Data Distributions

Overview

Data distributions describe how values in a dataset are spread across possible outcomes. In VCE General Mathematics, understanding distributions is the foundation for all statistical analysis — from reading graphs to calculating summary statistics and making inferences.

Types of Distribution Shape

When you display data in a histogram, dot plot, or stem-and-leaf plot, the shape tells you important information:

Shape	Description	Example
Symmetric	Mirror image left/right; mean ≈ median	Heights of adults
Positively skewed	Long tail to the right; mean > median	Income data
Negatively skewed	Long tail to the left; mean < median	Test scores near max
Bimodal	Two peaks	Heights of mixed gender group
Uniform	All values roughly equal frequency	Rolling a fair die

Centre and Spread

Every distribution is described by two key features:

Centre: where the data clusters (mean, median, mode)
Spread: how varied the data is (range, IQR, standard deviation)

KEY TAKEAWAY: Always describe a distribution’s shape, centre, and spread together — no single number tells the whole story.

Reading Frequency Histograms

A frequency histogram plots class intervals on the x-axis and frequency (count) or relative frequency on the y-axis.

Each bar represents an interval; bars touch (continuous data)
The modal class is the interval with the tallest bar
Skew is identified by which tail is longer

Example: A histogram of 30 students’ exam scores (out of 100):

Score interval	Frequency
40–49	2
50–59	5
60–69	10
70–79	8
80–89	4
90–99	1

This distribution is slightly negatively skewed (tail to the left), with modal class 60–69.

Stem-and-Leaf Plots

A stem-and-leaf plot preserves individual data values while showing the shape:

Stem | Leaf
  4  | 2 7
  5  | 1 3 5 8 9
  6  | 0 2 4 4 6 7 8
  7  | 1 3 5 6 8
  8  | 2 4 7 9
  9  | 1

Each row is a stem (tens digit); leaves are units digits. Reading left-to-right gives sorted data.

EXAM TIP: In a back-to-back stem-and-leaf plot, leaves on the left read right-to-left. Always state the key (e.g. “5 | 3 means 53”).

Dot Plots

A dot plot places one dot per observation above a number line. Best for small datasets. Gaps, clusters, and outliers are immediately visible.

Describing Distributions in Context

VCAA expects you to describe distributions using these four features:

Shape (symmetric / positively skewed / negatively skewed / bimodal)
Centre (typical value — quote the median or mean)
Spread (range or IQR)
Outliers (any unusually high or low values)

Example response: “The distribution of house prices is positively skewed with a median of \$650,000 and an IQR of \$180,000. There is one outlier at \$2.1 million.”

VCAA FOCUS: Always use the context of the question when describing distributions. Generic answers (“the data is spread out”) earn minimal marks — reference actual values and what they mean.

Common Mistakes

Confusing skew direction with where most data sits. Positive skew = tail to the right, even though most data is on the left.
Forgetting to mention outliers when describing a distribution.
Using mean to describe the centre of a skewed distribution (use median instead).

REMEMBER: Shape → Centre → Spread → Outliers. Use this order every time you describe a distribution.

Data Distributions Overview

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