Displaying & Describing Distributions

Classifying Data: The First Decision

Before you can choose a graph or a summary statistic, you must classify the variable. In VCE General Maths every variable is either categorical or numerical, and each of those splits again.

Categorical data places individuals into named groups. It divides into:

• Nominal — categories with no natural order, e.g. eye colour (blue, brown, green) or method of travel to school (car, train, bike, walk).
• Ordinal — categories that can be ranked, e.g. T-shirt size (S, M, L, XL) or a satisfaction rating (poor, fair, good, excellent).

Numerical data records quantities you can measure or count. It divides into:

• Discrete — counted values, usually whole numbers, e.g. number of pets, goals scored, siblings.
• Continuous — measured values on a continuous scale that can take any value in an interval, e.g. height (174.6 cm), time (12.38 s), mass.

A reliable test for discrete vs continuous: ask "can it ever be a fraction in a meaningful way?" You can have 1.7 metres of height (continuous) but not 1.7 children (discrete).

Why classification matters. The data type controls every later choice:

• Categorical data is summarised with frequency tables and displayed with bar charts (and you report the mode — the most common category — never a mean).
• Numerical data is displayed with dot plots, stem plots, histograms or boxplots, and summarised with the mean, median, range, IQR and standard deviation.

Worked classification. A survey records, for each student: (a) postcode, (b) number of text messages sent yesterday, (c) reaction time in seconds, (d) movie rating out of 5 stars. Classify each.

• (a) Postcode — although it is a number, it is a label; averaging postcodes is meaningless, so it is categorical, nominal.
• (b) Number of texts — a count, so numerical, discrete.
• (c) Reaction time — measured, can be any decimal, so numerical, continuous.
• (d) Star rating — ranked categories, so categorical, ordinal.

Postcode is the classic trap: a numeral printed on the data sheet is not automatically numerical data.

Dot Plots, Stem Plots & Histograms

These three displays are all for numerical data and let you read off shape, centre, spread and outliers.

Dot plots. Each data value is one dot stacked above a number line. Best for small data sets of discrete or rounded values. You can directly count to find the median and spot gaps and outliers.

Stem-and-leaf plots (stem plots). The leading digits form the stem and the final digit forms the leaf. They keep every original value (unlike a histogram) while showing shape. Always include a key, e.g. "6 | 3 means 63". Leaves should be ordered and the plot should look like a sideways bar chart.

Histograms. Data is grouped into equal-width class intervals; bar height is the frequency. Bars touch (unlike a bar chart) because the scale is continuous. Use a histogram for larger data sets.

Describing shape. Report three things in a sentence: shape, centre, spread, plus outliers.

• Symmetric — a single peak with roughly mirror-image tails.
• Positively skewed — the long tail points to the right (high values). Most data bunched at the low end.
• Negatively skewed — the long tail points to the left (low values).

A memory aid: the skew direction is named after where the tail points, not where the hump is.

Worked example — build a stem plot. Resting heart rates (bpm) of 15 students: 58, 62, 71, 65, 49, 73, 68, 60, 77, 55, 63, 81, 66, 70, 59.

Stems are the tens digit (4, 5, 6, 7, 8). Sort the values into stems, then order the leaves:

• 4 | 9
• 5 | 5 8 9
• 6 | 0 2 3 5 6 8
• 7 | 0 1 3 7
• 8 | 1

Key: 6 | 2 means 62 bpm. Reading the plot: the bulk of values sit in the 60s, there is a single peak there, and the tails are fairly even with one high value (81) and one low value (49). So the distribution is approximately symmetric, centred near the low-to-mid 60s, with no clear outliers (we confirm outliers formally in a later section). Counting to the 8th value (since n = 15, the median is the 8th) gives 65 bpm as the centre — consistent with our visual read.