26134 · Responsible Evidence-based Decisions
Exploring Data: Descriptive Statistics
Week 1 opens Module 1: how to summarise a raw dataset before you infer anything from it. You classify variables (qualitative vs quantitative, and the nominal/ordinal/interval/ratio scales), then compute measures of central tendency (mean, median, mode) and spread (range, variance, standard deviation, coefficient of variation), and read distribution shape. In the 26134 exam this shows up as compute-then-interpret items — calculate a mean and SD from a small sample, or explain why the median beats the mean under skew.
What this chapter covers
- 01Qualitative (nominal, ordinal) vs quantitative (discrete, continuous) variables and the four measurement scales
- 02Frequency, relative-frequency and cumulative-frequency tables; choosing class widths for a histogram
- 03Central tendency: mean x̄ = (Σxᵢ)/n, median (middle order statistic), mode
- 04Spread: range, sample variance s² = Σ(xᵢ − x̄)²/(n − 1), standard deviation s = √s²
- 05Coefficient of variation CV = (s/x̄) × 100% for comparing dispersion across scales
- 06Distribution shape: symmetric, right-skew (mean > median), left-skew, bimodal, uniform, J-shaped
- 07Why the mean alone is insufficient — two datasets can share a mean but differ in spread
- 08Excel descriptives: AVERAGE, MEDIAN, STDEV.S, VAR.S, MIN, MAX
Centre, spread and skew of a small salary sample
- +1Mean and median. Mean x̄ = (52 + 55 + 58 + 60 + 61 + 63 + 90)/7 = 439/7 = 62.71 ($000s). With n = 7 (odd) the median is the 4th ordered value = 60. There is no repeated value, so no mode.
- +1Sum of squared deviations from x̄ = 62.71: (−10.71)² + (−7.71)² + (−4.71)² + (−2.71)² + (−1.71)² + (0.29)² + (27.29)² ≈ 114.8 + 59.5 + 22.2 + 7.4 + 2.9 + 0.1 + 744.5 = 951.4.
- +1Sample variance s² = 951.4/(7 − 1) = 951.4/6 = 158.6, so s = √158.6 = 12.59 ($000s). Coefficient of variation CV = (12.59/62.71) × 100 = 20.1%.
- +1Interpret. The mean (62.71) sits above the median (60) because the single value 90 pulls the mean up — the sample is right-skewed. Report the median as the typical salary; it is robust to that outlier.
Key terms
- Qualitative vs quantitative variable
- Qualitative (categorical) values are labels — nominal (unordered, e.g. region) or ordinal (ordered, e.g. a satisfaction rating) — and are not meaningfully averaged. Quantitative values are numbers you can do arithmetic on, either discrete (counts) or continuous (measurements).
- Mean, median, mode
- Three measures of centre. The mean is the arithmetic average Σxᵢ/n; the median is the middle value of ordered data (robust to outliers); the mode is the most frequent value. Under right-skew, mean > median > mode.
- Sample variance and standard deviation
- Variance s² = Σ(xᵢ − x̄)²/(n − 1) measures average squared spread about the mean; the divisor n − 1 (degrees of freedom, Bessel's correction) makes it an unbiased estimator. The standard deviation s = √s² returns the spread to the data's own units.
- Coefficient of variation (CV)
- CV = (s/x̄) × 100%, a unitless relative measure of dispersion. It lets you compare variability between datasets measured on different scales or in different units, where comparing raw standard deviations would be meaningless.
- Skewness
- The asymmetry of a distribution. A long right tail (right/positive skew) pulls the mean above the median; a long left tail (left/negative skew) pulls it below. Symmetric distributions have mean ≈ median ≈ mode.
- Histogram
- A chart of grouped quantitative data with adjacent bars (no gaps), where bar height shows the frequency in each equal-width class. Its silhouette reveals distribution shape — symmetric, skewed, bimodal or uniform.
Exploring Data: Descriptive Statistics FAQ
When should I report the median instead of the mean?
Whenever the data are skewed or contain outliers. The mean is dragged toward extreme values, so for right-skewed data like income or salaries the median is the fairer 'typical' figure. Report both if you want to show the skew, and let the mean-vs-median gap signal its direction.
What is the difference between variance and standard deviation?
Variance s² is the average squared deviation from the mean, so its units are the data's units squared (e.g. dollars²), which is hard to interpret. The standard deviation s = √s² takes the square root to return to the original units, which is why you usually report s.
Why divide by n − 1 and not n for a sample?
Dividing by n − 1 (the degrees of freedom) corrects a bias: using the sample mean to centre the data 'uses up' one piece of information, so dividing by n would systematically underestimate the population variance. n − 1 gives an unbiased estimator. Use n only for a full population variance σ².
Can AI help me with descriptive statistics in 26134?
Yes, as a study aid. Sia can walk you through computing a mean, median, variance and CV from a small sample, explain why an outlier shifts the mean, and check your working and units on practice problems step by step. Use it to rehearse the method; it does not do your graded quizzes or exam for you, and the UTS academic-integrity policy applies — confirm assessment details on Canvas.
Exam move
Drill the descriptive workflow until it is automatic: classify each variable by type and scale, then compute centre (mean, median, mode) and spread (range, s², s, CV) and describe the shape in one sentence. Practise on small samples by hand so you understand each step, then reproduce it in Excel with AVERAGE, MEDIAN, STDEV.S and VAR.S so you are fast under the restricted open-book exam clock. Always pair a number with an interpretation — the exam rewards saying why the median beats the mean under skew, not just producing both. Keep one printed page of the descriptive formulas in your exam notes. When a step will not click, ask Sia to re-explain it and set a fresh practice sample; confirm any assessment details on Canvas.
Working through Exploring Data: Descriptive Statistics in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Exploring Data: Descriptive Statistics question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
