Statistics and Probability

Section 4 contributes 5 items to Paper 01 and one 20-mark question to Paper 02. The Paper 02 question typically combines statistical calculation with interpretation and a probability problem using diagrams.

Types of Data¶

Qualitative (categorical) data describes attributes without numerical value: eye colour, nationality, type of vehicle.

Quantitative data is numerical and divides into two types:

• Discrete: takes separate countable values (number of siblings, goals scored).
• Continuous: takes any value within an interval (height, temperature, time).

Measures of Central Tendency¶

$\bar{x} = \frac{\sum x}{n} \qquad \text{(ungrouped)} \qquad \bar{x} = \frac{\sum fx}{\sum f} \qquad \text{(grouped/frequency)}$

Measures of Spread¶

The simplest measure of spread is the range: $\text{range} = \text{maximum} - \text{minimum}$. It is easy to compute but sensitive to a single extreme value.

Quartiles and Interquartile Range¶

Ordered data is divided into four equal parts by the quartiles $Q_1$, $Q_2$ (median), and $Q_3$.

$Q_1 = \text{median of lower half}, \quad Q_2 = \text{median}, \quad Q_3 = \text{median of upper half}$

$\text{IQR} = Q_3 - Q_1 \qquad \text{Semi-IQR} = \frac{Q_3 - Q_1}{2}$

The IQR measures the spread of the middle 50% of the data. Unlike the range, it is not distorted by a single extreme value.

Percentiles¶

The $p$th percentile is the value below which $p\%$ of the data lies. Quartiles are specific percentiles: $Q_1 = P_{25}$, $Q_2 = P_{50}$, $Q_3 = P_{75}$.

Variance and Standard Deviation¶

Variance measures average squared distance from the mean. Standard deviation is its square root, restoring the original units.

$S^2 = \frac{\sum(x - \bar{x})^2}{n} \qquad S = \sqrt{\frac{\sum(x-\bar{x})^2}{n}} \qquad \text{(ungrouped)}$

$S^2 = \frac{\sum f(x-\bar{x})^2}{\sum f} \qquad \text{(grouped)}$

A small standard deviation means values cluster closely around the mean (consistent). A large standard deviation means values are widely spread (variable).

Stem-and-Leaf Diagrams¶

A stem-and-leaf diagram keeps the original data values while showing the distribution shape.

Constructing one:

1. Split each value: the stem is all digits except the last; the leaf is the last digit.
2. Write stems in a column and leaves in ascending order beside each stem.
3. Include a key.

1 | 7 9
2 | 2
3 | 1 3 8
4 | 4 7 9

Key: 1 | 7 means 17

A back-to-back stem-and-leaf diagram places two datasets side by side sharing a common stem, allowing direct visual comparison.

Box-and-Whisker Plots¶

A box plot summarises a dataset using five values: minimum, $Q_1$, median, $Q_3$, maximum.

Min    Q₁    Q₂    Q₃    Max
 |------[======|======]------|

The box spans from $Q_1$ to $Q_3$ and contains the middle 50% of the data. The whiskers extend to the minimum and maximum.

Reading Skewness from a Box Plot¶

For a positively skewed distribution: Mode $<$ Median $<$ Mean. For a negatively skewed distribution: Mean $<$ Median $<$ Mode.

Probability Theory¶

An experiment produces outcomes. The set of all possible outcomes is the sample space $S$. An event is a subset of the sample space.

$P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes in } S}$

This is classical probability, valid when all outcomes are equally likely.

Relative frequency gives an experimental estimate of probability when theoretical equally-likely outcomes cannot be assumed:

$P(A) \approx \frac{\text{number of times } A \text{ occurred}}{\text{total number of trials}}$

As the number of trials increases, the relative frequency approaches the true probability.

Basic Laws¶

• $0 \leq P(A) \leq 1$ for any event $A$.
• $\sum P(\text{all outcomes}) = 1$.
• $P(A') = 1 - P(A)$, where $A'$ is the complement of $A$ (the event that $A$ does not occur).

The Addition Rule¶

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

For mutually exclusive events ($A$ and $B$ cannot both occur): $P(A \cap B) = 0$, so:

$P(A \cup B) = P(A) + P(B)$

Conditional Probability¶

The conditional probability of $A$ given $B$ has occurred:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Rearranging: $P(A \cap B) = P(A|B) \cdot P(B)$.

Independent Events¶

Events $A$ and $B$ are independent if knowing that $B$ occurred gives no information about $A$:

$P(A|B) = P(A) \qquad \text{equivalently} \qquad P(A \cap B) = P(A) \cdot P(B)$

Probability Diagrams¶

Possibility Space Diagrams¶

A possibility space (sample space) diagram lists all outcomes in a grid. Useful for two-stage experiments like rolling two dice.

Tree Diagrams¶

Tree diagrams show sequential outcomes. Branches show each possible outcome at each stage; multiply along branches for intersection probabilities.

The syllabus restricts tree diagrams to two initial branches.

Venn Diagrams¶

Venn diagrams with two sets partition the sample space into four regions: $A$ only, $B$ only, $A \cap B$, and neither. The sum of all regions must equal $P(S) = 1$ (or $n(S)$ for counts).

The syllabus restricts Venn diagrams to two sets.