
8.6 The normal approximation to the binomial distribution
8.7 Estimators
8.8 Confidence intervals

9 The Poisson distribution Pois(λ)
9.1 Parameters
9.2 Description
9.3 Point probabilities
9.4 Expected value and variance
9.5 Addition formula
9.6 Significance probabilities for tests in the Poisson distribution
9.7 Example (significant increase in sale of Skodas)
9.8 The binomial approximation to the Poisson distribution
9.9 The normal approximation to the Poisson distribution
9.10 Example (significant decrease in number of complaints)
9.11 Estimators
9.12 Confidence intervals

10 The geometrical distribution Geo(p)
10.1 Parameters
10.2 Description
10.3 Point probabilities and tail probabilities
10.4 Expected value and variance

11 The hypergeometrical distribution HG(n, r,N)
11.1 Parameters
11.2 Description
11.3 Point probabilities and tail probabilities
11.4 Expected value and variance
11.5 The binomial approximation to the hypergeometrical distribution
11.6 The normal approximation to the hypergeometrical distribution

12 The multinomial distribution Mult(n, p1, . . . , pr)
12.1 Parameters
12.2 Description
12.3 Point probabilities
12.4 Estimatorer

13 The negative binomial distribution NB(n, p)
13.1 Parameters
13.2 Description
13.3 Point probabilities
13.4 Expected value and variance
13.5 Estimatorer

14 The exponential distribution Exp(λ)
14.1 Parameters
14.2 Description
14.3 Density and distribution function
14.4 Expected value and variance

15 The normal distribution
15.1 Parameters
15.2 Description
15.3 Density and distribution function
15.4 The standard normal distribution
15.5 Properties of Φ
15.6 Estimation of the expected value μ
15.7 Estimation of the variance σ2
15.8 Confidence intervals for the expected value μ
15.9 Confidence intervals for the variance σ2 and the standard deviation σ
15.10 Addition formula

16 Distributions connected with the normal distribution
16.1 The χ2-distribution
16.2 Student’s t-distribution
16.3 Fisher’s F-distribution

17 Tests in the normal distribution
17.1 One sample, known variance, H0 : μ = μ0
17.2 One sample, unknown variance, H0 : μ = μ0 (Student’s t-test)
17.3 One sample, unknown expected value, H0 : σ2 = σ2 0
17.4 Example
17.5 Two samples, known variances, H0 : μ1 = μ2
17.6 Two samples, unknown variances, H0 : μ1 = μ2 (Fisher-Behrens)
17.7 Two samples, unknown expected values, H0 : σ2 1 = σ2 2
17.8 Two samples, unknown common variance, H0 : μ1 = μ2
17.9 Example (comparison of two expected values)

18 Analysis of Variance (ANOVA)
18.1 Aim and motivation
18.2 k samples, unknown common variance, H0 : μ1 = · · · = μk
18.3 Two examples (comparison of mean values from 3 samples)

19 The chi-square test (or χ2-test)
19.1 χ2-test for equality of distribution
19.2 The assumption of normal distribution
19.3 Standardised residuals
19.4 Example (women with 5 children)
19.5 Example (election)
19.6 Example (deaths in the Prussian cavalry)

20 Contingency tables
20.1 Definition, method
20.2 Standardised residuals
20.3 Example (students’ political orientation)
20.4 χ2-test for 2 × 2 tables
20.5 Fisher’s exact test for 2 × 2 tables
20.6 Example (Fisher’s exact test)

21 Distribution free tests
21.1 Wilcoxon’s test for one set of observations
21.2 Example
21.3 The normal approximation to Wilcoxon’s test for one set of observations
21.4 Wilcoxon’s test for two sets of observations
21.5 The normal approximation to Wilcoxon’s test for two sets of observations

22 Linear regression
22.1 The model
22.2 Estimation of the parameters β0 and β1
22.3 The distribution of the estimators
22.4 Predicted values ˆyi and residuals ˆei
22.5 Estimation of the variance σ2
22.6 Confidence intervals for the parameters β0 and β1
22.7 The determination coefficient R2
22.8 Predictions and prediction intervals
22.9 Overview of formulas
22.10Example

A Overview of discrete distributions

B Tables

B.1 How to read the tables
B.2 The standard normal distribution
B.3 The χ2-distribution (values x with Fχ2(x) = 0.500 etc.)
B.4 Student’s t-distribution (values x with FStudent(x) = 0.600 etc.)
B.5 Fisher’s F-distribution (values x with FFisher(x) = 0.90)
B.6 Fisher’s F-distribution (values x with FFisher(x) = 0.95)
B.7 Fisher’s F-distribution (values x with FFisher(x) = 0.99)
B.8 Wilcoxon’s test for one set of observations
B.9 Wilcoxon’s test for two sets of observations, α = 5%

C Explanation of symbols

D Index