Module 10 — Synthesis, review and mock exam

Data Description | Part III — Two variables and synthesis

Modified

October 11, 2026

The business question. You have learned the full descriptive statistics toolkit: organising data (Modules 1–3), summarising one variable (Modules 4–7), and analysing two variables (Modules 8–9). This module puts it all together with a start-to-finish walkthrough on a fresh dataset, a common-pitfalls checklist, and an original mock exam mapped back to every module so you can pinpoint exactly what to revise.

Learning objectives

By the end of this module, you will be able to:

  • Perform a complete descriptive-analysis walkthrough on a new dataset: variable typing, frequency tables and charts, five-number summary and boxplot, centre, spread, shape, contingency table, scatter plot, correlation, regression.
  • Identify and avoid the most common pitfalls in descriptive statistics.
  • Self-assess exam readiness with an original mock exam; use the worked solutions to map every question to the module that covers it.

Time plan

Block Minutes What you do
Theory 30 Full walkthrough on Lakeside Bakery data, common pitfalls
Excel lab 30 Replicate the walkthrough in Excel
Exercises 120 90-min mock exam + 30-min debrief
Total 180

Theory

Complete walkthrough — Lakeside Bakery

The lakeside-bakery.csv dataset contains two outlets (Riverside, Hilltop) × 60 days = 120 rows. Variables: outlet, day, date, day_of_week, month, weather, temperature_c, foot_traffic, customer_count, items_sold, daily_revenue_eur, waste_kg, satisfaction_rating.

Step 1 — variable typing. outlet, weather, day_of_week, month (ordinal), satisfaction_rating (ordinal) are qualitative. temperature_c, daily_revenue_eur, waste_kg are quantitative continuous. day, foot_traffic, customer_count, items_sold are quantitative discrete.

Step 2 — frequency tables and charts. Build frequency tables for weather (nominal) and satisfaction_rating (ordinal) per outlet. Use bar charts (nominal) and ordered bar charts (ordinal). Identify the modal category for each.

Step 3 — five-number summary and boxplot. For daily_revenue_eur per outlet: compute min, Q1, median, Q3, max, IQR, fences. Draw side-by-side boxplots. Check for outliers.

Step 4 — centre, spread, shape. Compute mean, median, grouped mode, variance, standard deviation, CV, skewness for daily_revenue_eur per outlet. Compare outlets.

Step 5 — contingency table. Cross weather with daily_revenue_eur class (€50 bins) per outlet. Compute conditional distributions (revenue class given weather). Look for visible dependence.

Step 6 — scatter plot, correlation, regression. Plot temperature_c vs daily_revenue_eur per outlet. Compute covariance, correlation \(r\), \(r^2\), regression line. Interpret.

Common pitfalls checklist

WarningCommon mistakes to avoid
  1. Unequal-width histogram heights — use density \(n_i/h_i\), not raw counts, when class widths differ (Module 3).
  2. \(n\) vs \(n-1\) — use sample formulas (.S) by default; population formulas (.P) only when you have the entire population (Module 6).
  3. Correlation vs causation — \(r\) measures linear association, not cause and effect (Module 9).
  4. \(r\) vs \(r^2\) — \(r\) is correlation (direction + strength); \(r^2\) is explained share (always 0 to 1) (Module 9).
  5. QUARTILE.INC vs QUARTILE.EXC — the course uses exclusive (position \((n+1)p\)); do not mix them (Module 5).
  6. Ordinal treated as ratio — satisfaction_rating 1–5 has no equal intervals; median, not mean (Module 1).
  7. Extrapolation — regression predictions outside the observed \(X\) range are unjustified (Module 9).
  8. Raw counts vs conditional % — compare rows with different totals using conditional distributions (Module 8).
  9. Grouped mean/mode as exact — they are approximations; prefer raw data when available (Modules 3–4).
  10. Mean/median/mode ordering heuristic — for near-symmetric data, the gaps are tiny; trust the Fisher-Pearson skewness using all raw data (Module 7).
  11. Fence rule vs standard deviations — 1.5×IQR is for boxplots; 2σ is for normal distributions — different tools (Module 5).
  12. Intercept as real prediction — if \(X=0\) is not meaningful, the intercept is just a mathematical anchor (Module 9).
  13. Excel locale — French Excel uses ; and localised function names; use the Excel functions page to translate (Module 1).
  14. Line chart vs scatter plot — line charts imply sequence; scatter plots show relationship between two variables (Module 8).

Worked example — Lakeside Bakery (summary)

Statistic Riverside Hilltop
\(n\) 60 60
Mean revenue (€) 205.69 216.91
Median revenue (€) 184.94 195.16
\(Q_1\) / \(Q_3\) (€, QUARTILE.EXC) 156.66 / 261.25 151.62 / 265.92
Std. dev. (€) 83.48 81.96
CV 40.6% 37.8%
Skewness (SKEW) +0.93 +0.60
Outliers (1.5×IQR) 2 high none
\(r\)(temp, revenue) −0.087 0.125
Regression slope (€ per °C) −1.78 2.38

Both outlets are right-skewed (mean above median) with high relative variability (CV near 40%). Hilltop has a slightly higher average and slightly lower CV; Riverside is more skewed and has two unusually high days. Temperature tells us almost nothing about revenue at either outlet (\(|r| < 0.13\)): look at weather categories instead (Module 8).

Using Excel

All functions from Modules 1–9. The complete list is on the Excel functions page.

Proof / derivation

Let a dataset be partitioned into \(k\) groups (e.g., franchises). Group \(j\) has \(n_j\) observations with mean \(\bar{x}_j\). The overall mean is \[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{1}{n}\sum_{j=1}^{k} \sum_{i \in \text{group } j} x_i = \frac{1}{n}\sum_{j=1}^{k} n_j \bar{x}_j = \sum_{j=1}^{k} \frac{n_j}{n} \bar{x}_j. \]

This is exactly a weighted mean of the group means, with weights \(w_j = n_j/n\) (the group’s share of total observations). For “The Wandering Fork”, the overall 240-day mean revenue is the weighted mean of the Harbour 120-day mean and the Campus 120-day mean, with equal weights (120/240 = 0.5 each): \(0.5 \times 413.49 + 0.5 \times 455.03 = 434.26\) €, which is exactly what =AVERAGE(tblFork[daily_revenue_eur]) returns.

Visual intuition

The full walkthrough on Lakeside Bakery data is the visual consolidation of the entire course: the frequency tables and bar charts (Module 2) show the qualitative shape; the boxplots (Module 5) show the five-number summary and outliers at a glance; the histogram (Module 3) shows the revenue distribution shape; the scatter plots (Module 8) show the temperature–revenue relationship; and the regression lines (Module 9) quantify that relationship. Together, these pictures tell the complete story that the numbers alone cannot.

Interactive demo

Important

TODO: an interactive full-workflow dashboard (upload CSV, auto-generate all tables/charts) would be a capstone Shinylive demo.

Exercises

The exercises for this module are the mock exam below.

Mock exam — Modules 1–9

Important

This is an originally-authored mock exam on “The Wandering Fork” and Lakeside Bakery data, mirroring the format and difficulty of the Midterm Exam (35%, 45 min) and Final Exam (65%, 90 min) described in the syllabus: a multiple-choice section followed by computational exercises. It does not reproduce any verbatim question or answer key from the source exam files.

Part A — Multiple choice (1 point each)

“The Wandering Fork”’s satisfaction_rating (1–5) is: (a) Quantitative discrete (b) Qualitative ordinal (c) Qualitative nominal (d) Quantitative continuous Answer: (b). The categories have a natural order (1 < 2 < 3 < 4 < 5) but the distances between them are not necessarily equal.

Revise: Module 1.

In any relative frequency table for a qualitative variable, the percentages sum to: (a) 100% exactly (b) Approximately 100% (rounding may cause 99.9% or 100.1%) (c) The sample size \(n\) (d) 1 Answer: (b). Rounding often makes the sum 99.9% or 100.1%; the exact algebraic sum is 1 (or 100%), but printed percentages are rounded.

Revise: Module 2.

When class widths are unequal, the bar height in a histogram must represent: (a) Absolute frequency \(n_i\) (b) Relative frequency \(f_i\) (c) Density \(n_i/h_i\) (d) Cumulative frequency \(N_i^{+}\) Answer: (c). Area = height × width must equal frequency; with unequal widths, height = density = \(n_i/h_i\).

Revise: Module 3.

“The Wandering Fork” Harbour 30-day revenue has fences at 171.52 and 1065.07. The observed minimum is 407.59 and maximum is 886.56. How many outliers? (a) 0 (b) 1 (c) 2 (d) Cannot be determined Answer: (a). Both extremes lie inside the fences.

Revise: Module 5.

For a right-skewed distribution, the typical ordering is: (a) \(\bar{x} < Me < Mo\) (b) \(\bar{x} = Me = Mo\) (c) \(\bar{x} > Me > Mo\) (d) No consistent ordering Answer: (c). The long right tail pulls the mean above the median above the mode.

Revise: Module 4.

\(CV = 21.82\%\) for Harbour 30-day daily revenue. This means: (a) The standard deviation is 21.82% of the mean (b) 21.82% of days are within one standard deviation (c) The variance is 21.82% of the mean (d) The data is 21.82% skewed Answer: (a). \(CV = s/\bar{x} \times 100\%\).

Revise: Module 6.

Three skewness formulas give different signs for the same near-symmetric data. Which one uses all raw observations and is the course standard? (a) Pearson skewness coefficient (b) Yule-Kendall skewness coefficient (c) Fisher-Pearson sample skewness (Excel’s SKEW) (d) They are all equally authoritative Answer: (c). The Fisher-Pearson coefficient uses every raw observation; the others use grouped estimates or only the five-number summary.

Revise: Module 7.

To compare revenue distributions for Sunny vs Rainy days fairly, you should use: (a) Raw counts from the contingency table (b) Column percentages (c) Row percentages (conditional distribution of revenue given weather) (d) The grand total Answer: (c). Row percentages rescale each weather type to its own total, enabling fair comparison.

Revise: Module 8.

\(r = 0.621\) for temperature vs revenue (Harbour, 30 days). This means: (a) Temperature causes revenue to increase (b) 62.1% of revenue variability is explained by temperature (c) There is a moderate positive linear relationship (d) The regression slope is 0.621 Answer: (c). \(r\) measures linear association strength and direction; \(r^2 = 0.386\) is the explained share.

Revise: Module 9.

For Harbour’s first 30 days the regression line is \(\widehat{\text{revenue}} = -22.01 + 28.71 \times \text{temperature\_c}\). Predicting revenue at \(40\) °C is:

  1. Reliable — the line is the best fit
  2. Unreliable — extrapolation beyond the data range (16.4–27.1 °C)
  3. Exact — the line passes through all points
  4. Impossible — the intercept is negative Answer: (b). Extrapolation beyond observed \(X\) values is unjustified.

Revise: Module 9.

Part B — Computational exercises (6 points each)

Using the Harbour 120 days of wandering-fork.xlsx, compute the complete descriptive summary for daily_revenue_eur: (a) Class table with classes of width €100 starting at €0. (b) Five-number summary, IQR, fences, outliers. (c) Mean, median, grouped mode. (d) Sample variance, standard deviation, CV. (e) Fisher-Pearson skewness. (f) Comment on symmetry and variability using the course conventions. Solution. (a) Classes: \([0,100[\) (9), \([100,200[\) (11), \([200,300[\) (17), \([300,400[\) (24), \([400,500[\) (21), \([500,600[\) (10), \([600,700[\) (16), \([700,800[\) (7), \([800,900[\) (3), \([900,1000[\) (0), \([1000,1100[\) (2); total 120. Modal class \([300,400[\). (b) Min=51.21, Q1=267.89, Me=391.47, Q3=567.32, Max=1050.00, IQR=299.43, fences=-181.25/1016.45, 1 high outlier (1050.00 > 1016.45). (c) Mean=413.49, Me=391.47, grouped Mo ≈ 300 + 100×(24−17)/((24−17)+(24−21)) = 300 + 100×7/10 = 370. (d) \(s^2\)=46,661.10, \(s\)=216.01, CV=52.24%. (e) Skewness = 0.47 (mild right skew). (f) Mild right skew (skewness 0.47), high relative variability (CV 52.2% > 35%), one high outlier. Mean > median (413 > 391) consistent with right skew.

Revise: Module 3 (a), Module 5 (b), Module 4 (c), Module 6 (d), Module 7 (e–f).

Using the Campus 120 days of wandering-fork.xlsx: (a) Build a contingency table: weather (Rows) × daily_revenue_eur class (Columns, €100 bins). Show conditional distributions (% of row total). (b) Create a scatter plot of customer_count vs daily_revenue_eur. (c) Compute covariance, correlation \(r\), \(r^2\), regression line. (d) Predict revenue for a 30-customer day. Is this prediction reliable? (e) Verify the regression line passes through \((\bar{x},\bar{y})\). Solution. (a) Rows: Cloudy 48, Rainy 22, Sunny 50. 27.3% of Rainy days fall below €100 against 4.2% (Cloudy) and 4.0% (Sunny); all six days of €1,100 or more are Sunny. The row profiles differ: visible dependence. (b) Strong, upward, linear cloud with few outliers. (c) Cov=7,223.49, \(r\)=0.970, \(r^2\)=0.940, slope=12.30 € per customer, intercept=−22.75. (d) \(-22.75 + 12.30\times30 = 346.25\) €. Reliable: 30 customers lies inside the observed range (3–106), and \(r^2 = 0.94\). (e) \(\bar{x}\)=38.84, \(\bar{y}\)=455.03: \(-22.75 + 12.30\times38.84 = 454.98 \approx \bar{y}\) (the small gap is rounding of the coefficients). ✓

Revise: Module 8 (a–b), Module 9 (c–e).

Open lakeside-bakery.csv. For the Riverside outlet (60 days): (a) Classify all 13 variables by type and level of measurement. (b) Build frequency tables and charts for weather and satisfaction_rating. (c) Compute the five-number summary, boxplot, and outliers for daily_revenue_eur. (d) Compute mean, median, mode, variance, std, CV, skewness for daily_revenue_eur. (e) Build a contingency table: weather × daily_revenue_eur class (€50 bins). Compute conditional distributions. (f) Scatter plot, correlation, regression for temperature_c vs daily_revenue_eur. (g) Summarise the key differences between Riverside and Hilltop in three sentences. Solution. (a) Qualitative nominal: outlet, weather, day_of_week. Qualitative ordinal: month, satisfaction_rating. Quantitative discrete: day, foot_traffic, customer_count, items_sold. Quantitative continuous: temperature_c, daily_revenue_eur, waste_kg. (b) Weather: Sunny/Cloudy/Rainy frequencies; satisfaction: 1–5 frequencies. Bar charts for both. (c) Min=66.93, Q1=156.66, Me=184.94, Q3=261.25, Max=472.24, IQR=104.58, fences=−0.21/418.12, 2 high outliers (431.34, 472.24). (d) Mean=205.69, Me=184.94, modal class [150,200[ (22 days), grouped Mo ≈ 150 + 50×13/(13+14) ≈ 174.07, \(s\)=83.48, CV=40.6%, skew=+0.93 (clear right skew; Mo < Me < mean). (e) 63.6% of Rainy days earn below €150, against 20.0% of Cloudy and 6.9% of Sunny days: revenue clearly depends on weather. (f) \(r\)=−0.087, \(r^2\)=0.008, slope=−1.78, intercept=243.51: no linear link between temperature and revenue. (g) Hilltop earns slightly more on average (216.91 vs 205.69) with slightly lower relative spread (CV 37.8% vs 40.6%). Riverside is more right-skewed and has two outlier days; Hilltop has none. At both outlets weather matters, temperature does not.

Revise: Module 1 (a), Module 2 (b), Module 5 (c), Modules 4, 6 and 7 (d), Module 8 (e), Module 9 (f).

What comes next

This course describes data as a snapshot. Two natural sequels are time series (trend, seasonality, moving averages: the 120 days of “The Wandering Fork” are a time series) and index numbers (price and volume indices, base-year comparisons). Both are covered in the next course of the programme, together with probability and the inference tools that this course deliberately leaves out.

Common mistakes

WarningWatch out
  • Unequal-width histogram heights — use density \(n_i/h_i\), not raw counts, when class widths differ (Module 3).
  • \(n\) vs \(n-1\) — use sample formulas (.S) by default; population formulas (.P) only when you have the entire population (Module 6).
  • Correlation vs causation — \(r\) measures linear association, not cause and effect (Module 9).
  • \(r\) vs \(r^2\) — \(r\) is correlation (direction + strength); \(r^2\) is explained share (always 0 to 1) (Module 9).
  • QUARTILE.INC vs QUARTILE.EXC — the course uses exclusive (position \((n+1)p\)); do not mix them (Module 5).
  • Ordinal treated as ratio — satisfaction_rating 1–5 has no equal intervals; median, not mean (Module 1).
  • Extrapolation — regression predictions outside the observed \(X\) range are unjustified (Module 9).
  • Raw counts vs conditional % — compare rows with different totals using conditional distributions (Module 8).
  • Grouped mean/mode as exact — they are approximations; prefer raw data when available (Modules 3–4).
  • Mean/median/mode ordering heuristic — for near-symmetric data, the gaps are tiny; trust the Fisher-Pearson skewness using all raw data (Module 7).
  • Fence rule vs standard deviations — 1.5×IQR is for boxplots; 2σ is for normal distributions — different tools (Module 5).
  • Intercept as real prediction — if \(X=0\) is not meaningful, the intercept is just a mathematical anchor (Module 9).
  • Excel locale — French Excel uses ; and localised function names; use the Excel functions page to translate (Module 1).
  • Line chart vs scatter plot — line charts imply sequence; scatter plots show relationship between two variables (Module 8).

Further reading

Source Where What it adds Time Verified
OpenStax IBS 2e Chapter 2 Review (from book page 90: summary, practice, homework) Comprehensive review of all univariate topics ~10 pages checked against the PDF contents (T046)
OpenStax IBS 2e Chapter 13 Review (from book page 570: summary, practice, homework) Comprehensive review of bivariate topics ~7 pages checked against the PDF contents (T046)
Khan Academy Unit Exploring bivariate numerical data — practice Full practice set on correlation and regression ~30 min (estimate) link check pending (T048)
Khan Academy Unit Summarizing quantitative data — practice Full practice set on univariate summaries ~30 min (estimate) link check pending (T048)

OpenStax IBS 2e: Alexander Holmes, Barbara Illowsky and Susan Dean, Introductory Business Statistics 2e, OpenStax, Rice University, https://openstax.org/details/books/introductory-business-statistics-2e, licensed under CC BY-NC-SA 4.0. Sections are cited, not reproduced.