L-DS-2 Data Visualization

Descriptive statistics · lesson introduction

Data visualization

A graph is an argument built from data.

Choose · Construct · Audit · Interpret

Opening commitment · one dataset

Same values. Two questions.

One fictional student records their score (%) on 20 low-stakes quizzes, in quiz order.

Q1 50 Q2 63 Q3 58 Q4 67 Q5 65 Q6 71 Q7 69 Q8 74 Q9 68 Q10 78 Q11 72 Q12 75 Q13 81 Q14 77 Q15 84 Q16 79 Q17 76 Q18 86 Q19 88 Q20 98
A

How are the quiz scores distributed?

B

How did the scores change through quiz order?

Would one graph preserve what both questions need?

Preview · count and share

Same distribution. Two vertical scales.

What changes? What stays fixed?

Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)

8 of 2040%

Same boundaries and relative heights; different vertical language. Neither graph is cumulative: each bar describes one score interval.

Concrete construction routine

Seven checks make a graph stand alone.

What must a stand-alone graph make clear?

Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)
Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Relative-frequency histogram of fictional quiz scores Five touching bars for the same quiz-score intervals. Relative frequencies are 10, 25, 40, 20, and 5 percent, for a total of 100 percent. Fictional scores from 20 quizzes 0% 10% 20% 30% 40% 50 60 70 80 90 100 Quiz score (%) Relative frequency (%)

C1 · definition before use

A frequency distribution organizes observations into counts and running totals.

Frequency Distribution

A frequency distribution records each class or value, its frequency, its share of the whole, and the totals accumulated through that class.

Classthe interval fcount in one class fᵣ = f/nshare in one class cfrunning count cfᵣrunning share

One class uses f or fᵣ. Everything through a boundary uses cf or cfᵣ.

C1 · class design

Choose classes that cover every value once.

5063586765716974687872758177847976868898

Rangeminimum 50% · maximum 98% · choose five 10-point bands

Width Each band spans 10 percentage points. (Starting rule: range ÷ classes = (98 − 50) ÷ 5 = 9.6; round up to a convenient width of 10.)

Classes[50, 60) · [60, 70) · [70, 80) · [80, 90) · [90, 100]

50–<6060–<7070–<8080–<9090–100

Check: no gaps, no overlap, every observed value has exactly one home.

C1 · source → row

Tally every observation exactly once.

5063586765716974687872758177847976868898
ClassTallyf
50–<60||2
60–<70||||/5
70–<80||||/ |||8
80–<90||||4
90–100|1

2 + 5 + 8 + 4 + 1 = 20 quiz scores ✓

C1 · count → share → running total

Build one meaning at a time.

Classffᵣ = f/20cfcfᵣ
50–<6020.1020.10
60–<7050.2570.35
70–<8080.40150.75
80–<9040.20190.95
90–10010.05201.00
Σf = 20Σfᵣ = 1.00last cf = 20last cfᵣ = 1.00

C1 hinge · one class versus running share

What does cfᵣ = 0.75 mean?

Class [70, 80) · f = 8 · fᵣ = 0.40 · cf = 15 · cfᵣ = 0.75 · n = 20

A 75% of the scores are from 70% to below 80%. B 75% of the scores are below 80%. C 15% of the scores are below 80%. D The next quiz has a 75% chance of scoring below 80%.

B. Fifteen of twenty scores are in the first three classes, so 75% are below 80%.

C2 · definition before construction

A histogram preserves a quantitative number line.

Histogram

A histogram displays a quantitative frequency distribution with adjacent numerical intervals. In the equal-width histograms used here, bar height shows frequency or relative frequency.

Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)

Each bar spans exactly one class, boundary to boundary — bars touch because neighbouring classes share a boundary.

Here every class is 10 percentage points wide, so comparing bar heights compares class frequencies directly.

C2 · title and axes come before marks

Build the scaffold before the bars.

Score (%)f50–<60260–<70570–<80880–<90490–1001
Histogram scaffold for fictional quiz scores A descriptive graph title appears, followed by a labelled horizontal quiz-score axis with class boundaries from 50 to 100 percent, and a vertical frequency axis beginning at zero. No bars are drawn yet. Distribution of 20 Fictional Quiz Scores Quiz score (%) Frequency (quizzes) 02468 5060708090100

C2 · source row → plotted mark

Map each table row to one bar.

Score (%)f50–<60260–<70570–<80880–<90490–1001
Fictional quiz-score histogram construction A titled histogram with labelled axes. Five touching bars are revealed from source-table rows with frequencies 2, 5, 8, 4, and 1. Distribution of 20 Fictional Quiz Scores 02468 5060708090100 Quiz score (%)Frequency (quizzes)

C2 · construction is not finished at the last bar

Check before interpreting.

Rows → bars5 source rows → 5 touching bars

Total2 + 5 + 8 + 4 + 1 = 20

ClaimThe 70% to below-80% class is tallest: 8 of 20 scores.

LimitGrouping hides exact positions inside each class.

Title ✓ · x variable + unit ✓ · y quantity + unit ✓ · scale ✓ · source check ✓

C2 · class-number judgment

The data stay fixed. The visible detail changes.

5063586765716974687872758177847976868898
5 classes · width 10Boundaries every 10 points from 50 to 100
10 classes · width 5Boundaries every 5 points from 50 to 100
Changes: boundaries, counts per class, visual detailFixed: the observations, n, minimum, maximum

More classes do not reveal one “true” shape. Choose enough detail for the question without turning noise into a story.

C4 · time-series construction

To answer the second question, preserve quiz order.

1 50 2 63 3 58 4 67 5 65 6 71 7 69 8 74 9 68 10 78 11 72 12 75 13 81 14 77 15 84 16 79 17 76 18 86 19 88 20 98
Fictional quiz scores, quizzes 1 through 20A titled and labelled time-series plot maps each quiz number to its score, then connects adjacent quizzes in chronological order. Fictional Quiz Scores across 20 Quizzes Quiz number (order)Quiz score (%) 5060708090100

20 rows → 20 points · later scores are generally higher · this does not prove learning or predict quiz 21

C5 · purpose changes the graph

Same data. Different question.

DistributionHistogram

Preserves intervals and frequencies.

Change through orderTime series

Preserves quiz chronology and adjacent changes.

No graph is universally “best.” The best graph preserves the structure the question needs.

C3 · definition before construction

A bar chart compares distinct categories.

Bar Chart

A bar chart displays frequencies or relative frequencies for a qualitative variable. Equal-width bars remain separated because there is no meaningful value between categories.

Favourite Sport among 35 Students A titled bar chart with category labels, a frequency axis beginning at zero, visible gaps, and bars of heights 12, 9, 7, 4, and 3. Favourite Sport among 35 Students Favourite sport (category) Frequency (students) 0 4 8 12 Soccer Basketball Swimming Tennis Other

Equal-width bars stay separated — the gaps mark breaks between distinct categories.

Nominal categories may be reordered. Ordinal categories keep their natural order.

C3 · source category → separated mark

Build separated bars.

SportfSoccer12Basketball9Swimming7Tennis4Other3
Favourite Sport among 35 Students A titled bar chart with category labels, a frequency axis beginning at zero, visible gaps, and bars of heights 12, 9, 7, 4, and 3. Favourite Sport among 35 Students Favourite sport (category) Frequency (students) 0 4 8 12 Soccer Basketball Swimming Tennis Other

Five categories → five equal-width separated bars · total = 35 · Soccer is a plurality, not a majority.

C3 · order carries meaning

Nominal order can move. Ordinal order cannot.

Nominal · quiz error type
SignRoundingSetupInterpretation
SetupSignInterpretationRounding

A Pareto chart is a bar chart sorted by descending frequency — tallest bar first — so the largest contributor stands out. Here it could help a class decide which common error deserves attention first.

Ordinal · satisfaction
Very lowLowNeutralHighVery high

Keep the natural order—even if a middle category is tallest.

C3 hinge · numeric-looking labels

Do number labels make it quantitative?

1 very low · 2 low · 3 neutral · 4 high · 5 very high

A Histogram; the labels are numbers.B Time series; connect 1 through 5.C Bar chart in natural order 1 through 5.D Pareto chart sorted tallest to shortest.

C. The numbers code ordered categories; differences and in-between values are not numerical measurements.

C4 · pie construction

Build a pie only for a simple part-to-whole question.

A new survey of 40 students records one favourite sport each: Soccer 16 · Basketball 12 · Swimming 8 · Tennis 4.

Soccer 16/40 = 40% → 144°

Basketball 12/40 = 30% → 108°

Swimming 8/40 = 20% → 72°

Tennis 4/40 = 10% → 36°

100% · 360° ✓

Favourite Sport among 40 Students pie chartA titled pie chart is constructed from shares 40, 30, 20, and 10 percent, with cumulative angle boundaries at 144, 252, 324, and 360 degrees.Favourite Sport among 40 StudentsSoccer 40%Basketball 30%Swimming 20%Tennis 10%

Direct labels + descriptive title + 100% check. Use bars when precise comparisons matter more than the whole.

C4 · stem-and-leaf construction

For a small quantitative dataset, preserve every raw value.

Building a stem-and-leaf plot from fictional scores.

Fictional scores: 72, 68, 75, 81, 74, 69, 83, 77, 71, 76

Sort68, 69, 71, 72, 74, 75, 76, 77, 81, 83

Split tens | ones6 | 8 9

Place leaves7 | 1 2 4 5 6 7

Complete8 | 1 3

Key7 | 2 means 72 points

10 observations → 10 leaves ✓ · raw values remain recoverable

C4 · scatterplot construction

One paired row becomes one point.

Building a scatterplot from fictional paired data.

Study hScore152261358472568680776888
Fictional Study Hours and Exam Scores for Eight Students A titled scatterplot maps eight paired rows to eight points, each labelled with its study-hours and score pair and colour-matched to its source row. The cloud shows a positive association with local reversals; it does not establish causation or guarantee an individual score.Fictional Study Hours and Exam Scores (n = 8)Study time (hours)Exam score (points)506070809012345678(1, 52)(2, 61)(3, 58)(4, 72)(5, 68)(6, 80)(7, 76)(8, 88)

8 paired rows → 8 unconnected points · positive whole-cloud association · no causal or individual guarantee

C5 · variable structure + purpose

What must the graph show?

QualitativeCompare categories

Bar chart · default for counts or shares

Pie chart · optional for a few categories when part-to-whole is the point

Pareto: rank priorities · Ordinal: keep natural order
One quantitative variableShow a distribution

Histogram · default for continuous or many-valued data

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

Stem-and-leaf · small n when every raw value matters

Few discrete counts: separated frequency bars can work
Quantitative through ordered timeTime serieschange from one time point to the next
Two quantitative variablesScatterplotrelationship between paired values

One qualitative + one quantitative needs a comparison display taught later—not an automatic bar chart of raw observations.

C5 hinge · one anonymous class vote

Choose the best graph for all four situations.

1Compare category sizes for 200 students’ transport modes.
2Show the distribution of commute times for 80 students.
3Show how median clinic wait changed month by month.
4Explore sleep hours and reaction time as paired values.
A 1 Bar · 2 Histogram · 3 Time series · 4 Scatterplot B 1 Pie · 2 Stem-and-leaf · 3 Histogram · 4 Time series C 1 Histogram · 2 Bar · 3 Scatterplot · 4 Pie D 1 Bar · 2 Time series · 3 Histogram · 4 Scatterplot

A. Qualitative categories → bar; one quantitative distribution → histogram; ordered time → time series; paired quantitative relationship → scatterplot.

C6 · integrity begins before interpretation

Audit before you trust the graph.

Check whether the design makes the data easier to judge—or distorts the impression.

  1. 1 · FamilyDoes it answer the stated question?
  2. 2 · AxesLabels, units, baseline, and scale
  3. 3 · GeometryWidths, areas, and point placement
  4. 4 · Order / windowMeaningful order and disclosed time range
  5. 5 · Context / claimSource, n, denominator, and a defensible sentence

C6 hinge · correct marks are not enough

Which graph can stand alone?

A Plot option A Weekly wait-time points are plotted correctly. No descriptive title. Axis names and units are missing. No sample or source context is shown. 35 40 45 50 W1 W8
B Plot option B Weekly wait-time points are plotted correctly. Wait Time Axis names and units are missing. No sample or source context is shown. Wait Time 35 40 45 50 W1 W8
C Plot option C Weekly wait-time points are plotted correctly. Median Clinic Wait by Week Both axes are named and include units. n = 240 visits · clinic records Median Clinic Wait by Week 35 40 45 50 W1 W8 Week Median wait (minutes) n = 240 visits · clinic records
D Plot option D Weekly wait-time points are plotted correctly. No descriptive title. Axis names and units are missing. Blue line = data 35 40 45 50 W1 W8 Blue line = data

C. A graph needs a descriptive title, named axes, units, and enough context to interpret the marks.

C6 · integrity audit 1

Which graph answers each rainfall question?

Evidence: rainfall totals for Jan–Dec are 42, 38, 51, 47, 63, 71, 69, 58, 55, 49, 44, 40 mm.

Question 1 How did rainfall change from January to December?

Question 2 How many months fall in each 10-mm interval?

Question 1 preserves month orderTime seriespreserves chronology
Question 2 counts interval frequenciesHistogrampreserves intervals and frequencies

Sorting the months wettest-to-driest would answer a ranking question—but destroy time order.

C6 · integrity audit 2

Which picture exaggerates the 9.7% increase—and why?

Zero baseline: Annual Revenue A bar chart comparing 196 million dollars in Year 1 with 215 million dollars in Year 2. The vertical scale begins at 0 million dollars. Annual Revenue · Zero baseline 0 50 100 150 200 250 $196M $215M Year 1 Year 2 Revenue ($ millions) Truncated baseline: Annual Revenue A bar chart comparing 196 million dollars in Year 1 with 215 million dollars in Year 2. The vertical scale begins at 180 million dollars. Annual Revenue · Truncated baseline 180 190 200 210 220 $196M $215M Year 1 Year 2 Revenue ($ millions)

From $196M to $215M: 19/196 ≈ 9.7%. On the truncated axis, 35 ÷ 16 ≈ 2.19, so the visible lengths exaggerate the change.

Bar length ordinarily needs a zero baseline. A clearly labelled line graph may zoom when small changes are the purpose.

C6 · integrity audit 3

A graph can be accurate and still omit what the claim needs.

  1. OrderChronology and ordinal categories remain meaningful.
  2. WindowThe displayed time range is disclosed; no convenient slice stands for the full history.
  3. ContextPopulation/source, n, denominator, missingness, and units are visible when needed.
  4. ClaimThe sentence stays inside what the graph and data establish.

Audit: family/purpose · title/labels · baseline/scale · order/window · context/claim.

Return · the same 20 fictional quiz scores

Now each question has an honest display.

Frequency histogram of fictional quiz scores Five touching bars for quiz-score intervals from 50 to 100. Frequencies are 2, 5, 8, 4, and 1, for a total of 20 quiz scores. Fictional scores from 20 quizzes 0 2 4 6 8 50 60 70 80 90 100 Quiz score (%) Frequency (quizzes)
Claim: 8 of 20 scores are from 70% to below 80%.
Limit: grouping hides exact scores and quiz order.
Fictional Quiz Scores across 20 Quizzes A complete time-series graph with quiz number on the horizontal axis, quiz score percent on the vertical axis, labelled ticks, twenty points, and a line connecting quizzes in order. Fictional Quiz Scores across 20 Quizzes 506070 8090100 1510 1520 Quiz number (order) Quiz score (%) Claim: later scores are generally higher, with local drops.
Limit: the graph does not prove learning or predict quiz 21.

Same values. Different preserved structure. Different defensible claim.

Independent transfer · tree hidden

Choose, build, check, and interpret both displays.

Change across eight weeks

Weekly median wait (min)

W1 46W2 45W3 44W4 45W5 43W6 42W7 41W8 40
Relationship in eight audited visits

(wait min, satisfaction /100)

(12,93)(18,88)(22,86)(29,79)(35,77)(41,69)(48,64)(55,59)
  1. Choose one graph for each purpose.
  2. Write a descriptive title and label both axes with units.
  3. Plot every source row; run a row/mark check.
  4. Write one supported claim and one limit for each.

Delayed transfer model

Compare your decisions—not your drawing style.

Time seriesWeekly Median Clinic Wait (Weeks 1–8)

x: week · y: median wait (minutes)

Overall, not steady, decrease from 46 to 40 min.Limit: eight displayed weeks; no causal explanation.
ScatterplotWait Time and Satisfaction in Eight Audited Visits

x: wait (minutes) · y: satisfaction (points/100)

Negative association in these eight visits.Limit: no causation or individual guarantee.

Revise one title, label, mark, check, claim, or limit on your own work.

Objective exit A · C1/C2

Which option builds the correct histogram and reports the share below 30 minutes?

Commute time (min): [0,10) 2 · [10,20) 5 · [20,30) 8 · [30,40) 4 · [40,50) 1 · n = 20

A Touching bars · heights 2, 5, 8, 4, 1 · 75% below 30 min B Touching bars · heights 2, 7, 15, 19, 20 · 40% below 30 min C Separated bars · heights 2, 5, 8, 4, 1 · 75% below 30 min D Touching bars · heights 10%, 25%, 40%, 20%, 5% · 40% below 30 min

A. Equal-width quantitative intervals use touching bars with frequency heights; (2 + 5 + 8)/20 = 0.75.

Objective exit B · choose the family, then audit the claim

What is wrong with the 80% conclusion?

Monthly Median Clinic Wait, January to December A complete time-series graph with median wait falling from 48 minutes in January to 40 minutes in December. The labelled vertical axis is zoomed from 38 to 48 minutes. A caption claims an 80 percent decrease. Monthly Median Clinic Wait · Jan–Dec 38 40 42 44 46 48 JanJunDec Month (ordered time)Median wait (minutes) Claim: “waits fell 80%”
A Use a histogram; time order is irrelevant.B The time series is appropriate, but the 80% claim confuses distance from a nonzero baseline with percentage change in wait time.C Any nonzero y-axis makes the line graph dishonest.D Use a pie chart because twelve months form a whole year.

B. The family fits ordered time and a disclosed zoom can be legitimate. The actual decrease is (48 − 40)/48 ≈ 16.7%, not 80%.

Anonymous exit reflection · confidence, not mastery

How confident are you with each idea?

C1

I can build a complete frequency distribution table.

C2

I can construct a histogram from a frequency table.

C3

I can build a bar chart and tell nominal from ordinal data.

C4

I can build a pie chart, stem-and-leaf plot, scatterplot, or time series.

C5

I can select the correct graph from a variable's type and purpose.

C6

I can identify a misleading graph and name what makes it misleading.

0 Not yet 1 Very unsure 2 Beginning 3 Mostly understand 4 Confident 5 I could explain it

One habit to carry forward

A graph is an argument.

Choose for the question→Title + label→Map every mark→Check→Claim + limit

Make the structure visible. Make the claim honest.

L-DS-3 · Central Tendency Measures