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VCE General Mathematics (Year 12) · Data Analysis · 14m read

Investigating Data Distributions

Every General Maths exam opens with data. Before you can compare, model or predict anything, you need to look at one variable and say what it is doing: is it symmetric or skewed, where does it sit, how spread out is it, and is anything weird? Master this "describe the distribution" routine and you will pick up easy marks on Exam 1 and Exam 2 every year.

Learning objective: Classify variables, choose suitable displays and describe a distribution using shape, outliers, centre and spread.

By the end of this lesson you can:

  • I classify variables as nominal, ordinal, discrete or continuous.
  • I build and read histograms, dot plots and boxplots (including log scales).
  • I describe a distribution using shape, outliers, centre and spread.

A distribution is a group photo. Centre, spread, shape and outliers describe the whole photo — where people stand, how spread out, and who's off to the side.

Variables, displays and the SOCS routine

TYPES OF VARIABLES:
• CATEGORICAL — NOMINAL (labels with no order: eye colour, state) or ORDINAL (ordered labels: low / medium / high).
• NUMERICAL — DISCRETE (counts: number of siblings) or CONTINUOUS (measurements: height, time).
• A number can still be categorical — a postcode is nominal.

CHOOSING A DISPLAY:
• Categorical → bar chart (or segmented bar chart).
• Numerical, small data set → dot plot or stem plot.
• Numerical, large data set → histogram or boxplot.
• Values spanning several powers of ten (incomes, populations) → histogram with a LOG10 scale.

LOG SCALES:
• log10(10) = 1, log10(100) = 2, log10(1000) = 3.
• A value of 2.5 on a log axis means 10^2.5 ≈ 316.
• A log transformation can make a strongly positively skewed distribution look symmetric.

DESCRIBING A DISTRIBUTION (SOCS):
• SHAPE — symmetric, positively skewed (tail to the right), negatively skewed (tail to the left); note if bimodal.
• OUTLIERS — any values outside the fences?
• CENTRE — median (or mean if symmetric).
• SPREAD — IQR (or SD if symmetric), plus range for context.

OUTLIER FENCES:
• Lower fence = Q1 − 1.5 × IQR.
• Upper fence = Q3 + 1.5 × IQR.
• In a boxplot, outliers are plotted as separate points and the whisker stops at the last value inside the fence.

Picking centre and spread

  • Symmetric, no outliers: Report mean and standard deviation.
  • Skewed or outliers present: Report median and IQR — they resist extreme values.
  • Always: Include units and state which measure you used and why.

Five-number summary and fences

  • Interquartile range: \text{IQR} = Q_3 - Q_1
  • Lower fence: Q_1 - 1.5 \times \text{IQR}
  • Upper fence: Q_3 + 1.5 \times \text{IQR}

Quick reference

| Feature | Describe with |
|---|---|
| Shape | Symmetric, positively/negatively skewed |
| Centre | Median (skewed) or mean (symmetric) |
| Spread | IQR (skewed) or SD (symmetric) |
| Outliers | Q1 − 1.5×IQR, Q3 + 1.5×IQR |
| Display | Histogram, boxplot, dot plot |

Worked example (Medium)

Ten students recorded their weekly hours of part-time work: 12, 15, 15, 17, 18, 19, 21, 22, 24, 40. Find the five-number summary and decide whether 40 is an outlier.

  1. Median. n = 10, so the median is the mean of the 5th and 6th values: (18 + 19) ÷ 2 = 18.5 hours.
  2. Quartiles. Lower half 12, 15, 15, 17, 18 → Q1 = 15. Upper half 19, 21, 22, 24, 40 → Q3 = 22.
  3. IQR. IQR = 22 − 15 = 7.
  4. Fences. Upper fence = 22 + 1.5 × 7 = 32.5. Lower fence = 15 − 10.5 = 4.5.
  5. Decide. 40 > 32.5, so 40 is an outlier. The upper whisker ends at 24.

Answer: Min 12, Q1 15, median 18.5, Q3 22, max 40; 40 hours is an outlier.

Worked example (Easy)

A histogram of town populations uses a log10 scale. The tallest column covers 3.5 to 4.0. Describe the populations in that column.

  1. Undo the log. 10^3.5 ≈ 3162 and 10^4.0 = 10 000.
  2. Interpret. The most common towns have populations between about 3200 and 10 000.

Answer: Between roughly 3200 and 10 000 people.

Common mistakes

  • Calling a postcode or jersey number numerical Numbers used as labels cannot be averaged meaningfully. Fix: Ask: does arithmetic on it make sense? If not, it is categorical.
  • Mixing up the direction of skew Skew is named after the TAIL, not the peak. Fix: Long tail on the right (high values) = positive skew.
  • Drawing the whisker to the outlier Whiskers stop at the most extreme value still inside the fence. Fix: Plot outliers as separate dots.

Choosing mean or median

Outliers and skew drag the mean towards the tail, but barely move the median.

That's why house prices and incomes are usually reported as medians — a few mansions or billionaires would distort the mean.

Practice questions

  1. Customer satisfaction is recorded as "poor", "fair", "good" or "excellent". This variable is:

    1. Nominal
    2. Ordinal
    3. Discrete numerical
    4. Continuous numerical

    (1 marks)

  2. A data set has Q1 = 40 and Q3 = 56. Is a value of 82 an outlier? Show working.

    (2 marks)

  3. On a log10 scale, a value is plotted at 2. What is the actual value?

    1. 2
    2. 20
    3. 100
    4. 1000

    (1 marks)

Summary

  • Classify variables before choosing a display.
  • Use SOCS: shape, outliers, centre, spread.
  • Outlier fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.
  • Log scales turn powers of ten into evenly spaced values.

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