ETX5900 · Business Statistics
Descriptive Statistics II: Distribution, Summary Measures & Pivot Tables
This is Module 2 (Week 2) of ETX5900 Business Statistics at Monash University, the numerical half of descriptive statistics (Berenson Ch 3). You learn to put a single number on a data column — the centre (mean, median, mode), the spread (range, IQR, sample variance s², standard deviation s, and the unit-free coefficient of variation CV), and the shape (skewness and the empirical rule) — then summarise groups with PivotTables and cross-tabulation. These measures are the foundation of every later topic in the unit, from the normal distribution to the Chi-square assignment task.
What this chapter covers
- 01Central tendency: mean x̄, median and mode — and when to use each
- 02The mean-median gap as a fast read of distribution shape
- 03Measures of variation: range, interquartile range (IQR = Q₃ − Q₁)
- 04Sample variance s² and standard deviation s — the n−1 (Bessel) divisor vs population N
- 05Coefficient of variation CV = (s / x̄) × 100% for comparing relative spread across differently-scaled series
- 06Distribution shape: symmetric vs left/right skew; the empirical 68–95–99.7 rule (bell-shaped data only)
- 07The five-number summary and the boxplot
- 08PivotTables and cross-tabulation for group summaries; descriptive vs predictive data mining
Worked example: a full numerical summary of a small sample
- +1Mean. Σx = 12+14+15+15+18+21+23+30 = 148, so x̄ = 148 / 8 = 18.5 ($000).
- +1Median & mode. The data is ordered; with n = 8 the median is the average of the 4th and 5th values = (15 + 18)/2 = 16.5. The value 15 occurs twice and nothing else repeats, so the mode = 15.
- +1Squared deviations. Σ(x−x̄)² = 42.25+20.25+12.25+12.25+0.25+6.25+20.25+132.25 = 246.
- +1Sample variance & SD (divide by n−1 = 7). s² = 246 / 7 = 35.14; s = √35.14 = 5.93 ($000). Dividing by 8 would be the wrong, population, formula.
- +1Coefficient of variation. CV = (s / x̄) × 100% = (5.93 / 18.5) × 100% = 32.0% — moderately variable relative to its mean.
- +1Shape. mean 18.5 > median 16.5, and the high value 30 sits far above the pack, so the distribution is right- / positively skewed; report the median (16.5) as the typical month.
Key terms
- Mean (x̄)
- The arithmetic average, x̄ = (Σxᵢ)/n. Best on roughly symmetric data; it is pulled toward a long tail, so on skewed money data it overstates the 'typical' value.
- Median
- The middle value of the ordered data (the average of the two middle values when n is even). Robust to outliers, so it is the fairer centre for skewed variables such as incomes, prices and sales.
- Sample variance (s²) & standard deviation (s)
- s² = Σ(xᵢ − x̄)² / (n−1) and s = √s². The n−1 divisor (Bessel's correction) removes the downward bias in estimating the population spread; a population uses divisor N instead.
- Coefficient of variation (CV)
- CV = (s / x̄) × 100% — a unit-free, percentage measure of relative spread. It lets you compare variability across series with different means or units, where the raw standard deviation cannot.
- Interquartile range (IQR)
- IQR = Q₃ − Q₁, the spread of the middle 50% of the ordered data. Being built only from the quartiles, it is resistant to outliers, unlike the range or the standard deviation.
- Skewness
- A measure of asymmetry. Right (positive) skew has a long high tail and mean > median; left (negative) skew has a long low tail and mean < median; a symmetric distribution has mean ≈ median.
- Empirical (68–95–99.7) rule
- For approximately bell-shaped data, about 68% of values lie within x̄ ± 1s, about 95% within x̄ ± 2s, and about 99.7% within x̄ ± 3s. It does not apply to clearly skewed data.
- PivotTable / cross-tabulation
- An Excel tool that summarises one variable (as a count, sum or average) within groups of another, laid out as a table. It is the descriptive-analytics workhorse behind the unit's group summaries and the Chi-square assignment task.
Descriptive Statistics II: Distribution, Summary Measures & Pivot Tables FAQ
When do I divide by n and when by n−1?
Divide by n−1 for a sample variance or standard deviation (s², s) — the default, since you almost always work from a sample. Divide by N only when the data is the entire population (σ², σ). Using n for a sample is one of the most common mark-losers on this topic, so read whether the question says 'sample' or 'population'.
Standard deviation or coefficient of variation — which shows more variability?
The standard deviation is in the data's own units, so you cannot use it to compare two series with different means or units. For that, use the coefficient of variation, CV = (s / x̄) × 100%. A larger s does not automatically mean more variable: a series with a bigger s but a proportionally bigger mean can have a lower CV, and so be relatively less variable.
Can AI help me with descriptive statistics and pivot tables in ETX5900?
Yes — Sia can explain the concepts step by step: how to choose between the mean and the median, why the sample variance divides by n−1, how to read skew from a boxplot, and how a PivotTable summarises groups. It walks you through a method and checks your reasoning, but it does not do your assessed quiz, assignment or exam for you, and it cannot promise any grade or mark — the point is to build the understanding you carry into the invigilated e-exam.
Exam move
Drill the three questions you ask of any data column: where is the centre, how spread out is it, and what shape is it. Compute the mean, median, mode, s and CV by hand on a few small samples until the n−1 divisor is automatic, and always attach units and a one-line interpretation to each number — that interpretation is where easy marks are won and lost. Learn to read the mean-median gap and the boxplot for skew, and remember the empirical rule is for bell-shaped data only. Because the Final Examination is worth 50% and provides a formula sheet and statistical tables, the marks reward choosing the right measure and one clean calculation, not memorising formulae. As the exam date is 'to be advised', budget your time in proportion to the marks on each question and confirm the exam length on Moodle.
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