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Week 2 · Chapter

Chapter 2 — Summarizing Data with Tables & Graphs

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
The module's primary reading · ~25–35 minutes · Objective 2 (the tables-and-graphs half) · Read this before (or alongside) the week's lecture — everything else in the module builds on it.

Why you already care about this

Here is a small experiment you can run on yourself. Look at a list of twenty-five raw numbers — say, the finishing times from a neighborhood race — and try to say one true thing about them. Most people can't. Now look at the same twenty-five numbers as a histogram, and suddenly you can say four true things in four seconds: where the values cluster, how far they spread, which side trails off, and whether anything sits suspiciously alone. Nothing about the data changed. The display changed, and the display is what your brain can actually read.

That power cuts both ways. A picture communicates faster than a paragraph, which means a dishonest picture deceives faster than a dishonest paragraph. The same bar chart that summarizes a month of prices can be quietly redrawn — same numbers, different axis — to triple a difference in your eye. Somebody, somewhere, is counting on you not to check.

By the end of this chapter you will be able to: build a frequency table and a relative-frequency table for any variable; choose and read the right display — bar chart, pie chart, histogram, dot plot, or stem plot; describe a distribution's shape in the standard vocabulary (symmetric, skewed, uniform, bimodal) and flag informal outliers; and name the classic tricks behind a misleading graph, with the fix for each. The one-line version of the whole week: every dataset has a shape — your job is to see it, and to notice when someone's hiding it.

Where we are

Week 1 was about where data come from: populations and samples, how the sample was picked, and what was actually recorded (the variables and their NOIR levels). This week we take a trustworthy column of data and do the first real statistical act: summarize it — first in a table, then in a picture. One Week-1 idea does heavy lifting here: knowing whether a variable is categorical or quantitative, because that single classification decides which tables and graphs are even possible.

1. Frequency tables: counting and sharing

Start with the plainest tool in statistics: counting.

  • The frequency of a category (or class) is simply how many data values fall in it — a count.
  • The relative frequency is that count's share of the whole: count ÷ total. It answers "what fraction?" instead of "how many?"
  • A frequency table lists each category with its frequency; add a relative-frequency column and you can compare datasets of different sizes on equal footing. (A 40-song playlist and a 4,000-song library can both be "35% pop.")

Relative frequencies come with a built-in error check, and you should use it every time: the shares must sum to 1 — as percents, to 100%. A column that totals 1.10 is confessing that something was double-counted; one that totals 0.90 says something was missed. (A sum of 0.99 or 1.01 is usually just rounding — note it and move on.)

Worked example. A student's workout playlist holds 40 songs. Tallying by genre: pop 14, hip-hop 10, rock 8, country 5, other 3.

First check the counts: 14 + 10 + 8 + 5 + 3 = 40 ✓. Then convert each count to a share:

  • pop: 14 ÷ 40 = 0.35
  • hip-hop: 10 ÷ 40 = 0.25
  • rock: 8 ÷ 40 = 0.20
  • country: 5 ÷ 40 = 0.125
  • other: 3 ÷ 40 = 0.075

Check the shares: 0.35 + 0.25 + 0.20 + 0.125 + 0.075 = 1.000 ✓ — as percents, 35% + 25% + 20% + 12.5% + 7.5% = 100% ✓. Now read one line both ways, because both sentences matter: "14 songs are pop" (the frequency) and "35% of the playlist is pop" (the relative frequency). A memory hook to keep: frequency counts it; relative frequency shares it.

Categorical data arrive pre-grouped — every song already has a genre. Quantitative data need one extra move: numbers don't come in groups, so we cut the number line into classes (your spreadsheet will call them bins or buckets) of equal width, then count values into the classes. One convention keeps everything tidy: each class includes its left edge and excludes its right edge. In classes labeled 20–<25 and 25–<30, a value of exactly 25 belongs to 25–<30 — left edge in, right edge out — so nothing is ever counted twice. You'll see this in action in Section 3.

In practice (your tools). Nobody tallies by hand for long. In Google Sheets or Excel, with categories in column B, =COUNTIF(B2:B41,"pop") returns the frequency instantly; for a quantitative class like 25–<30, =COUNTIFS(A2:A26,">=25",A2:A26,"<30") does the counting with the boundary convention built in. Divide each count by the total for the relative frequencies, and =SUM(...) over the share column is your sums-to-1 check.

2. Pictures for categories: bar charts and pie charts

Once a categorical variable is tabled, two displays draw it.

A bar chart gives each category one bar, and the bar's length shows its frequency (or relative frequency). Two features are load-bearing:

  • The bars are separated. The gaps are a message: "these are separate labels, not points on a number line." You may order the bars however communicates best — alphabetically, or biggest-first.
  • The count axis starts at zero. The bar's entire length is the data; a bar that starts its life at 90 instead of 0 is telling a partial truth (Section 5 is about exactly this move).

A pie chart shows how one whole splits into parts: the circle is the total, and each slice's share of the circle is that category's relative frequency. The playlist works perfectly as a pie: every song belongs to exactly one genre, and the slices — 35%, 25%, 20%, 12.5%, 7.5% — total 100%.

But the pie has fine print, and most real-world pie abuse comes from ignoring it. A pie chart is only legitimate when the categories are non-overlapping parts of one whole:

  • If people can belong to several categories at once (club memberships, streaming subscriptions, badges earned), the counts overlap and don't form a whole — no pie. Use bars.
  • If the slice percents don't total 100, something was double-counted or left out — the chart is broken, not just ugly.
  • And practically: beyond a handful of categories, skinny slices become unreadable. Bars stay readable at any count.

The rule worth posting over your desk: pie = parts of one whole; anything else = bars.

Misconception check. "A histogram is just a bar chart with a different name." No — and the difference is the whole point of the next section. A bar chart displays categories: separated bars, reorder them freely. A histogram displays a number line: touching bars in a fixed order. You can sort a bar chart tallest-first; sorting a histogram would scramble the number line itself. The giveaway is the gaps: bars apart = categories; bars touching = a number line.

3. Pictures for numbers: dot plots, histograms, and stem plots

Quantitative variables live on a number line, and their displays should too. Three tools, from smallest data to biggest.

The dot plot is the display you can draw in thirty seconds: one dot per data value, stacked above a number line. Nothing is hidden — every individual value stays visible — which makes it ideal for small datasets.

Worked example. Ask 15 people how many books they read last year: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 14. The dot plot stacks one dot at 2, two at 3, three at 4, four at 5, two at 6, one each at 7 and 8 — and one dot alone at 14, past a visible gap. (Count the dots: 1 + 2 + 3 + 4 + 2 + 1 + 1 + 1 = 15 ✓.) One glance now reports: values cluster from 2 to 8, the tallest stack sits at 5, and the 14 sits apart from everything — our first outlier sighting, formally handled in Section 4.

The histogram is the workhorse for anything bigger. Cut the number line into equal-width classes, count each class (that's Section 1's table), and draw one touching bar per class. The touching is meaningful: bars share edges because their classes share edges on the number line.

Worked example. Twenty-five finishing times, in minutes, from a neighborhood 5K race:

21, 23, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 41, 43.

Using classes five minutes wide, count each class (remember: left edge in, right edge out):

The table below is the frequency and relative-frequency table for the 25 finishing times.

Class (minutes) Frequency Relative frequency
20–<25 3 3 ÷ 25 = 0.12
25–<30 8 8 ÷ 25 = 0.32
30–<35 7 7 ÷ 25 = 0.28
35–<40 5 5 ÷ 25 = 0.20
40–<45 2 2 ÷ 25 = 0.08

Checks: 3 + 8 + 7 + 5 + 2 = 25 ✓ and 0.12 + 0.32 + 0.28 + 0.20 + 0.08 = 1.00 ✓. The histogram is five touching bars with heights 3, 8, 7, 5, 2 — and one glance now says what twenty-five raw numbers couldn't: a single peak in the upper 20s, thinning toward the slow side.

How many classes should you use? Rule of thumb: 5 to 15 equal-width classes, and let your tool help you experiment. Too few classes mash the shape into a blob; too many shatter it into splinters.

The stem plot (stem-and-leaf plot) is the histogram that keeps its digits. Split each value into a stem (leading digits) and a leaf (final digit); list each stem once, then hang its leaves, sorted. The same 25 race times:

2 | 1 3 4 5 6 6 7 8 8 9 9
3 | 0 1 1 2 3 4 4 5 6 7 8 9
4 | 1 3

Here the stem is the tens digit, so 3 | 0 1 1 … reads as 30, 31, 31, … Count the leaves: 11 + 12 + 2 = 25 ✓. Turn the plot sideways and it is a histogram — but unlike a histogram, every original value is still readable. (Three stems is admittedly coarse; a split-stem version — each stem listed twice, leaves 0–4 on the first line and 5–9 on the second — would reproduce our five-class histogram exactly.)

Choosing among the three: dot plot for small sets where every value matters; histogram for anything big; stem plot when you want the picture and the raw digits at once.

In practice (your tools). In Google Sheets: select the data column → Insert ▸ Chart → choose Histogram as the chart type → Customize ▸ Histogram ▸ Bucket size to experiment with class widths. In Excel: select the column → Insert ▸ Statistic Chart ▸ Histogram, then right-click the axis → Format Axis to set the bin width. Try a few widths: an honest shape survives reasonable bucket choices. You'll do exactly this to real penguin data in Data Lab 2.

4. Shape: reading what you drew

Once data sit on a number line, the graph has a shape, and shape is information. The standard vocabulary:

  • Symmetric — the left and right halves are rough mirror images. The famous special case is the bell shape: values pile up evenly around a middle, like the weights of same-variety apples in an orchard bin.
  • Skewed right — the long, thin tail stretches toward the large values. Classic example: household incomes — most values bunch low-to-middle, while a few enormous ones stretch the tail rightward.
  • Skewed left — the tail stretches toward the small values. Classic example: retirement ages — most cluster in the 60s, with a trickle of early retirees trailing down into the 40s.
  • Uniform — flat; every value about equally common, like the last digit of phone numbers.
  • Bimodal — two distinct peaks. Two peaks usually mean two groups hiding in one column — a restaurant's customer count by hour peaks at lunch and again at dinner.

Misconception check. "The tall bars are on the left, so it's skewed left." This is the single most common shape error, and it has a one-line cure: skew is named for the tail, not the peak. Our 5K histogram (heights 3, 8, 7, 5, 2) peaks early and trails off toward the high times — the tail points right, so the distribution is skewed right, even though the tall bars crowd the left of the picture. Say the hook twice: the tail tells the tale.

Informal outliers. An outlier is a value that sits far from the bulk of the data, usually past a visible gap — like the 14-book reader in Section 3's dot plot, whose nearest neighbor is 8. This week, the eye test is enough (Week 3 adds an arithmetic rule — a fence built at 1.5 × IQR). What matters now is what you do with one:

  • A real but unusual value (a genuine bookworm) stays in the data — and gets mentioned in your summary.
  • A data-entry error (a typo of 14 for 4) gets fixed — and the fix gets documented.
  • What is never allowed: silently deleting a value because it spoils the picture. Investigate, don't delete.

Finally, the reading checklist for any distribution — this week by eye, next week with numbers: shape (which of the five words fits?), center (roughly where does the middle sit?), spread (how far do values run?), and surprises (outliers, gaps, second peaks).

5. Misleading graphs: the same data, told slant

Week 1's villain was the biased sample — data broken before anyone drew anything. This week's villain draws a chart that makes honest data lie. The defense is knowing the tricks, and the first one is worth computing in full.

Worked example — the truncated axis. An ad compares two internet providers' average download speeds: Provider K at 95 Mbps, Provider L at 100 Mbps. The ad's bar chart starts its vertical axis at 90 instead of 0.

  • As drawn, K's bar rises 95 − 90 = 5 units above the axis floor; L's rises 100 − 90 = 10 units.
  • Visual impression: 10 ÷ 5 = 2.0 — L's bar is twice as tall. "Twice the speed!"
  • The honest arithmetic: 100 ÷ 95 ≈ 1.05 — L is about 5% faster.

One axis choice inflated a 5% edge into a visual 100%. Redraw the same two bars from zero and they look nearly identical — which is the truthful picture. The principle: a bar's length encodes its value, so bar charts start the axis at zero, always.

The full catalog of classic tricks — you will find all four in the wild within a month:

  1. Truncated axis — bars start above zero, so small gaps look giant (the example above).
  2. Area tricks — a pictogram doubles a value by doubling both dimensions of a picture, so the ink quadruples (2 × 2 = 4). Your eye reads area; the data was only ever height.
  3. Missing labels — no axis numbers, no total, no sample size. A graph you cannot check isn't evidence; it's decoration.
  4. Cherry-picked classes — bucket widths tuned until a dip disappears or a peak appears. (An honest shape survives reasonable bucket choices — which is why you experimented with bucket sizes in Section 3.)

One honest nuance so the rule doesn't overreach: the start-at-zero law is about bar charts, where length is the data. A line chart tracking small-but-real changes may legitimately zoom its axis — with the axis loudly labeled so nobody mistakes the magnification. When zooming is honest and when it's a con is genuinely debatable — which is why it's this week's discussion topic. The pocket rule: bars start at zero; if you zoom, label it loudly.

In practice (your tools). Every charting tool will happily draw the dishonest version — Sheets and Excel let you set an axis minimum in two clicks (in Sheets: Customize ▸ Vertical axis ▸ Min). The software never protects you; the habit of checking the axis floor does. In this week's lab, you'll build the lying version of a real chart on purpose, then fix it — the fastest known vaccine.

Key terms

  • Frequency — the count of data values in a category or class.
  • Relative frequency — a category's share of the whole: count ÷ total; all shares sum to 1.
  • Frequency table — a table listing each category (or class) with its frequency (and often its relative frequency).
  • Class (bin, bucket) — an equal-width interval of the number line used to group quantitative data; includes its left edge, excludes its right.
  • Bar chart — separated bars for categories; bar length = count (axis starts at zero).
  • Pie chart — slices of one whole; only valid for non-overlapping parts summing to 100%.
  • Histogram — touching bars over classes on a number line; the standard shape display for quantitative data.
  • Dot plot — one dot per value stacked on a number line; keeps every value visible.
  • Stem plot (stem-and-leaf) — leading digits as stems, final digits as leaves; a histogram that keeps its digits.
  • Distribution — the overall pattern of a variable's values: what values occur and how often.
  • Shape — the distribution's form: symmetric, skewed right/left, uniform, or bimodal.
  • Skewed right / skewed left — the long, thin tail stretches toward large / small values (skew is named for the tail).
  • Outlier (informal) — a value far from the bulk of the data, usually past a visible gap; investigate, don't delete.
  • Misleading graph — a display whose design (truncated axis, area tricks, missing labels, cherry-picked classes) distorts honest numbers.

Summary

  • Count, then share: a frequency table counts each category or class; relative frequencies (count ÷ total) make shares comparable and must sum to 1 — your built-in error check.
  • Match the display to the variable: categories → bar chart (or pie, only for parts of one whole totaling 100%); quantitative → dot plot (small), histogram (big), stem plot (picture + digits). Bars apart = categories; bars touching = a number line.
  • Read shape with the standard words — symmetric, skewed right, skewed left, uniform, bimodal — then note center, spread, and surprises. Skew is named for the tail: the tail tells the tale.
  • Outliers are flags, not garbage: real values get reported, errors get fixed, nothing gets silently deleted.
  • Graphs can lie about honest data: truncated axes, area tricks, missing labels, cherry-picked buckets. Bars start at zero; if you zoom, label it loudly.

Check your understanding

Work these before looking at the answers — they're ungraded, and the answers (with reasoning) are at the bottom of the page.

  1. A food hall logs 40 lunch orders by cuisine: tacos 16, noodle bowls 10, sandwiches 8, salads 6. Build the relative-frequency column, verify it passes the sums-to-1 check, and say in one sentence what the tacos number means in "share" language.
  2. Choose the best display for each, with a one-line reason: (a) how a nonprofit's total annual donations split across its five programs; (b) the floor areas (in square feet) of 300 homes listed in one region.
  3. A plant nursery measures the heights of its tree saplings. Most measure 30–40 cm, while a handful of stunted saplings measure near 10 cm. Which shape word fits this distribution, and why — argue from the tail, not the peak.
  4. A juice ad's bar chart compares vitamin C per serving: its own brand at 110 mg versus a rival at 100 mg, with the vertical axis starting at 95. Compute what the chart makes your eye see versus what the numbers say, name the trick, and state the fix.

Answers (worked reasoning — read after attempting)

  1. Shares: tacos 16 ÷ 40 = 0.40; noodle bowls 10 ÷ 40 = 0.25; sandwiches 8 ÷ 40 = 0.20; salads 6 ÷ 40 = 0.15. Check: 0.40 + 0.25 + 0.20 + 0.15 = 1.00 ✓. In share language: "40% of the lunch orders were tacos" — the count 16 says how many; the relative frequency 0.40 says what fraction of all orders.
  2. (a) Pie chart (a bar chart also works): the five programs are non-overlapping parts of one whole — total donations — so slices summing to 100% are legitimate here. (b) Histogram: 300 values of a quantitative variable need classes on a number line; with n = 300, a dot plot would be an unreadable dot blizzard, and a bar chart has no number line at all.
  3. Skewed left. The bulk of the heights sits at 30–40 cm and the long, thin tail stretches toward the small values (the stunted saplings near 10 cm). The peak's position is irrelevant to the name — skew follows the tail, and the tail points left. (If you answered "skewed right because most trees are on the right," that's precisely the misconception: the tail tells the tale.)
  4. As drawn, the brand's bar rises 110 − 95 = 15 units and the rival's rises 100 − 95 = 5 units, so the eye sees 15 ÷ 5 = 3.0 — "three times the vitamin C." The honest arithmetic: 110 ÷ 100 = 1.10, just 10% more. The trick is a truncated axis (the bars start at 95, not 0). The fix: redraw the bar chart with the axis starting at zero — the two bars then look, correctly, almost the same.

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page for optional deeper dives — and when you're ready to practice, Lecture Tutorial 2 turns your chatbot into your personal Week 2 tutor.