National Year 11 Statistics
Chapters: 2
1. 2. Processing, representing and analysing data (part 2)
2e Scatter diagrams and correlation · 2f Time series · 2g Quality assurance (H) · 2h Estimation
- Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.
- Time Series: Moving Averages, Trend and Seasonal Variation – A time series is a set of data recorded at equal time intervals, such as sales every quarter or rainfall every month. We plot it as a line graph with time on the horizontal axis. Real series go up and down with the seasons (seasonal variation), in longer waves (cyclic variation) and with random noise. To see the long-term direction (the trend) we smooth the data with moving averages: the mean of each group of consecutive values, for example 4 values for quarterly data. A line of best fit through the moving averages is the trend line. Seasonal variation = actual value − trend value; the mean seasonal variation for each season, added to the trend, gives a forecast.
- Sampling Distributions and the Central Limit Theorem – A statistic (like a sample mean x̄ or a sample proportion p̂) changes from sample to sample. If you took every possible sample and plotted the statistic, you would get its sampling distribution. Its centre is the true population value (μ or p), its spread is the standard error (σ/√n for means, √(p(1−p)/n) for proportions), and by the Central Limit Theorem its shape becomes close to normal when n is large enough.
- Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.
2. 3. Probability
Probability from data and theory · Combined events · Probability distributions (H)
- Probability: Chances of a Single Event – Probability tells how likely something is, as a number from 0 to 1. When all outcomes are equally likely, P(event) = number of favourable outcomes ÷ total number of outcomes. An impossible event has probability 0 and a sure event has probability 1.
- Combined Events, Tree Diagrams and Conditional Probability – When two things happen, list every outcome in a sample space, a table, a Venn diagram or a tree diagram. Multiply along tree branches (the multiplication rule) and add the paths you want. If the first event changes the second, the events are dependent: picking without replacement is the classic case. Conditional probability P(A | B) is the chance of A when we already know B happened: P(A | B) = P(A and B) ÷ P(B).
- Probability Distributions of Discrete Random Variables – A random variable X turns each outcome of an experiment into a number. Its probability distribution lists every value x with its probability P(X = x); each P is between 0 and 1 and they add to 1. The mean E(X) = Σx·P(x) is the long-run average (balance point). The variance Var(X) = Σ(x − μ)²P(x) = E(X²) − μ² measures spread; σ = √Var. Special models: uniform, binomial B(n, p) with mean np and variance np(1 − p), and Poisson with mean = variance = λ.