Data and chance: recap

Discrete data is counted: it takes separate values, such as 0, 1, 2 siblings. Continuous data is measured: it can be any value in a range, such as a 100 m time of 14.37 s.

Measured values are always rounded: 152 cm to the nearest centimetre means from 151.5 cm up to 152.5 cm. Decimals alone do not make data continuous: shoe sizes (7, 7.5, 8) are discrete.

Four numbers sum up a data set:

• mean: add the values, then divide by how many there are (3, 5, 5, 6, 11: 30 ÷ 5 = 6);
• median: sort the values and take the middle one; for an even count, halfway between the two middle values;
• mode: the most common value (there can be two, or none);
• range: highest minus lowest (11 − 3 = 8), a measure of spread.

The mean, median and mode can disagree. Look for outliers first (values far from the rest):

• An outlier pulls the mean towards it, so with one the median usually describes a typical value better ($5, 6, 6, 7, 8, 40$: mean 12, median 6.5).
• With no outliers, the mean is a fair summary because it uses every value.
• For "which is most common?", use the mode.

A stem-and-leaf plot keeps every value and shows the shape. For two-digit numbers the stem is the tens digit and the leaf is the units digit; leaves go in order along their stem's row. Include every stem from the smallest to the largest, even with no leaves, and give a key: $3 \mid 5$ means $35$.

Read the median by counting leaves to the middle; the range is the last value minus the first.

Describe a data set by its shape (symmetrical, or skewed with a tail; clusters, gaps, outliers), its centre (median or mean) and its spread (range).

To compare two sets, compare the centres and the spreads, with numbers and in context: "Similar medians (15 and 16), but Class A is more consistent (range 6 against 16)." One value, such as the highest score, does not describe a set.

A statistical investigation asks a question data can answer, plans the variable and a fair sample, collects and displays the data, summarises its shape, centre and spread, and reports an answer with its limits.

A sample that is small or chosen for convenience may be biased, so the report says who was measured and how.

The sample space is the list of every possible outcome. If the outcomes are equally likely,

$P(\text{event}) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$, so $P(\text{even on a dice}) = \tfrac{3}{6} = \tfrac{1}{2}$.

If they are not (a half-red spinner, a bag with more of one colour), use the sizes or the counts: $P(\text{red}) = \tfrac{1}{2}$. The probabilities of all the outcomes add to 1; 0 means impossible, 1 means certain.

A probability predicts what happens over many trials: expected count = probability × number of trials. In 60 rolls of a fair dice, expect about $60 \times \tfrac{1}{6} = 10$ sixes.

Then test it: run the experiment and record the counts. The relative frequency is count ÷ trials. In a short experiment the results vary: 7 or 12 sixes in 60 rolls is normal, so do not judge a dice on a few rolls.

Over a few trials, the relative frequency (count ÷ trials) jumps about. Over many trials it settles close to the probability: the law of large numbers. One more head moves 7/10 a lot, but 501/1000 hardly at all.

It is not because results balance out: a coin has no memory, and after 5 tails heads is still exactly 1/2. Large experiments are how we estimate probabilities we cannot work out, such as a drawing pin landing point-up.

The complement of an event A, written not A, is every other outcome. A and not A cover every outcome exactly once, so P(A) + P(not A) = 1, and P(not A) = 1 − P(A).

This makes at least one easy: count the outcomes with none, and subtract from 1. Two events that just can't happen together are not complementary unless they cover every outcome too.

Sorting a group by two yes/no properties makes four groups: one only, the other only, both and neither. Everyone is in exactly one.

A Venn diagram shows them as two overlapping circles in a box; a two-way table shows them as four cells with totals. Write both first, then take it from each total.

Two events one after the other make pairs. A tree diagram splits each first outcome into every second outcome; a grid puts one event down the side and the other across. Both list every pair once, so each is equally likely.

The number of outcomes is the two counts multiplied. Order matters: HT and TH are different outcomes. Then P(event) = outcomes in the event ÷ all outcomes.

A and B: both happen (the overlap). A or B: at least one happens, both included (everything inside the circles). Not A: everything outside A. Neither: outside both.

Count each outcome once. From totals: P(A or B) = P(A) + P(B) − P(A and B), because the overlap is in both totals.

A simulation repeats a chance experiment many times with random numbers. Record each result in a frequency table, then relative frequency = count ÷ trials.

Compare it with the probability from the sample space. A short run can be well off; over many trials the relative frequency settles close to the probability. A simulation estimates; the grid gives the exact value.

The population is everyone the question is about. A census measures all of them (exact, but often slow and costly); a sample measures some (quicker and cheaper, an estimate).

An experiment changes something on purpose and measures the effect; an observation watches without changing anything. Choose the technique that fits the question, the cost and the time.

Bias is anything in how data is collected that pushes the result one way. Sampling bias: the sample isn't like the population (only people at a gym). Response bias: the question or setting pushes answers (leading questions, no privacy). Measurement bias: the tool or method is off.

A bigger sample doesn't fix bias. Random choice, neutral questions, anonymous answers and checked tools do.

Simple random: every member has the same chance. Systematic: every kth member of a list from a random start (k = population ÷ sample size). Stratified: split into groups and choose at random within each, in proportion, so each group gets its fair share. Convenience: whoever is easiest, often biased. Quota: fixed shares, but the collector chooses, so it can be biased too.

To report on a sample: describe its shape (symmetric or skewed, clusters, outliers), centre (mean, median) and spread (range). Then give an estimate for the population, with the sampling method and size, using words like "about".

Primary data you collect yourself; secondary data someone else collected. Either way, say where it came from.

Sampling variation: random samples from the same population give different results, because each holds different members. It isn't a mistake.

Bigger samples vary less, so their results are more reliable. Fair samples scatter around the population's value; bias pushes them all to one side.

An investigation with samples asks a question about a population, plans a fair and ethical sample (random, consent, anonymous, neutral questions), collects and describes the data, then makes an inference.

A good inference is about the population sampled, gives an estimate ("about"), and states the uncertainty: the sample's size and method, and that another sample might differ a little.