Statistics and probability

• Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
• Grouped data: mid-interval values, interval width, upper and lower interval boundaries.
• Mean, variance, standard deviation.

Not required:

• Estimation of mean and variance of a population from a sample.

• For examination purposes, in papers 1 and 2 data will be treated as the population.
• In examinations the following formulae should be used:

• TOK: The nature of mathematics. Why have mathematics and statistics sometimes been treated as separate subjects?
• TOK: The nature of knowing. Is there a difference between information and data?
• Aim 8: Does the use of statistics lead to an overemphasis on attributes that can easily be measured over those that cannot?
• Appl: Psychology SL/HL (descriptive statistics); Geography SL/HL (geographic skills); Biology SL/HL 1.1.2 (statistical analysis).
• Appl: Methods of collecting data in real life (census versus sampling).
• Appl: Misleading statistics in media reports.

• Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
• The probability of an event A as P( A) .
• The complementary events A and A′ (not A).
• Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.

• Aim 8: Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (eg economics)?
• Int: The development of the mathematical theory of probability in 17th century France.

• Conditional probability; the definition
• Independent events; the definition
• Use of Bayes’ theorem for a maximum of three events.

• Appl: Use of probability methods in medical studies to assess risk factors for certain diseases.
• TOK: Mathematics and knowledge claims. Is independence as defined in probabilistic terms the same as that found in normal experience?

• Concept of discrete and continuous random variables and their probability distributions.
• Definition and use of probability density functions.
• Expected value (mean), mode, median,variance and standard deviation.
• Applications.

• For a continuous random variable, a value at which the probability density function has a maximum value is called a mode.
• Examples include games of chance.

• TOK: Mathematics and the knower. To what extent can we trust samples of data?
• Appl: Expected gain to insurance companies.

• Binomial distribution, its mean and variance.
• Poisson distribution, its mean and variance.

Not required:

• Formal proof of means and variances.

• Link to binomial theorem in 1.3.
• Conditions under which random variables have these distributions.

• TOK: Mathematics and the real world. Is the binomial distribution ever a useful model for an actual real-world situation?

• Normal distribution.
• Properties of the normal distribution.
• Standardization of normal variables

.

• Probabilities and values of the variable must be found using technology.
• The standardized value (z) gives the number of standard deviations from the mean.
• Link to 2.3.

• Appl: Chemistry SL/HL 6.2 (collision theory);Psychology HL (descriptive statistics); Biology SL/HL 1.1.3 (statistical analysis).
• Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and conclusions?
• TOK: Mathematics and knowledge claims. To what extent can we trust mathematical models such as the normal distribution? Int: De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.