Populations and samples, random and non-random sampling methods, and presenting and interpreting data with measures of location, spread, histograms and box plots.

What this dot point is asking

WJEC wants you to distinguish a population from a sample, to describe and compare sampling methods (random, systematic, stratified, quota, opportunity), and to present and interpret data using measures of location (mean, median, mode), measures of spread (range, interquartile range, standard deviation), histograms and box plots. This is the descriptive backbone of the statistics section before the probability and distribution work.

The answer

Populations and samples

A census measures every member of a population; a sample measures a subset. Sampling is used when a census is too costly, too slow, or destroys the item tested. A good sample should be representative so that conclusions generalise to the population.

Sampling methods

Stratified sampling is the one most often calculated: find the sampling fraction $\dfrac{\text{sample size}}{\text{population}}$ and apply it to each stratum.

Measures of location and spread

The mean is $\bar{x} = \dfrac{\sum x}{n}$; the median is the middle value when ordered; the mode is the most frequent. For spread, the range is max minus min, the interquartile range is $Q_3 - Q_1$ (resistant to outliers), and the standard deviation measures typical distance from the mean.

Histograms and box plots

A histogram displays continuous grouped data with bars whose area equals the frequency, so the vertical axis is frequency density $= \dfrac{\text{frequency}}{\text{class width}}$. A box plot marks the five-number summary and makes outliers and skew visible at a glance.

Examples in context

Example 1. Comparing two box plots. Two classes sit the same test. Class A has median $58$ and IQR $14$; class B has median $62$ and IQR $26$. Class B scored higher on average but was far more spread out, so its results were less consistent. Box plots let you compare centre and spread in one picture.

Example 2. A biased frame. A council surveys residents using the electoral roll, which omits people under $18$ and the unregistered. Conclusions about "all residents" are biased because the sampling frame is incomplete. Identifying the gap between frame and population is a standard exam discussion point.

Try this

Q1. A factory takes every 50th item off a production line for testing. Name this sampling method. [1 mark]

• Cue. Systematic sampling (every $k$th unit from a start point).

Q2. A class of 30 has $\sum x = 1500$ for a test. Find the mean. [1 mark]

• Cue. $\bar{x} = \dfrac{1500}{30} = 50$.

Q3. A histogram class spans $10$ to $20$ with frequency $35$. Find the frequency density. [2 marks]

• Cue. Class width is $10$, so frequency density $= \dfrac{35}{10} = 3.5$.