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VCE General Mathematics (Year 11) · Data Investigation · 12m read

Summarising Univariate Data

Ever wonder how to turn a messy pile of numbers into something that actually makes sense? Summarising univariate data means calculating key statistics—like the mean, median, and range—that capture what a single dataset is really telling you. You'll use these tools constantly in General Maths assessments and real situations like understanding test score patterns or analysing sales figures, so mastering them now means you're set up for success across the entire subject.

Learning objective: Calculate and interpret summary statistics for numerical data.

By the end of this lesson you can:

  • I find mean, median, mode, range, IQR, SD.
  • I choose the right measure of centre and spread.
  • I describe the shape of a distribution.

Summary statistics are a selfie of the data. One snapshot — centre, spread and shape — tells you what the whole data set looks like.

Measuring centre and spread

DATA TYPES:
• CATEGORICAL — labels (hair colour, yes/no).
• NUMERICAL — discrete (counts) or continuous (measurements).

MEASURES OF CENTRE:
• MEAN — arithmetic average. Sensitive to outliers.
• MEDIAN — middle value. Robust to outliers.
• MODE — most frequent. Useful for categorical.

MEASURES OF SPREAD:
• RANGE = max − min.
• INTERQUARTILE RANGE (IQR) = Q3 − Q1 — middle 50%.
• STANDARD DEVIATION (SD) — average deviation from mean.

QUARTILES:
• Q1 = 25th percentile.
• Q2 = median.
• Q3 = 75th percentile.

FIVE-NUMBER SUMMARY:
• Min, Q1, Median, Q3, Max.
• Used to build BOXPLOTS.

OUTLIERS:
• Value outside Q1 − 1.5 × IQR or Q3 + 1.5 × IQR.
• Investigate before removing.

DISTRIBUTION SHAPE:
• SYMMETRIC — similar on both sides.
• POSITIVELY SKEWED — tail to the right.
• NEGATIVELY SKEWED — tail to the left.

CHOOSING MEASURES:
• Symmetric, no outliers → mean + SD.
• Skewed OR outliers → median + IQR.

CALCULATOR SKILLS:
• Use 1-Var Stats on your VCE-approved CAS.
• Input list → read x̄, sx, Q1, Q3, etc.

REPORTING:
• Include units.
• Round sensibly (1-2 decimal places).
• Always STATE the measure you chose.

VCAA EMPHASIS:
• Interpretation > mechanics.
• "Compare these two distributions" questions are standard.

Which measures?

  • Symmetric: Mean + SD.
  • Skewed: Median + IQR.
  • Outliers: Use median + IQR.

Univariate summary

  • Mean: \bar x = \dfrac{\sum x}{n}
  • Standard deviation: s = \sqrt{\dfrac{\sum (x - \bar x)^2}{n - 1}}
  • IQR: Q_3 - Q_1

Quick reference

| Statistic | Meaning |
|---|---|
| Mean | Average |
| Median | Middle |
| Range | Max − min |
| IQR | Q3 − Q1 |
| SD | Spread around mean |
| Shape | Symmetric, positively/negatively skewed |

Worked example (Medium)

Data: 2, 3, 3, 4, 5, 5, 6, 7, 25. Find median, IQR. Discuss centre.

  1. Sort. Already sorted.
  2. Median. Middle value (5th of 9) = 5.
  3. Q1. Median of lower half (2, 3, 3, 4) = 3.
  4. Q3. Median of upper half (5, 6, 7, 25) = 6.5.
  5. IQR. Q3 − Q1 = 6.5 − 3 = 3.5.
  6. Outlier. 25 is far above Q3 + 1.5×IQR = 11.75 → outlier. Median + IQR preferable to mean + SD.

Answer: Median = 5, IQR = 3.5; presence of outlier makes median + IQR the appropriate summary.

Worked example (Hard)

Two factories report daily outputs. Factory A: 50, 52, 48, 53, 51, 49, 120. Factory B: 80, 82, 78, 85, 79, 81, 83. Compare using appropriate summary statistics.

  1. Outlier check A. Q1 = 49, Q3 = 53, IQR = 4. Upper fence = 53 + 6 = 59. Value 120 > 59 → outlier in Factory A.
  2. Choose measures. Factory A has outlier → median + IQR. Factory B is symmetric → mean + SD.
  3. Compute A. Median A = 51, IQR A = 4. (Mean would be ~60.4 — distorted by 120.)
  4. Compute B. Mean B = 81.1, SD B ≈ 2.3.

Answer: Factory A: median 51, IQR 4 (outlier present). Factory B: mean 81.1, SD 2.3. B has higher typical output and consistent results; A needs investigation of the outlier.

Common mistakes

  • Using mean + SD when data has outliers The mean is pulled by extreme values, giving a misleading centre. Fix: Always check for outliers first (1.5 × IQR rule). If outliers exist, report median + IQR.
  • Confusing IQR with range Range = max − min (entire spread); IQR = Q3 − Q1 (middle 50% only). Fix: IQR focuses on the central portion of data, ignoring extremes.
  • Forgetting to justify the choice of summary VCAA marks require reasoning, not just numbers. Fix: State: "Because the distribution is skewed / contains outliers, median + IQR is used."

Why standard deviation matters beyond school

Standard deviation (SD) measures how spread out data values are from the mean. In VCE General Maths, you use it alongside the mean for symmetric distributions — but its applications extend far beyond Year 11.

In finance, SD is used to measure volatility — how much an investment's returns fluctuate. A stock with SD of 20% is riskier than one with SD of 5%. Portfolio managers use SD to balance risk and return.

In quality control, manufacturers set tolerance limits based on SD. The famous Six Sigma methodology aims for fewer than 3.4 defects per million — achieved when the process mean is six standard deviations from the nearest specification limit.

In medicine, clinical trials report results as mean ± SD to show both the typical outcome and how much variation exists. A drug that lowers blood pressure by 10 ± 2 mmHg is far more reliable than one showing 10 ± 15 mmHg.

The empirical rule (68-95-99.7%) states that for approximately normal distributions, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. This connects directly to your VCE studies.

Practice questions

  1. Which summary pair is best for SKEWED data?

    1. Mean + SD
    2. Median + IQR
    3. Mode + Range
    4. Mean + Range

    (1 marks)

  2. Why is the MEAN sometimes misleading?

    (3 marks)

Summary

  • Find mean, median, mode, range, IQR, SD.
  • Choose the right measure of centre and spread.
  • Describe the shape of a distribution.

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