IB AA: Statistics and Probability

This chapter introduces the methods for collecting, analysing, interpreting, and presenting data. It is divided into two main parts: descriptive statistics and probability. You will begin with descriptive statistics, learning to calculate measures of central tendency (mean, median, mode) and measures of dispersion (range, interquartile range, standard deviation). This involves working with both raw data and frequency tables, and representing data visually using histograms, cumulative frequency graphs, and box-and-whisker plots.

The second part of the chapter covers probability theory. It builds from the basic definition of probability to more complex scenarios involving combined events, mutually exclusive events, and conditional probability. You will use Venn diagrams and tree diagrams to model situations and calculate probabilities. The chapter then introduces discrete random variables and two important probability distributions: the binomial distribution and the normal distribution. By the end, you will be expected to model real-world problems using these distributions and use a calculator to find associated probabilities.

Many students find conditional probability and knowing when to apply the addition versus the multiplication rule challenging at first. Distinguishing between the conditions for a binomial distribution and a normal distribution is another common difficulty. The chapter relies on a good foundation in basic algebra for manipulating formulas, but the concepts themselves are more about logical reasoning than complex algebraic skill.

Formulas for IB AA: Statistics and Probability

Mean of a data set: $\bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$ x is the data value and f is its frequency. For ungrouped data, all f_i are 1.
Standard deviation: $\sigma = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}}$ This is the square root of the variance, measuring the spread of data around the mean.
Interquartile range (IQR): $IQR = Q_3 - Q_1$ Q_3 is the upper quartile (75th percentile) and Q_1 is the lower quartile (25th percentile).
Probability of combined events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ This is the probability of event A or event B (or both) occurring.
Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ This is the probability of event A, given that event B has already occurred.
Independent events: $P(A \cap B) = P(A) \times P(B)$ If this equation holds, events A and B are independent of each other.
Expected value of a discrete random variable X: $E(X) = \sum x P(X=x)$ This is the mean or weighted average of all possible values of the variable.
Binomial distribution probability: $P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}$ Calculates the probability of r successes in n independent trials, where p is the success probability.
Expected value (mean) of a binomial distribution: $E(X) = np$ n is the number of trials and p is the probability of success.
Variance of a binomial distribution: $\text{Var}(X) = np(1-p)$ n is the number of trials and p is the probability of success.
Standardisation of a normal variable (Z-score): $Z = \frac{X - \mu}{\sigma}$ Converts a value X from a normal distribution with mean μ and standard deviation σ to the standard normal distribution.