Week 2 · Chapter

Chapter 2 — Summarizing Data with Tables & Graphs

Introduction to Statistics Free sample edition

The module's primary reading · ~30–40 minutes · Objective 2 (summarize and display a single quantitative or categorical variable with appropriate tables and graphs, and describe its distribution's shape, center, and spread with the right numerical summaries) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

A list of 344 penguins, or 12 months, or the 36 ways two dice can land is hard to take in all at once. This week you learn to turn a long list into a short table and a clear picture. You will count how often each value shows up, which gives a frequency (a count of how many individuals fall in a group). You will draw the counts as bars, slices, dots, or a histogram (a bar picture of numbers sorted into equal-width ranges). Then you will learn to describe the picture's shape out loud, and to notice when a graph has been drawn to fool you.

Why you already care about this

Between 2007 and 2009, researchers in Antarctica's Palmer Archipelago logged 344 penguins of three species on three islands. Last week you sorted the variables in that field log. This week you make the log readable. Nobody can look at 344 rows and see a pattern. A table of counts and one good graph can show the pattern in seconds.

You already read pictures of data every day. A weather app shows a bar for each day's chance of rain. A store page shows a row of stars and a bar for each star rating. A news story puts a chart beside a headline. Each picture makes a claim about the numbers behind it. Most of the time the claim is fair. Sometimes the picture stretches the truth, and the numbers would tell a calmer story.

This chapter gives you the habits to read a table or a graph the way a careful analyst does. What was counted? What is being compared? Where is the pile, and where is the tail? Does the axis start at zero?

By the end of this chapter you will be able to:

count 344 penguins by species, turn the counts into shares, and draw them as bars or slices;
draw twelve month lengths three ways and pick the picture that fits the data;
describe a picture's lean, its peaks, and any value that sits far away;
read the counts and totals in a table that sorts penguins by species and island;
catch the two most common tricks that make a chart mislead.

Carry this sentence with you: name the skew by the tail — not by the pile.

Where we are

Already covered: Week 1, Foundations & Types of Data, taught you to tell a population from a sample and to sort every variable by type and level of measurement.
This week: tables and graphs for one variable, the language of shape, a first look at tables that count two categories, and the tricks that make a graph mislead.
How they connect: the type of a variable decides its picture: labels get bars or slices, and numbers get dots, stems, or bins.
Where we're going next: Week 3, Numerical Summaries: Center & Spread, puts numbers on the middle and the spread that you only describe in words this week.

Week 1 is the one this chapter leans on. Recall the two broad kinds of variable. A categorical variable is one that places each individual in a group, like species or island. A quantitative variable is one that records a number that counts or measures, like body mass or the days in a month. Recall too that n is the number of individuals in a sample and N is the number in a whole population. You will use both letters again today.

1. Tables and charts for categories: count, share, and picture

Picture a volunteer at a neighborhood food bank after a donation drive. Hundreds of items sit on the tables, and the manager wants to know what came in. The volunteer could read every label. Or the volunteer could sort the items into piles, one for canned goods, one for pasta, one for cereal, and count each pile. That second move is the first tool of the week.

A frequency is the number of individuals that fall in one category or one bin of a variable.
A frequency table is a table that lists each category or bin of a variable beside its frequency, the count of individuals in it.

We write a frequency as f. So f (a category's count) is a whole number, and the frequencies in a table add up to the total.

Counts alone can be hard to compare. Forty cans out of 50 donations is a lot; 40 out of 1,000 is not. So we often turn each count into a share of the whole.

A relative frequency is a frequency divided by the total number of individuals, written as a proportion or a percentage, so it shows each category's share of the whole.
A relative-frequency table is a table that lists each category or bin of a variable beside its relative frequency; its shares add to 100%, apart from rounding.

In symbols, for the 344 penguins below, a relative frequency is f ⁄ n: the count for one category divided by the total count.

Worked example — counting the penguin species in the field log. The field log records a species for every bird. Sort the 344 birds into three piles and count each one.

Count. There are 152 Adelie, 68 Chinstrap, and 124 Gentoo penguins.
Check the total. 152 + 68 + 124 = 344, so n (the sample size) is 344.
Divide each count by the total. Adelie: 152 ⁄ 344 ≈ 44.2%. Chinstrap: 68 ⁄ 344 ≈ 19.8%. Gentoo: 124 ⁄ 344 ≈ 36.0%. Each is rounded to one decimal at the end.
Check the shares. 44.2 + 19.8 + 36.0 = 100.0, so the shares account for every bird.

Here is the result as one table.

Species of the 344 penguins in the Palmer field log: counts and percentages
Species	Frequency (count)	Relative frequency (percent)
Adelie	152	44.2%
Chinstrap	68	19.8%
Gentoo	124	36.0%
Total	344	100.0%

In words: a little under half of the logged penguins are Adelie, about a fifth are Chinstrap, and a little over a third are Gentoo.

Honesty note: rounded shares do not always land on exactly 100.0. Three shares rounded to one decimal can add to 99.9 or 100.1. That small gap is rounding, not a mistake in the counts, and a careful writer says so.

Two pictures for categories

Once you have the 344 penguins counted, you can draw the counts.

A bar chart is a graph of a categorical variable that draws one separate bar per category, with each bar's height showing that category's frequency or relative frequency.
A pie chart is a circle cut into slices, one per category, where each slice's share of the circle equals that category's relative frequency; it fits only when the categories together make up one whole.

On a bar chart of the species, the Adelie bar is tallest at 152, the Gentoo bar comes next at 124, and the Chinstrap bar is shortest at 68. The bars stand apart with gaps between them, because the categories are separate groups with no order. You could list Gentoo first and nothing would change.

On a pie chart of the same counts, Adelie takes the largest slice. Its slice is 44.2% of the circle, a bit less than half. Chinstrap's slice is the smallest, about a fifth of the circle.

Which picture should you choose? A bar chart is better for comparing categories with each other, because the eye compares lengths well. A pie chart is better for showing parts of one whole, and only when every individual sits in exactly one slice. Each penguin has one species, so both pictures are honest here.

Misconception check. Students often reach for a pie chart whenever they see percentages. Picture a campus survey that asks, "Which streaming services do you use? Check all that apply." One student can check three boxes, so the percentages can add up to far more than 100%. Those answers are not slices of one whole, and a pie chart of them is meaningless. The cure: before you draw a pie, check that every individual lands in exactly one category and that the shares add to 100%. If not, draw a bar chart.

In practice (your tools). Load the field log into Google Sheets with the same =IMPORTDATA(...) formula you used in Week 1, so species is in column A. In an empty cell, =COUNTIF(A2:A345,"Adelie") returns 152. Change the name in quotes to count Chinstrap and Gentoo. To get a share, divide by the total: =COUNTIF(A2:A345,"Adelie")/344, then use Format, then Number, then Percent, and it shows 44.2% after you set one decimal. To draw the picture, select your small table of species and counts and use Insert, then Chart; in the Chart editor, under Setup, pick Column chart for bars or Pie chart for slices. In Excel, the same =COUNTIF formula works, and Insert, then the Column or Pie chart button, draws the chart.

Memory hook for the food bank and the penguins alike: bars compare; pies split one whole.

2. Graphs for numbers: dot plots, stem plots, and histograms

Now think about a number instead of a label. A runner logs the minutes of each morning jog for a month. A list of thirty numbers says little. But if you place each jog on a number line, you see at once where most runs land and which ones were unusual.

That picture of where the runner's 30 values fall has a name.

A distribution is the pattern of a variable's values: which values occur and how often each one occurs. The runner's jog times have one; so do the 12 month lengths.

Three graphs show the distribution of a quantitative variable, from 12 values to 342. They differ mainly in how many values they can handle.

A dot plot is a graph of a quantitative variable that places one dot above a number line for each individual, stacking the dots that share a value.
A stem plot is a display of a quantitative variable that splits each value into a stem (its leading digits) and a leaf (its last digit), then lists the leaves in order beside their stems.
A histogram is a graph of a quantitative variable that splits the number line into equal-width bins and draws touching bars whose heights show how many values fall in each bin.
A bin is one of the equal-width intervals that a histogram splits a quantitative variable into; every value is counted in exactly one bin.

Worked example — a dot plot of the month lengths. The twelve months of a common year have these lengths, in days: 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31. Because these are all twelve months, they form a whole population, and N (the population size) is 12.

Draw a number line from 28 to 31.
Place one dot per month above its length. February gives 1 dot at 28. April, June, September, and November give 4 dots at 30. The other seven months give 7 dots at 31.
Check the count. 1 + 4 + 7 = 12, which matches N.

Dot plot of the twelve month lengths, written as the number of dots above each value
Days in the month	Number of dots
28	1
29	0
30	4
31	7

In words: the dots pile up at 31, a smaller stack sits at 30, and one lonely dot sits at 28. Nothing sits at 29 in a common year.

Worked example — a stem plot of the same months. Split each length into a stem (the tens digit) and a leaf (the ones digit). The value 28 has stem 2 and leaf 8; the value 31 has stem 3 and leaf 1.

2 | 8
3 | 0 0 0 0 1 1 1 1 1 1 1
How to read it: 2 | 8 stands for 28 days

Stem 2 holds one leaf, the 8. Stem 3 holds four 0s and seven 1s. Count the leaves: 1 + 4 + 7 = 12. A stem plot is a dot plot turned on its side that keeps every original value readable.

Worked example — a histogram of penguin body mass. The field log records a body mass for 342 penguins, so a dot per bird would be a crowded mess. A histogram groups them. The lightest bird weighs 2,700 g and the heaviest weighs 6,300 g.

Choose a bin width. Take bins 500 g wide, starting at 2,500 g.
Count the bins needed. The last bin must reach past 6,300 g, so the bins run from 2,500 g up to 6,500 g. That is (6,500 − 2,500) ⁄ 500 = 8 bins.
Place the extremes. The 2,700 g bird lands in the first bin, 2,500 to under 3,000 g. The 6,300 g bird lands in the last bin, 6,000 to under 6,500 g.
Draw touching bars. Each bar's height is the number of birds in its bin. The bars touch, because body mass runs continuously along the number line.

In words: eight bars, each 500 g wide, cover every one of the 342 measured birds. Your software counts the birds in each bin for you; your job is to choose a sensible width and read the shape.

How do you choose among the three? A dot plot suits a small dataset, like twelve months. A stem plot suits a small-to-medium dataset when you want to keep every value. A histogram suits a large dataset, like 342 penguins, where only the overall shape matters.

Misconception check. Students often treat a histogram as a bar chart with the gaps closed. The two look alike, but they answer different questions. A bar chart's bars stand for separate categories, so you can reorder them and leave gaps. A histogram's bars stand for neighboring ranges of one number line, so they must stay in order, they must touch, and they must all be the same width. If you widen one bin, its bar collects more values and looks more important than it is.

Caveat: the same data can look a little different with a different bin width. Very wide bins blur the shape into one or two blocks; very narrow bins make it jagged. Try two or three widths before you describe the shape.

In practice (your tools). In Google Sheets, body mass is column F of the field log. Select F1:F345, then use Insert, then Chart. In the Chart editor, under Setup, set Chart type to Histogram chart. Under Customize, open Histogram and set Bucket size to 500. The two NA entries are text, not numbers, so they are left out of the bars. In Excel, select the same column and use Insert, then Insert Statistic Chart, then Histogram; then right-click the horizontal axis, choose Format Axis, and set Bin width to 500.

Memory hook for 12 months versus 342 penguins: a few values, use dots; many values, use bins.

3. Shape: read the tail, the peaks, and the stragglers

Think about a café that tracks when its customers walk in. If most arrive around the lunch rush, the arrival times form one big hump. If a crowd arrives before morning classes and another at lunch, the times form two humps. The picture's shape tells a story before you compute anything.

When you describe a distribution's shape, you answer three questions. Is it balanced or lopsided? How many peaks does it have? Does any value sit far from the rest?

Symmetric is the word for a distribution whose left and right sides are close to mirror images of each other around its center, like the 11 sums of two dice in the table below.
A tail is the thin end of a distribution, where the values are few and spread out, on the far side from the main pile of values.
Right-skewed is the word for a distribution whose long tail stretches to the right, toward the larger values.
Left-skewed is the word for a distribution whose long tail stretches to the left, toward the smaller values.
Unimodal is the word for a distribution with one clear peak, a single most crowded region of values.
Bimodal is the word for a distribution with two clear peaks, which often signals that two different groups have been mixed together.
An outlier is a value that falls far from the overall pattern of the rest of the data; unless a stated rule is applied, calling a value an outlier is an informal judgment.

Worked example — three shapes side by side. Here are three datasets you already know, each with a different shape.

A symmetric shape: the sums of two dice. Two fair dice can land in 36 equally likely ways, from 1 and 1 up to 6 and 6. Add the two faces in each way and count how often each sum appears.

Number of ways two fair dice can produce each sum, out of 36 equally likely outcomes
Sum of the two dice	Number of ways
2	1
3	2
4	3
5	4
6	5
7	6
8	5
9	4
10	3
11	2
12	1

Check the total: 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36. The single peak sits at 7, with 6 ways. Moving out from 7, the two sides match step for step: 6 and 8 each have 5 ways, and 2 and 12 each have 1 way. So the shape is symmetric and unimodal.

A left-skewed shape: the month lengths. Go back to the dot plot. The tallest stack is at 31 days, and the lone 28 sits by itself in the left tail. The tail points left, toward the smaller values, so the month lengths are left-skewed.

A right-skewed shape: penguin body mass. The histogram of the 342 masses shows its main pile in the middle bins and a long, thinning tail of heavy birds stretching to the right. Two numbers back up what the eye sees. The middle value of the 342 masses, with half the birds lighter and half heavier, is 4,050 g. The average is 4,201.75 g, which sits above the middle. Skew pulls the average toward the long tail, so an average above the middle points to a tail on the right. Body mass is mildly right-skewed.

In words: the dice sums balance like a tent, the months lean their tail to the left, and the penguins lean their tail to the right. A field biologist choosing a single "typical" mass has to decide which of those two numbers to report, and next week gives that choice its rules.

Misconception check. Students often name the skew by where the pile is. In the month lengths, the pile is on the right, at 31, so it is tempting to call the shape right-skewed. That reverses it. Skew is named for the tail, the long thin side, and the months' tail points left, toward February. Trace the thin side with your finger: the direction your finger points is the name of the skew.

Peaks and stragglers

The number of peaks matters as much as the lean. The dice sums have one peak, so they are unimodal. The café with a morning crowd and a lunch crowd would be bimodal. Two peaks are a clue to ask a question: are two different groups hiding in one graph? Splitting the data by group often turns one bimodal picture into two unimodal ones.

Now look for stragglers. Is February an outlier among the months? It sits alone, but it is only 2 days below the nearest value, 30. Whether that counts as "far" is a judgment call.

Limitation: without a stated rule, "outlier" is only an opinion. So say what you see, such as "one month sits apart in the left tail," and do not claim more. Next week you will learn a rule that makes the call for you.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of something you measure in your own life, such as sleep hours, commute minutes, or monthly phone data; what shape do you expect its distribution to have, and why?
(b) Think of a place where two different groups use the same space, such as a gym, a bus line, or a dining hall; which measurement there might give a bimodal picture?

4. Two-way tables: read two categories at once

A food truck owner might count orders by menu item and, at the same time, by lunch or dinner. One count per item would hide the lunch-versus-dinner story. A table with a row for each item and a column for each meal shows both stories at once.

A two-way table is a table that counts individuals by two categorical variables at once, with the categories of one variable as rows and the categories of the other as columns.
A cell is one box inside a two-way table, holding the count of individuals who belong to one row category and one column category at the same time.

The row totals give the count for each row category. The column totals give the count for each column category. Both sets of totals add up to the same grand total.

Worked example — the penguins by species and island. Every logged penguin has both a species and an island. Count them both ways at once.

Palmer penguins counted by species (rows) and island (columns), with totals
Species	Biscoe	Dream	Torgersen	Total
Adelie	44	56	52	152
Chinstrap	0	68	0	68
Gentoo	124	0	0	124
Total	168	124	52	344

Check every total.

Rows. Adelie: 44 + 56 + 52 = 152. All 68 Chinstrap are on Dream, and all 124 Gentoo are on Biscoe.
Columns. Biscoe: 44 + 124 = 168. Dream: 56 + 68 = 124. Torgersen holds 52 birds, all Adelie.
Grand total. 168 + 124 + 52 = 344, and 152 + 68 + 124 = 344. The two routes agree.

Now read the 12 cells. Adelie is the only species logged on all three islands. Chinstrap appears only on Dream, and Gentoo only on Biscoe. Torgersen holds only Adelie. In words: the table shows that species and island travel together in this field log. Because the log is an observational study, that pattern is an association, not a cause.

Misconception check. Students often grab a number before reading its labels. In this table, 124 appears twice. It is the Gentoo row total, and it is also the Dream column total. Those are two different groups that happen to share a count: 124 Gentoo birds on Biscoe, and 124 birds of two species on Dream. The cure: before you use any number from a two-way table, say its row label and its column label out loud.

In practice (your tools). In Google Sheets, species is column A and island is column B. To fill one cell, =COUNTIFS(A2:A345,"Adelie",B2:B345,"Biscoe") returns 44. To build the whole table at once, select the data and use Insert, then Pivot table. In the editor, add species under Rows, island under Columns, and species again under Values, summarized by COUNTA. In Excel, use Insert, then PivotTable, and drag the same fields into Rows, Columns, and Values.

Memory hook for any table: read the row, read the column — then the number.

5. Misleading graphs: catch the trick before it catches you

Picture a phone ad with two bars. The new model's battery bar towers over the old one. You might think the battery lasts several times longer. Then you look at the side of the chart and see the axis starts high, so a small gain is drawn as a giant one.

A misleading graph is a graph whose design, rather than its data, leads a reader to a wrong impression of the numbers.
A truncated axis is a graph axis that starts at a value other than zero, which can make small differences between bars look large.
An area distortion is a picture graph that scales an icon in both height and width, so the icon's area grows much faster than the amount it stands for.

Worked example — the shrinking February on a cropped chart. Draw a bar chart of month lengths, but start the vertical axis at 27 days instead of 0.

February's bar. Its drawn height is 28 − 27 = 1 unit.
A 31-day month's bar. Its drawn height is 31 − 27 = 4 units.
What the eye sees. February's bar is a quarter as tall, so February looks like a quarter of a month.
What is true. 28 ⁄ 31 ≈ 90.3%, so February is only 3 days shorter than a 31-day month.

In words: the picture shows 4 units versus 1 unit, a 4-fold gap, while the real gap is 31 days versus 28. A student reading only the bars would decide February is tiny. With the axis started at 0, the two bars are nearly the same height, which matches the truth.

Worked example — the doubled icon in a coffee ad. A coffee chain's ad shows its regular cup as a small icon and its new large cup as the same icon drawn twice as tall and twice as wide. The large cup holds twice as much. But the icon's area grows by 2 × 2 = 4. So the picture shows 4 times the area for an amount that only doubled. Your eye reads area, so the new cup looks four times bigger.

Here is a short checklist for any graph you meet.

Does a bar chart's axis start at zero? A bar's length is its message, so it must start at 0.
Are the axes labeled, with units?
Are a histogram's bins all the same width?
If icons stand for amounts, did only one dimension grow?

Caveat: not every axis must start at zero. A dot plot or a histogram shows where values fall, so its number line can start near the data. The zero rule is for bars, because their length carries the comparison.

Misconception check. Students often trust a chart because software drew it. The software draws exactly what it is told, and its automatic settings can pick an axis start for you that is not zero. A spreadsheet will happily draw a truncated axis. The cure is the checklist: you, not the software, are responsible for checking where the axis starts.

In practice (your tools). In Google Sheets, open the Chart editor on a column chart, go to Customize, then Vertical axis, and type 0 in the Min box. In Excel, right-click the vertical axis, choose Format Axis, and under Bounds set Minimum to 0.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Find a chart in an ad, a news story, or a social media post; run the checklist on it and say whether it passes.
(b) Imagine you had to make a small difference look big on purpose; which trick from this section would you use, and how would a careful reader catch you?

Memory hook for every bar chart: check the axis first — then trust the bars.

Key terms

frequency — The number of individuals that fall in one category or one bin of a variable. (first used: 1. Tables and charts for categories: count, share, and picture)
frequency table — A table that lists each category or bin of a variable beside its frequency, the count of individuals in it. (first used: 1. Tables and charts for categories: count, share, and picture)
relative frequency — A frequency divided by the total number of individuals, written as a proportion or a percentage, so it shows each category's share of the whole. (first used: 1. Tables and charts for categories: count, share, and picture)
relative-frequency table — A table that lists each category or bin of a variable beside its relative frequency; its shares add to 100%, apart from rounding. (first used: 1. Tables and charts for categories: count, share, and picture)
bar chart — A graph of a categorical variable that draws one separate bar per category, with each bar's height showing that category's frequency or relative frequency. (first used: 1. Tables and charts for categories: count, share, and picture)
pie chart — A circle cut into slices, one per category, where each slice's share of the circle equals that category's relative frequency; it fits only when the categories together make up one whole. (first used: 1. Tables and charts for categories: count, share, and picture)
distribution — The pattern of a variable's values: which values occur and how often each one occurs. (first used: 2. Graphs for numbers: dot plots, stem plots, and histograms)
dot plot — A graph of a quantitative variable that places one dot above a number line for each individual, stacking the dots that share a value. (first used: 2. Graphs for numbers: dot plots, stem plots, and histograms)
stem plot — A display of a quantitative variable that splits each value into a stem (its leading digits) and a leaf (its last digit), then lists the leaves in order beside their stems. (first used: 2. Graphs for numbers: dot plots, stem plots, and histograms)
histogram — A graph of a quantitative variable that splits the number line into equal-width bins and draws touching bars whose heights show how many values fall in each bin. (first used: 2. Graphs for numbers: dot plots, stem plots, and histograms)
bin — One of the equal-width intervals that a histogram splits a quantitative variable into; every value is counted in exactly one bin. (first used: 2. Graphs for numbers: dot plots, stem plots, and histograms)
symmetric — Describes a distribution whose left and right sides are close to mirror images of each other around its center. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
tail — The thin end of a distribution, where the values are few and spread out, on the far side from the main pile of values. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
right-skewed — Describes a distribution whose long tail stretches to the right, toward the larger values. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
left-skewed — Describes a distribution whose long tail stretches to the left, toward the smaller values. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
unimodal — Describes a distribution with one clear peak, a single most crowded region of values. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
bimodal — Describes a distribution with two clear peaks, which often signals that two different groups have been mixed together. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
outlier — A value that falls far from the overall pattern of the rest of the data; unless a stated rule is applied, calling a value an outlier is an informal judgment. (first used: 3. Shape: read the tail, the peaks, and the stragglers)
two-way table — A table that counts individuals by two categorical variables at once, with the categories of one variable as rows and the categories of the other as columns. (first used: 4. Two-way tables: read two categories at once)
cell — One box inside a two-way table, holding the count of individuals who belong to one row category and one column category at the same time. (first used: 4. Two-way tables: read two categories at once)
misleading graph — A graph whose design, rather than its data, leads a reader to a wrong impression of the numbers. (first used: 5. Misleading graphs: catch the trick before it catches you)
truncated axis — A graph axis that starts at a value other than zero, which can make small differences between bars look large. (first used: 5. Misleading graphs: catch the trick before it catches you)
area distortion — A picture graph that scales an icon in both height and width, so the icon's area grows much faster than the amount it stands for. (first used: 5. Misleading graphs: catch the trick before it catches you)

Summary

A frequency table counts each category; a relative-frequency table shows each count as a share of the total.
A bar chart compares categories; a pie chart shows parts of one whole and needs shares that add to 100%.
Dot plots suit small datasets, stem plots keep every value, and histograms group many values into equal-width bins.
A histogram's bars touch and stay in order; a bar chart's bars stand apart.
Describe shape by balance, by the number of peaks, and by any values that sit far away.
Name the skew by its tail: a tail to the right is right-skewed, a tail to the left is left-skewed.
In a two-way table, read the row label and the column label before you use a count.
A bar chart's axis must start at zero, and an icon should grow in one dimension only.

Check your understanding

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

A food truck records 40 orders in one lunch hour: 18 tacos, 12 rice bowls, 6 salads, and 4 drinks only. Build the relative-frequency table, and say whether a pie chart would be an honest picture.
Ten riders report their commute times in minutes: 12, 15, 18, 21, 22, 24, 27, 33, 41, 58. Make a stem plot with the tens digit as the stem, and describe the shape.
A used-car lot lists 50 cars. Most are priced between $6,000 and $12,000, and a thin tail of a few cars stretches out to $40,000. A classmate calls the distribution left-skewed "because the bars are piled up on the left." What is the right name, and why?
A gym's flyer shows a bar of 480 members last year and a bar of 520 members this year, with the vertical axis starting at 460. How many times taller does this year's bar look, and what is the real change?
A library counts 50 visitors by purpose and time of day: 12 came to study in the morning, 18 came to study in the evening, 15 came to borrow in the morning, and 5 came to borrow in the evening. Build the two-way table with totals, and say how many visitors came in the evening.

Answers (worked reasoning — read after attempting)

Check the total first: 18 + 12 + 6 + 4 = 40. Now divide each count by 40. Tacos: 18 ⁄ 40 = 45.0%. Rice bowls: 12 ⁄ 40 = 30.0%. Salads: 6 ⁄ 40 = 15.0%. Drinks only: 4 ⁄ 40 = 10.0%. The shares add to 45.0 + 30.0 + 15.0 + 10.0 = 100.0. Each order falls in exactly one category, so the four shares split one whole and a pie chart is honest. A bar chart would work too, and it makes the comparison between tacos and bowls easier to see.
The stem plot is:
```
1 | 2 5 8
2 | 1 2 4 7
3 | 3
4 | 1
5 | 8
How to read it: 2 | 1 stands for 21 minutes
```
Count the leaves: 3 + 4 + 1 + 1 + 1 = 10, one per rider. Most commutes pile up in the 10s and 20s, and the stems thin out toward the larger values. The tail points right, so the shape is right-skewed and unimodal. The 58-minute ride sits 17 minutes above the next value, 41. It looks like a possible outlier, but without a stated rule that is a judgment, so say "one long commute sits apart in the right tail."
The right name is right-skewed. The pile of cars sits on the left, at the lower prices, but skew is named for the tail, and the thin tail stretches right toward $40,000. Your classmate named the skew by the pile, which is the classic reversal.
The drawn heights are 480 − 460 = 20 units and 520 − 460 = 60 units. Divide: 60 ⁄ 20 = 3, so this year's bar looks 3 times as tall. The real change is 520 − 480 = 40 members, and 40 ⁄ 480 ≈ 8.3%. So membership grew by about 8.3%, not threefold. The truncated axis made a modest gain look huge.
Put purpose in the rows and time in the columns. Study row: 12 + 18 = 30. Borrow row: 15 + 5 = 20. Morning column: 12 + 15 = 27. Evening column: 18 + 5 = 23. Grand total: 30 + 20 = 50, and 27 + 23 = 50, so the two routes agree. The evening total is 23 visitors.

Library visitors counted by purpose (rows) and time of day (columns), with totals
Purpose	Morning	Evening	Total
Study	12	18	30
Borrow	15	5	20
Total	27	23	50

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week a rule for outliers joins numbers for center and spread.

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