Marta presents the satisfaction survey results to the executive committee next month, and she puts you in charge of the materials: "Nobody in that room is going to read a table of quartiles. I want charts that can be understood in five seconds and that don't mislead anyone." This lesson closes the module by teaching you exactly that: how to organize data into frequency tables (the skeleton of every chart), which chart suits each type of variable, how to read the shape of a distribution on a histogram, and how to recognize — and never commit — the tricks of misleading charts. A good chart is the bridge between your analysis and the decision; a bad one can trigger the wrong decision with perfectly correct data.

1. Frequency tables: absolute, relative and cumulative
2. Grouping into intervals
3. Which chart for which variable
4. Charts for categorical variables: bars and pies
5. Charts for quantitative variables: histogram and frequency polygon
6. The shape of the distribution
7. Comparing and relating: comparative box plots, lines and scatter
8. Good practice and misleading charts

Before the chart comes the frequency table: the organized count of how many times each value appears. Three columns make it up:

• Absolute frequency (\( n_i \)): the number of observations with that value.
• Relative frequency (\( f_i = n_i / n \)): the proportion or percentage of the total. Essential for comparing groups of different sizes.
• Cumulative frequencies (\( N_i \), \( F_i \)): the sum of the frequencies up to and including that value. They only make sense if the values can be ordered (ordinal or quantitative variables).

You finally process the full survey. The distribution of the satisfaction score (0–10):

(The percentages are rounded to one decimal place, which is why some partial sums may be a tenth off.)

This table already answers business questions without any chart:

• Mode: 9 (170 responses, 26.6%).
• Median: the cumulative column crosses 50% at score 8 (\( F_8 = 52.2,% \)) → median 8. Consistent with what you have learned: the mean (8.1) sits below the mode and very close to the median, with the tail of low scores pulling it down.
• Cumulative reading: only 7.5% score 5 or lower; 71.2% (100 − 28.8) score 8 or higher. In satisfaction management, these cumulative figures ("% detractors", "% promoters") are often the indicator that gets tracked, more than the mean.

With continuous variables (amounts, times) hardly any value repeats and the previous kind of table breaks down: you must group into intervals (or classes).

The notation [15 – 30) means "from 15 inclusive up to but not including 30": that way every receipt falls into exactly one interval, with no ambiguity at the boundaries.

Practical grouping criteria:

• Number of intervals: between 5 and 15 usually works; a common reference is Sturges' rule, \( k \approx 1 + 3.3 \cdot \log_{10}(n) \) (with \( n = 200 \), \( k \approx 8.6 \) → about 7–9 intervals). Too few intervals hide the shape; too many shatter it into noise.
• Constant width whenever possible (here, €15): intervals of unequal width complicate the reading and are a classic source of misleading charts.
• "Round" limits (0, 15, 30…) that the reader can process effortlessly.
• Grouping loses detail (that is the price of clarity): compute the exact measures (mean, quartiles) on the original data, not on the grouped table, if you have them.

The first charting decision is not aesthetic: it depends on the type of variable (revisit Types of Data) and on the question you want to answer.

As a quick decision rule:

One bar per category; the height is the frequency (absolute or relative). It is the safest chart in existence: the human eye compares lengths with great precision.

Example: share of sales by product category at NovaMarket, a bar chart built on this table:

Good practice: bars starting from zero, always (the length is the data); nominal categories sorted from most to least frequent (ordinal ones in their natural order: never reorder the 0–10 scores by frequency); bars separated by gaps (the gap visually distinguishes bars from a histogram, whose rectangles touch).

The classic "pie": each category is a slice proportional to its frequency. It only works when three conditions hold: the parts add up to a whole (100%), there are few categories (4–6 at most) and the differences are large. The split of the 640 responses by channel — in-store 70% (448), online 20% (128), app 10% (64) — is a good candidate. The share of 15 product categories is not: the eye compares angles far worse than lengths. When in doubt, use bars; never use pies to compare values that are close to one another.

The histogram is the chart of the interval table: one rectangle per interval, drawn touching each other (the variable is continuous, there are no gaps), whose area represents the frequency. With intervals of constant width, the height is directly the frequency; if you ever use unequal widths, you must plot density (frequency ÷ width) or you will distort the picture.

On the histogram of the 200 Valencia-Ruzafa receipts (previous table) you would see: a rapid climb up to the [15 – 30) interval, which is the modal interval (64 receipts), and a gradual descent with a long tail to the right, down to the handful of €90–105 receipts. One glance and you already know the business: many small and medium purchases, few large ones.

The frequency polygon connects with straight lines the midpoints of the top of each histogram bar (for the receipts: the points (7.5, 28), (22.5, 64), (37.5, 52)…). It conveys the same as the histogram, but being a line it lets you overlay several distributions without them hiding each other: for instance, the distribution of Saturday receipts against Tuesday receipts on the same grid.

You do not just look at a histogram: you read it. Three questions guide the reading:

1. Is it symmetric or skewed?

Here every piece of the module clicks into place: satisfaction has mean 8.1, median 8 and mode 9 — the mean below, dragged down by the left tail of unhappy customers — while receipts have their mean above the median because of their right tail of large purchases. The skewness you used to diagnose by comparing mean and median in Measures of Central Tendency, you can now see.

2. How many peaks does it have? A unimodal distribution has a single peak; a bimodal one, two. Bimodality almost always betrays two mixed populations: the histogram of customer footfall by hour at Madrid-Centro has two clear peaks (12:00–14:00 and 18:00–20:00) because it mixes two kinds of visit — the midday shop and the after-work shop. Faced with a bimodal histogram, the right question is not "what is the mean?" (it would land in the valley, where there is hardly any data) but "which two groups am I mixing that I should separate?".

3. Are there isolated values? Stray bars far from the main body: the outlier candidates of the previous lesson, now visible.

To compare a continuous variable across groups, the histogram falls short (overlaying more than two is unreadable). The tool is the comparative box plot: the boxes of all the groups lined up on the same axis. For example, the e-commerce delivery times by quarter:

With the three boxes drawn side by side, the story reads itself: in the third quarter the median drops (faster deliveries), the box narrows (more consistency) and there is still the odd outlier to investigate. One chart, three distributions compared on center, dispersion, skewness and outliers: that is why the comparative box plot is one of analysts' favorite charts.

When the horizontal axis is time (days, months, quarters), the line chart shows the evolution: the e-commerce monthly sales, the mean satisfaction per survey wave. Consecutive points joined by segments; the eye picks up trends, seasonality (NovaMarket's December peak) and breaks. Two rules: the time axis in chronological order with uniform spacing, and no connecting of points across data gaps. The in-depth analysis of trend and seasonality has its own lesson on Time Series; here it is enough to know how to present the evolution.

The scatter plot shows two quantitative variables at once: each observation is a point with coordinates (x, y). Example: sales-floor area against average daily sales for eight stores — Cuenca (850 m², €19,200), Bilbao-Casco (1,100 m², €44,000), Zaragoza-Centro (1,150 m², €44,000), Valencia-Ruzafa (1,200 m², €44,100), Barcelona-Gràcia (1,250 m², €46,100), Madrid-Chamberí (1,300 m², €47,500), Sevilla-Nervión (1,400 m², €52,300), Madrid-Centro (2,000 m², €68,400). Once the cloud is drawn, a clear pattern emerges: more floor area, more sales. Quantifying the strength of that relationship (and its pitfalls) is the subject of Correlation Analysis; for now, keep the chart as a presentation and first-exploration tool.

A chart can lie without containing a single false number. The most frequent tricks — which you must learn to spot as a reader and avoid as an author:

1. The truncated axis. The classic. Sevilla-Nervión sold €44,100 yesterday and Zaragoza-Centro €43,900: a 0.5% difference. On a bar chart whose axis starts at €43,800, Sevilla's bar looks three times taller; with the axis starting at 0, the two bars are visually identical (which is the truth). Rule: bars always start at zero, because their length encodes the value. On a line chart a clipped axis is acceptable to reveal the detail of the variation, but it must be clearly flagged.

2. Areas and pictograms. To show that online sales have doubled, someone draws a shopping cart twice as tall… but by scaling height and width together, the area quadruples and the brain reads "×4". With 3D volumes, ×8.

3. Decorative 3D. Pies and bars in 3D with perspective distort systematically: the slices at the front look bigger. 3D adds no information; only error.

4. The conveniently chosen time window. "Sales have been growing since March" can be true and at the same time hide a 12% year-on-year decline. Picking the time range that favors the desired conclusion is cherry-picking. Show enough context and compare like-for-like periods.

5. Unequal interval widths without adjustment in a histogram: an interval twice as wide accumulates more frequency and looks like an artificial peak (which is why the height must be density in that case).

6. Overload. Twelve colors, a double vertical axis, rotated labels, endless decimals. Every element that adds no information steals attention from the ones that do.

The good-chart checklist, before sending it to Marta:

• A title that states the message ("Q3 shortens and stabilizes deliveries"), not just the content ("Delivery times").
• Labeled axes with units; bars from zero; data source and \( n \) visible.
• The right chart type for the variable type (the table in section 3).
• A meaningful order (categories by frequency, ordinals on their own scale, time in chronological order).
• Can it be understood in five seconds without reading the fine print? If not, simplify.

• Choosing the chart for its looks rather than the variable type. A pie for 12 categories or bars for a continuous variable confuse people, however "pretty" they look.
• Truncating the axis of a bar chart to magnify tiny differences. It is the most common mistake (or trick) in business reports.
• Confusing a bar chart with a histogram. Separated bars for categories; touching rectangles for intervals of a continuous variable. It is not a typographic whim: it communicates whether the variable is discrete/categorical or continuous.
• Ignoring the shape. Reporting only the mean of a bimodal or heavily skewed distribution is describing a customer who does not exist; you look at the histogram before choosing the numerical summaries.
• Using absolute frequencies to compare groups of different sizes. 150 complaints from the online channel versus 90 from the app look worse for online… until you express them as a percentage of each channel's orders. Compare with relative frequencies.
• Tip: for every chart you publish, first write the message it must convey in one sentence; if the finished chart does not shout that sentence, change it.

From 20 receipts at the Madrid-Chamberí store you note the payment method: card, card, cash, mobile, card, cash, card, card, mobile, card, cash, card, card, mobile, cash, card, card, cash, card, mobile.

1. Build the table of absolute and relative frequencies. Do the cumulative frequencies make sense?
2. Which chart(s) would you use, and in what order would you place the categories?

Delivery times of 400 e-commerce orders, grouped:

1. Compute the relative frequencies and the cumulative relative frequencies.
2. Describe the shape the histogram would have (peaks, symmetry). What relationship do you expect between mean and median?
3. Logistics wants to promise "delivery in under 36 h". What percentage of orders meets that promise today?

A software vendor shows Marta this chart: 3D bars comparing the mean satisfaction of customers "with our app" (8.3) and "without it" (8.0), with the vertical axis running from 7.9 to 8.4, and no indication of how many customers are in each group. Identify at least three problems and propose how it should be redone.

Exercise 4.1:

1. The count: card 12 (60%), cash 5 (25%), mobile 4 (20%)… careful! 12 + 5 + 4 = 21 > 20: recount. Correct: card 11 (55%), cash 5 (25%), mobile 4 (20%); total 20 and 100%. (This stumble is deliberate: always check that the frequencies add up to \( n \); it is the most basic quality control for a table.) The cumulative frequencies make no sense: payment method is nominal — there is no order to accumulate along.
2. A bar chart (or a pie, acceptable here: 3 categories that make up the whole), with the categories sorted from most to least frequent: card, cash, mobile. Never a histogram: this is not a continuous variable.

Exercise 4.2:

1. Relative (%): 9, 32, 31, 16, 7, 3, 2. Cumulative (%): 9, 41, 72, 88, 95, 98, 100.
2. Unimodal, with the peak at [18 – 24) — closely followed by [24 – 30) — and right-skewed: the tail stretches out towards the slow deliveries (up to 54 h) while the reach on the left is short. Consequently, you should expect mean > median. Common mistake: calling it "left-skewed" because the peak is on the left; skewness is named after the side of the tail, not the peak.
3. Orders under 36 h qualify: the cumulative frequency through [30 – 36) is 88%. If the commercial commitment demands 90%, it is not being met today (though only just): actionable information for logistics.

Exercise 4.3:

Problems: (1) a truncated axis (7.9–8.4): it magnifies a 0.3-point difference until it looks enormous — with bars, the axis must start at 0; (2) decorative 3D that distorts the comparison; (3) missing group sizes (\( n \)): without them you cannot tell whether the 8.3 comes from 30 customers or from 3,000; (4) moreover, the "with app / without app" groups may differ in everything else (customers who install the app are probably the most loyal ones), so the difference does not prove the app's effect — distinguishing association from causation will come back in Module 6. Redo it as: 2D bars from 0 (where 8.3 and 8.0 will be seen to be nearly equal), stating each group's \( n \) and, better still, showing each group's full distribution of scores, not just the means.

With this lesson you close Module 2 and your descriptive toolbox is complete: frequency tables as the skeleton, the right chart for each variable type (bars and pies for categories; histogram, polygon and box plot for continuous variables; lines for time; scatter for pairs of variables), the reading of shape — symmetry, tails, number of peaks — that gives meaning to the measures of the previous lessons, and the critical eye to spot truncated axes, inflated areas and the other visual traps. Marta has her presentation: satisfaction with a median of 8 and a left tail of unhappy customers, receipts with a right tail, deliveries improving quarter by quarter.

But in the committee someone will ask the inevitable question: "That 8.1 comes from 640 responses out of 380,000 members… how much can we trust it?" To answer, describing is not enough: you have to measure uncertainty, and the language of uncertainty is probability. That is what Module 3 is about, starting with the Basic Concepts of Probability: random experiments, events and the different ways of attaching a number to the phrase "it is likely that".