Collecting, Displaying and Misusing Data

In short: Data is either categorical (groups, like colours) or numerical (counts or measurements). A sample must be chosen fairly to represent the whole population, and graphs like histograms, dot plots and box plots show data clearly when their scales are not misleading.

Starting point: Years 9 and 10

Types of data

• Categorical data is in groups: favourite sport, eye colour. Nominal has no order, ordinal has an order (small, medium, large).
• Numerical data is numbers: discrete is counted (number of pets), continuous is measured (height).

Census or sample?

A census collects data from every member of a population. A sample uses part of it. Samples are cheaper and faster, but only useful if they represent the population.

Sampling methods

• Simple random: everyone has an equal chance, like drawing names from a hat.
• Systematic: choose every $k$th person from a list, such as every 10th student.
• Stratified: split the population into groups (strata), then sample each group in proportion to its size.

Bias

A sample is biased if some people are more likely to be chosen, or if the questions push people towards an answer. The canteen survey was biased: only hungry students on one day were asked. Leading questions like "Don't you agree the wrap is delicious?" also cause bias.

Displays

• Dot plot and column graph for small data sets or categories.
• Stem-and-leaf plot: the stem is the tens digit, the leaf the units. It keeps every value and shows the shape. A back-to-back stem-and-leaf plot compares two groups.
• Histogram: bars touching, for grouped numerical data.

This shows 12, 15, 18, 20, 23, 23, 27, 29, 31, 34. The mode is 23 and the median is the average of 23 and 23, so 23.

Misleading graphs

Watch for an axis that does not start at zero, uneven scales, pictures where both width and height grow, and missing labels. They can make a small change look huge.

Time-series data

A time series records the same thing over time, like monthly rainfall or daily temperatures. Plot it as a line graph with time across the bottom.

Quick summary

• Data is categorical (nominal or ordinal) or numerical (discrete or continuous).
• A census asks everyone. A sample asks some, and must be representative.
• Sampling: simple random, systematic, stratified. Bias makes results untrustworthy.
• Displays: dot plots, column graphs, stem-and-leaf plots, histograms. Watch for misleading scales.
• Time series show trend, seasonal and irregular patterns.

Quiz