A good picture lets the data speak — and an honest one never lets it lie.
Numbers arrive in a pile. Thirty quiz scores, a year of monthly sales, the way every student gets to school — written out as a list, the meaning hides. A picture lets the pile speak: the eye finds the peak, the gap, the trend in an instant. But a chart is an argument, and an argument can be honest or rigged. This lesson builds the four workhorse charts — the bar, the pie, the line, and the frequency histogram — learns which job each one does, and learns to catch the favorite trick of every misleading graph: a chopped axis that turns a molehill into a mountain.
Before a chart there is a count. Take a class's thirty quiz scores. Sort them into ranges — here five ranges ten points wide — and tally how many land in each. The tally is the frequency of that range. Divide by the total n and you get the relative frequency, the range's share of the class:
relative frequency f = count in the rangen
| Score range | Tally (frequency) | Relative frequency f |
|---|---|---|
| 50 – 60 | 2 | 2/30 ≈ 0.067 |
| 60 – 70 | 6 | 6/30 = 0.200 |
| 70 – 80 | 10 | 10/30 ≈ 0.333 |
| 80 – 90 | 8 | 8/30 ≈ 0.267 |
| 90 – 100 | 4 | 4/30 ≈ 0.133 |
| Total | 30 | 1.000 |
Two facts hold for every frequency table, and they are the whole reason the table is trustworthy: the frequencies add up to n, and the relative frequencies add up to 1. (A value on a boundary needs a rule — here the lower edge belongs to a range, so a 70 counts in 70 – 80, not 60 – 70.) The table is the honest skeleton; every chart in this lesson is just that skeleton wearing different clothes.
A frequency is a raw count; a relative frequency is that count over the total. Relative frequencies always sum to 1 — that is what makes them shares.
Suppose 200 students each name how they get to school: Walk 64, Bus 80, Bike 24, Car 32. These are categories, not measurements, and three charts can carry them — but each answers a different question.
Bus is 80 of 200, so its share is 80/200 = 0.40 = 40%, and its slice spans 0.40 × 360° = 144°. The four shares — 32%, 40%, 12%, 16% — sum to 100%, exactly as a pie demands.
Switch the chart type for the same commute data. Watch which question each chart answers — and notice a pie hides the actual counts while a bar shows them.
A pie chart compares slices within one circle. Setting two pies of different totals side by side and comparing slice to slice is a classic deception — a 30% slice of a small pie is a smaller count than a 20% slice of a huge one. Shares are not amounts.
The commute data was categorical. But quiz scores, heights, and waiting times are continuous — they live on a number line, with no natural gaps. To picture them we group the line into touching intervals (called bins) and draw a bar over each. That is a histogram, and its bars touch precisely because the data is continuous: there is no empty space between "70 – 80" and "80 – 90", so there is no gap between their bars. (A bar chart's gaps shout "separate categories"; a histogram's touching bars shout "one continuous scale.")
Here is the rule that makes a histogram more than a pretty bar chart. We let each bar's area, not its height, equal that bin's relative frequency. A bar of width w and height density has area = width × height, so we set
height (density) = relative frequencybin width w, so that area = relative frequency.
Add up every bar's area and you add up every relative frequency — which we already know is 1. The total area under a histogram is exactly 1. That single fact is the bridge to Stage 33's probability density and the bell curve you will meet again in 34.5 — Statistical inference: a continuous distribution is the shape a histogram approaches as the bins grow thin.
Change the bin width for thirty test scores. The bars always touch, and the relative-frequency areas always sum to 1 — no matter how coarse or fine the bins.
With width 10, the busiest bin 70 – 80 holds 10 of 30 scores. Its relative frequency is 10/30 ≈ 0.333; its density (bar height) is 0.333 / 10 ≈ 0.0333; its area is 10 × 0.0333 ≈ 0.333 — back to the relative frequency, as designed. Sum the five areas: 0.067 + 0.200 + 0.333 + 0.267 + 0.133 = 1.
Picking a chart is two quick questions. First: is the variable categorical (named groups: walk, bus, bike) or continuous (a measurement on a number line: a score, a height)? Second: am I comparing amounts, showing shares of a whole, or tracking a trend? The two answers point straight at the chart.
| You want to… | Data is… | Use | Bars… |
|---|---|---|---|
| Compare amounts across groups | categorical | bar chart | have gaps |
| Show each part's share of one whole | categorical | pie chart | — |
| Track a value over time | ordered / time | line chart | — |
| Show the shape of a measurement | continuous | histogram | touch |
Reading a chart well means reading the axis first: where does it start, what are the units, what is the bar width? A histogram's tallest bar marks the most common range; its long thin tail marks rare extremes. A bar chart's tallest bar is simply the biggest group. The shape is the message — but only once you trust the scale.
Using a bar chart (gaps, categorical) where a histogram (touching, continuous) belongs throws away the most important fact about measurements: that they sit in order on a line. Gaps between "60 – 70" and "70 – 80" pretend those ranges are unrelated categories. They are not.
Every honest chart starts its amount-axis at zero, because a bar's height is meant to stand for its value. Cut the bottom off the axis — start it at 90, or 100 — and the bars no longer represent the numbers; they represent only the little bit above the floor. A whisker of a difference becomes a canyon. This is the single most common trick in misleading graphics, and it is everywhere in advertising and headlines.
Two years of revenue: 2023 = 102, 2024 = 108. The true ratio is 108 / 102 ≈ 1.06 — growth of about 6%. Start the axis at 90 and the bars stand 12 and 18 tall: the visual ratio is 18 / 12 = 1.5. Start it at 100 and the bars are 2 and 8: the visual ratio is 8 / 2 = 4 — a 6% rise made to look like quadrupling.
Slide the axis floor upward. The data never changes — only where the axis starts. The readout reports the true ratio (always the same) against the exaggerated visual ratio (in red).
A truncated (chopped) amount-axis is the headline crime — but it has cousins: a 3-D pie that tilts the front slices fatter, a stretched pictogram where a "twice-as-big" image covers four times the area, an axis with no zero and no units. When the picture astonishes, check the scale before you believe it. (A line chart of a trend may start above zero — but it should say so plainly.)
| Tool | What it shows | The honest rule |
|---|---|---|
| Frequency table | counts & shares of each range | relative freqs sum to 1 |
| Bar chart | amounts across categories | gaps; axis at 0 |
| Pie chart | shares of one whole | slices sum to 100% |
| Line chart | a trend over time | points in order |
| Histogram | the shape of continuous data | bars touch; total area = 1 |
| Chopped axis | (the trap) | start at 0 or you lie |
Next, in 34.3 — Measures of center, we squeeze the whole pile down to a single number for its middle: the mean, the median, and the mode.
In a survey of 40 people, 10 chose tea, 22 chose coffee, and 8 chose water. Find the relative frequency of each, and the angle of the coffee slice in a pie chart.
Tea 10/40 = 0.25; coffee 22/40 = 0.55; water 8/40 = 0.20 — these sum to 1.00. The coffee slice spans 0.55 × 360° = 198°.
A friend draws a bar chart with gaps for the heights of 50 students, grouped into 150 – 160 cm, 160 – 170 cm, and so on. What is wrong, and what should the chart be?
Height is continuous, so the ranges sit in order on a number line with no gaps between them. The correct chart is a histogram whose bars touch; gaps would falsely suggest the height bands are separate, unrelated categories.
Twenty reaction times (in seconds) are grouped into bins of width 0.2 from 0.2 to 1.0. One bin holds 8 of the times. What relative frequency does that bin have, and what is its bar's height (density) so that its area equals that relative frequency?
Relative frequency = 8/20 = 0.40. Density = relative frequency ÷ width = 0.40 / 0.2 = 2.0. Check the area: width × height = 0.2 × 2.0 = 0.40, the relative frequency. ✓
A headline shows a bar chart of a stock: last month $50, this month $53, with the y-axis starting at $48. The bars look like 2 and 5 units tall. State the true ratio and the visual ratio, and name the trick.
True ratio = 53/50 = 1.06 (a 6% rise). Visual ratio = 5/2 = 2.5. The trick is a truncated (chopped) axis: starting at $48 instead of $0 makes a 6% change look like a 150% jump.
For each goal, name the best chart: (a) the share each continent holds of the world's population; (b) a city's monthly rainfall across a year; (c) comparing the populations of five cities; (d) the distribution of 200 light-bulb lifetimes in hours.
(a) pie chart (shares of one whole); (b) line chart (a trend over the ordered months); (c) bar chart (compare amounts across separate categories); (d) histogram (the shape of a continuous measurement, bars touching).
A histogram of test scores has bins of width 10 with relative frequencies 0.10, 0.25, 0.40, 0.20, and a final bin left blank. What must the last relative frequency be, and why are you sure without seeing the data?
The relative frequencies (the bar areas) must sum to 1. So the last one is 1 − (0.10 + 0.25 + 0.40 + 0.20) = 1 − 0.95 = 0.05. The total-area-equals-1 rule pins it down with no further information.
Six questions to lock it in. Tap the answer you think is right.
This lesson represents data with the plots students must master under CCSS HSS-ID.A.1 — dot plots, bar/pie/line charts, and the frequency histogram — and reads relative-frequency data toward HSS-ID.B.5. The non-negotiable ideas: a histogram's bars touch and its relative-frequency areas sum to 1 (height is a density, area is the relative frequency); a bar chart is for categories and a histogram for continuous measurements; and a truncated amount-axis is a deception, not a style choice. A good follow-up is to bring in a real advertisement or news graphic and ask students to find the chopped axis and compute the true-versus-visual ratio.