Essentials of Statistics

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
   1. Probability space, probability function, sample space, event
   2. Conditional probability
   3. Independent events
   4. The Inclusion-Exclusion Formula
   5. Binomial coefficients
   6. Multinomial coefficients
3. Random variables
   1. Random variables, definition
   2. The distribution function
   3. Discrete random variables, point probabilities
   4. Continuous random variables, density function
   5. Continuous random variables, distribution function
   6. Independent random variables
   7. Random vector, simultaneous density and distribution function
4. Expected value and variance
   1. Expected value of random variables
   2. Variance and standard deviation of random variables
   3. Example (computation of expected value, variance and standard deviation)
   4. Estimation of expected value µ and standard deviation s by eye
   5. Addition and multiplication formulas for expected value and variance
   6. Covariance and correlation coefficient
5. The Law of Large Numbers
   1. Chebyshev’s Inequality
   2. The Law of Large Numbers
   3. The Central Limit Theorem
   4. Example (distribution functions converge to F)
6. Descriptive statistics
   1. Median and quartiles
   2. Mean value
   3. Empirical variance and empirical standard deviation
   4. Empirical covariance and empirical correlation coefficient
7. Statistical hypothesis testing
   1. Null hypothesis and alternative hypothesis
   2. Significance probability and significance level
   3. Errors of type I and II
   4. Example
8. The binomial distribution Bin(n, p)
   1. Parameters
   2. Description
   3. Point probabilities
   4. Expected value and variance
   5. Significance probabilities for tests in the binomial distribution
   6. The normal approximation to the binomial distribution
   7. Estimators
   8. Confidence intervals
9. The Poisson distribution Pois(?)
   1. Parameters
   2. Description
   3. Point probabilities
   4. Expected value and variance
   5. Addition formula
   6. Significance probabilities for tests in the Poisson distribution
   7. Example (significant increase in sale of Skodas)
   8. The binomial approximation to the Poisson distribution
   9. The normal approximation to the Poisson distribution
   10. Example (significant decrease in number of complaints)
   11. Estimators
   12. Confidence intervals
10. The geometrical distribution Geo(p)
    1. Parameters
    2. Description
    3. Point probabilities and tail probabilities
    4. Expected value and variance
11. The hypergeometrical distribution HG(n, r,N)
    1. Parameters
    2. Description
    3. Point probabilities and tail probabilities
    4. Expected value and variance
    5. The binomial approximation to the hypergeometrical distribution
    6. The normal approximation to the hypergeometrical distribution
12. The multinomial distribution Mult(n, p1, . . . , pr)
    1. Parameters
    2. Description
    3. Point probabilities
    4. Estimatorer
13. The negative binomial distribution NB(n, p)
    1. Parameters
    2. Description
    3. Point probabilities
    4. Expected value and variance
    5. Estimatorer
14. The exponential distribution Exp(?)
    1. Parameters
    2. Description
    3. Density and distribution function
    4. Expected value and variance
15. The normal distribution
    1. Parameters
    2. Description
    3. Density and distribution function
    4. The standard normal distribution
    5. Properties of F
    6. Estimation of the expected value µ
    7. Estimation of the variance s2
    8. Confidence intervals for the expected value µ
    9. Confidence intervals for the variance s2 and the standard deviation s
    10. Addition formula
16. Distributions connected with the normal distribution
    1. The ?2-distribution
    2. Student’s t-distribution
    3. Fisher’s F-distribution
17. Tests in the normal distribution
    1. One sample, known variance, H0 : µ = µ0
    2. One sample, unknown variance, H0 : µ = µ0 (Student’s t-test)
    3. One sample, unknown expected value, H0 : s2 = s2 0
    4. Example
    5. Two samples, known variances, H0 : µ1 = µ2
    6. Two samples, unknown variances, H0 : µ1 = µ2 (Fisher-Behrens)
    7. samples, unknown expected values, H0 : s2 1 = s2 2
    8. Two samples, unknown common variance, H0 : µ1 = µ2
    9. Example (comparison of two expected values)
18. Analysis of Variance (ANOVA)
    1. Aim and motivation
    2. k samples, unknown common variance, H0 : µ1 = · · · = µk
    3. Two examples (comparison of mean values from 3 samples)
19. The chi-square test (or ?2-test)
    1. ?2-test for equality of distribution
    2. The assumption of normal distribution
    3. Standardised residuals
    4. Example (women with 5 children)
    5. Example (election)
    6. Example (deaths in the Prussian cavalry)
20. Contingency tables
    1. Definition, method
    2. Standardised residuals
    3. Example (students’ political orientation)
    4. ?2-test for 2 × 2 tables
    5. Fisher’s exact test for 2 × 2 tables
    6. Example (Fisher’s exact test)
21. Distribution free tests
    1. Wilcoxon’s test for one set of observations
    2. Example
    3. The normal approximation to Wilcoxon’s test for one set of observations
    4. Wilcoxon’s test for two sets of observations
    5. The normal approximation to Wilcoxon’s test for two sets of observations
22. Linear regression
    1. The model
    2. Estimation of the parameters ß0 and ß1
    3. The distribution of the estimators
    4. Predicted values ˆyi and residuals ˆei
    5. Estimation of the variance s2
    6. Confidence intervals for the parameters ß0 and ß1
    7. The determination coefficient R2
    8. Predictions and prediction intervals
    9. Overview of formulas
    10. Example
23. A Overview of discrete distributions
24. B Tables
    1. B.1 How to read the tables
    2. B.2 The standard normal distribution
    3. B.3 The ?2-distribution (values x with F?2(x) = 0.500 etc.)
    4. B.4 Student’s t-distribution (values x with FStudent(x) = 0.600 etc.)
    5. B.5 Fisher’s F-distribution (values x with FFisher(x) = 0.90)
    6. B.6 Fisher’s F-distribution (values x with FFisher(x) = 0.95)
    7. B.7 Fisher’s F-distribution (values x with FFisher(x) = 0.99)
    8. B.8 Wilcoxon’s test for one set of observations
    9. B.9 Wilcoxon’s test for two sets of observations, a = 5%
25. C Explanation of symbols
26. D Index