How does a histogram differ from a bar chart?

A bar chart shows CATEGORIES — a mode of transport, a brand, a colour — and its bars are separated by gaps, because there is nothing in between two categories. A histogram shows one NUMERIC quantity cut into intervals, and its bars touch, because the classes touch. Hence the second difference: the bars of a chart may be reordered freely, the classes of a histogram may not — they lie on a number line in the only order there is.

How many classes should a histogram have?

Enough for the shape of the distribution to show and not so many that the noise does. A practical rule is five to twelve classes for a few dozen observations; too few and the histogram collapses into one bar, too many and it turns into a comb of empty and single-item classes. The classes must be equally wide, unless the heights are drawn as densities — the count divided by the class width.

When is a pie chart the right choice?

When you are showing how a WHOLE splits into a few disjoint parts whose shares differ clearly. A circle is no good for comparing two data sets, for quantities that do not add up to a whole, or for shares that are close to one another — the eye judges lengths far better than angles, so 24 and 26 per cent look identical on a pie and visibly different on a bar chart.

Why is quoting the mean sometimes an abuse?

Because one extreme observation can drag the mean and cannot drag the median. Nine property deals priced between 380 and 480 thousand plus one at 2400 thousand give a mean of 648 thousand and a median of 430 thousand. The sentence "the average price is 648 thousand" is true and still misleading, because not one of the nine ordinary deals comes anywhere near that number. For a skewed distribution the honest measure is the median.

What is the three-dimensional effect on a chart?

It is a drawing adding a dimension the data does not have. A pie slice in perspective shows more visible area when it sits at the front, and a bar drawn as a solid grows in height, width and depth at once, so doubling a value produces a block that looks several times bigger. The third dimension carries no information here — it exists purely to distort proportions, which is why no serious report has one.

How can I check quickly whether a chart is lying?

Four questions cover most cases. Where does the vertical axis start? Are the classes or categories equally wide? Are the units and the scale labelled, and do they stay the same all the way along the axis? Is the range shown a slice chosen to hide the rest of the trajectory? A chart that survives those four may still be ugly, but it can no longer mislead by accident.

Intermediate level

Presenting data and misleading charts

Q: Does the vertical axis always have to start at zero?

On a bar chart, yes: there the information is carried by the HEIGHT of the bar, and truncating the axis changes the proportions immediately. On a line chart showing change over time, truncation is often justified — a body temperature between 36 and 40 degrees gains nothing from an axis starting at zero. So the working rule is: comparing quantities, axis from zero; showing a trajectory, an axis fitted to the range but with its starting point clearly marked.

The same data set can be shown honestly or used to lie, without a single false number being written down. The frequency table, the bar chart, the histogram and the pie chart — how they differ, when to reach for which, and five tricks that make a drawing say something the data underneath it does not.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

A chart in the newspaper
A story about a poll shows two bars, 42 and 46 per cent, on a vertical axis starting at 40. The drawn heights are 2 and 6 units, so the leader’s bar is three times taller. The true ratio is 46 : 42, a lead of 9.5 per cent of the value — the drawing overstates it more than thirtyfold. One glance at the number at the bottom of the axis settles whether the picture can be trusted.
An election infographic
A campaign publishes a support chart: 34 per cent against 31, axis from 30. The bar heights are 4 and 1, so a three-point gap looks like four to one. With a polling margin of error around 3 points that gap is often not significant at all — and the chart shows it as a landslide.
A sales report
A sales team reports 320 deals in the quarter. A histogram of their values in 500-unit classes shows 214 of them below 1500 and 12 above 5000. The mean deal is 1840 and the median 1120 — and it is the median that says what a typical customer is worth. A commission forecast built on the mean would overstate the quarter by 64 per cent.
Planning a commute
A histogram of forty employees’ commute times in ten-minute classes has counts 3, 9, 14, 8, 4 and 2. Under 30 minutes sits 3 + 9 + 14 = 26 people, or 65 per cent of the team. An employer who wants 90 per cent of them at their desk on time has to budget not 30 minutes but 50 — that is where the cumulative count first reaches 38 out of 40.

All formulas

Total of the frequencies
$\sum_{i} n_i = n$
the table check: class counts must add up to the sample size
Relative frequency
$w_i = \frac{n_i}{n}$
the class's share; times 100 it is a percentage
Angle of a pie slice
$\alpha_i = 360^\circ \cdot \frac{n_i}{n}$
a full circle is 360°, so the angle is the share in degrees
Histogram class width
$h = \frac{x_{\max} - x_{\min}}{k}$
k — the number of classes; the range of the variance lesson cut into equal pieces
Frequency density
$g_i = \frac{n_i}{h_i}$
the bar height for UNEQUAL classes — not the count itself
Apparent ratio on a truncated axis
$\frac{H_B}{H_A} = \frac{x_B - c}{x_A - c}$
c — where the axis starts; at c = 0 it collapses to x_B / x_A

The two lessons before this one reduced a data set to numbers: the mean and the median, and the standard deviation. This lesson is about the opposite move — showing the whole set at once, so the reader can see its shape. And about the fact that the same set can be shown honestly or dishonestly without a single false number being written down.

The frequency table

Everything starts with counting. Forty people said how they get to work:

mode of transport	count $n_i$	frequency $w_i$	per cent
car	$18$	$18/40$	$45\%$
bus	$9$	$9/40$	$22.5\%$
bicycle	$7$	$7/40$	$17.5\%$
on foot	$6$	$6/40$	$15\%$
total	$40$	$1$	$100\%$

Two columns and one check. The count $n_i$ is simply how many times something occurred, the relative frequency $w_i = n_i/n$ is its share, and the "total" row is the table's own test:

\sum_{i} n_i = n, \qquad \sum_{i} w_i = 1

If the counts do not add up to the sample size, an observation went missing somewhere — and no chart built on that table will be right.

The bar chart

The table is exact, but its shape is invisible. A bar chart shows the same split as heights:

Four categories, four separated bars. The gaps between them are not decoration: they say that there is nothing halfway between a car and a bus.

Two things in this drawing are convention rather than choice:

The bars are separated. Categories are not a continuum, so bars that touched would suggest something the data does not contain.
The count axis starts at zero. Here the information is carried by height, and a height only means anything measured from zero. We come back to this in the second half of the lesson, because it is the most frequently broken rule in all of descriptive statistics.

The order of the bars, on the other hand, is free: alphabetical, descending, or whatever order is natural for the subject. Descending reads most easily when the question is "what dominates".

The pie chart

The same split can be shown as parts of a whole. The angle of a slice follows straight from its share:

\alpha_i = 360^\circ \cdot \frac{n_i}{n}

The same four numbers as a division of a circle. The drawing computes the angles from the values, so a slice cannot disagree with the data: 18 out of 40 is 162 degrees and nothing else.

How many degrees does the bus slice take?

A pie has a narrow range of use, and it is worth knowing: it works only when the parts add up to a whole, when there are at most a handful of them, and when their shares differ clearly. The eye compares lengths far better than angles, so two slices of $24\%$ and $26\%$ look identical, where on a bar chart the difference would be obvious at once.

The histogram

So far the data has been categorical. When the measured quantity is a number — a time, a height, a price — there are no categories and you have to make them, by cutting the range into classes of equal width:

h = \frac{x_{\max} - x_{\min}}{k}

The commute times of those same forty people run between $0$ and $60$ minutes. Six classes of ten minutes each:

class	$[0;10)$	$[10;20)$	$[20;30)$	$[30;40)$	$[40;50)$	$[50;60)$
count	$3$	$9$	$14$	$8$	$4$	$2$

A histogram: the bars meet edge to edge, because the classes are adjacent on the time axis. One peak and a mild skew to the right — long commutes are rarer than short ones, but they reach further.

Only this drawing shows the shape of the distribution, which neither of the two previous lessons does: a single peak, a moderate right skew, no outliers. The mean and the standard deviation describe that shape with two numbers; the histogram shows all of it.

How a histogram differs from a bar chart

This is the point people most often trip over, so it is worth settling in a table:

	bar chart	histogram
what is on the $x$ axis	categories	a numeric quantity (classes)
do the bars touch	no, there are gaps	yes, the classes adjoin
may they be reordered	yes	no — they lie on a number line
what carries the information	the height	the area of the bar (with equal classes, the height)
example	transport mode, brand	time, height, price, test score

The row about area is the one people forget, and it comes back in a moment as the second of the abuses. With classes of equal width, area and height are proportional, so the difference never surfaces — it only shows once the classes stop being equal.

Which one to use is settled by a single question: is the horizontal axis a number? If it is — a histogram. If it is not — a bar chart. A pie chart only if, on top of that, what interests you is the division of a whole rather than a comparison of quantities.

How to lie with a chart

The second half of this lesson is about the fact that every rule above can be broken without stating a single false number. The five tricks below are classics; each one has a name, because a named abuse is easier to spot.

1. The truncated axis

The most common and the most effective. The vertical axis does not start at zero, so a difference in height stops matching the difference in value. Two poll results, $42\%$ and $46\%$ :

Axis from zero. A four-point gap looks exactly like what it is — a modest lead.

The same two numbers, axis from 40. Bar B is drawn three times taller than A. No number changed — only the bottom end of the axis did.

The arithmetic is simple and worth doing in your head. If the axis starts at $c$ , the drawn heights are $x_A - c$ and $x_B - c$ , so the apparent ratio is

\frac{H_B}{H_A} = \frac{x_B - c}{x_A - c} = \frac{46 - 40}{42 - 40} = \frac{6}{2} = 3

while the true ratio is $46 : 42 \approx 1.10$ . The drawing overstates the lead more than thirtyfold.

The correction: on a bar chart the count axis starts at zero — always, with no exception, because the information there is carried by height. On a line chart showing a trajectory truncation can be justified (a body temperature between $36$ and $40$ degrees gains nothing from an axis starting at zero), but then the starting point has to be clearly labelled.

2. Unequal histogram classes

A histogram of ages: class $[0;10)$ with $8$ people, $[10;20)$ with $9$ , and $[20;60)$ with $36$ . Drawn with ordinary bars of equal width, the last one is four times taller than the first and looks like the dominant group. In fact it is simply four times wider:

g_1 = \frac{8}{10} = 0.8, \qquad g_2 = \frac{9}{10} = 0.9, \qquad g_3 = \frac{36}{40} = 0.9

The densities are nearly identical — there is no dominance in the data at all.

The correction: with unequal classes the height of a bar is the frequency density $g_i = n_i/h_i$ , not the raw count, because it is the area that carries the information. The simplest way out, though, is a different one: equal class widths, and the problem disappears.

3. The three-dimensional effect

A pie or a bar chart drawn "in perspective" adds a dimension the data does not have. A slice at the front shows more visible area than an identical slice at the back, and a bar drawn as a solid grows in height, width and depth at once, so doubling a value yields a block that looks several times larger.

The correction: a third dimension on two-dimensional data carries no information, so its only effect is to distort proportions. A flat chart is always the more honest one — and that is the whole rule.

4. Cherry-picking the scale and the range

The same time series shown over the last three months is rising, and shown over three years is falling. Nothing in the data changed; what changed is the slice the author chose to draw. A close relative is switching the unit or the scale halfway along an axis, and a time axis with unequal spacing ( $2010$ , $2015$ , $2016$ , $2017$ placed at even intervals).

The correction: the range and the scale are part of the claim a chart makes. An honest chart shows the whole trajectory available, or says outright why it shows a slice; a time axis has spacing proportional to time.

5. A mean where the median belonged

The last trick is not about the drawing but about the caption under it. Nine property deals, in thousands:

380,\ 400,\ 410,\ 420,\ 430,\ 450,\ 460,\ 480,\ 2400

The mean is $5830 : 9 \approx 648$ thousand, the median $430$ thousand. The sentence "the average price in this district is 648 thousand" is true and still misleading: not one of the nine deals comes near that number, because a single extreme value pulled the mean up by over $200$ thousand.

The correction: for a skewed distribution, or one with outliers, the measure of a typical value is the median, exactly as the lesson on measures of location settles it. An honest report gives both and shows the histogram that explains why they differ.

Exercises

The first set drills the arithmetic of a truncated axis. The prompt gives where the axis starts and both bar values, and asks how many times taller bar B looks — the ratio of the heights measured from the truncated base, not from zero. Subtracting the base before dividing is the whole point here; the answer is a number.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

How many times taller B looks — y axis starts at 20: A = 24, B = 28

The second set is authored, because it asks about things the generator deliberately does not draw: reading a table and a histogram, the angle of a pie slice, and the truncated-axis calculation run backwards — from the apparent proportion back to the true value. Every answer is a number; give the angle in degrees and the density as a decimal.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 6Score: 0

Pie chart, n = 40. The bus slice is 9 people. Angle of the slice → ° =

Judging "what is wrong with this chart" has no single number for an answer, so it is not an exercise — it is content, and you have it written out in the five points above. The four questions in the last FAQ entry are the practical version of that list.

Common mistakes

Drawing a bar chart on a truncated axis — on bars the information is carried by height, so truncating the axis always distorts the comparison; a line chart plays by different rules than a bar chart.
Confusing a histogram with a bar chart — a gap between bars means "these are categories". Categories drawn without gaps suggest a continuity that is not there; histogram classes drawn with gaps hide a continuity that is.
Unequal classes with heights equal to the counts — with differing widths the information is carried by area, so the height has to be the density $n_i/h_i$ .
A pie chart for quantities that do not add up to a whole — a circle claims to be showing a division; used to compare independent numbers it lies in its very shape.
Too many slices — past six or seven parts a pie stops being readable; small categories get merged into one "other".
Quoting only the mean for a skewed distribution — a single outlier can push the mean outside the entire main range of the data; for such a distribution the typical value is described by the median.
Treating a pretty chart as evidence — a third dimension, a gradient and a drop shadow add no information, and are often exactly what draws the eye away from the truncated axis.

Formula card

Topic: Presenting data

Total of the frequencies
$\sum_{i} n_i = n$
the table check: class counts must add up to the sample size
Relative frequency
$w_i = \frac{n_i}{n}$
the class's share; times 100 it is a percentage
Angle of a pie slice
$\alpha_i = 360^\circ \cdot \frac{n_i}{n}$
a full circle is 360°, so the angle is the share in degrees
Histogram class width
$h = \frac{x_{\max} - x_{\min}}{k}$
k — the number of classes; the range of the variance lesson cut into equal pieces
Frequency density
$g_i = \frac{n_i}{h_i}$
the bar height for UNEQUAL classes — not the count itself
Apparent ratio on a truncated axis
$\frac{H_B}{H_A} = \frac{x_B - c}{x_A - c}$
c — where the axis starts; at c = 0 it collapses to x_B / x_A

A histogram of the commute times of forty people, in ten-minute classes. The bars touch because the classes touch — the x axis is continuous, not split into categories.

The same poll result on an axis that starts at zero: 42 per cent against 46 per cent. A four-point gap looks exactly like what it is.

The same two numbers on an axis truncated to the range 40 to 48. Bar B is drawn three times taller than A, although support differs by under ten per cent of its value.

Frequently asked questions

Luminous histogram bars settling into the soft bell of a distribution, one outlying point brighter than the rest, against a cosmic nebula in violet and magenta

Statistics and probability

The average person does not exist

All of descriptive statistics rests on a single trade: you give up information about a crowd and receive one number in return. The trade is often worth making, but it is never free — and it pays to know exactly what gets lost in it. From the mortality tables of 1662, through Quetelet's average man, to the aircraft nobody ever saw.

16 min

Presenting data and misleading charts

Before you start

Where this is used

All formulas

The frequency table

The bar chart

The pie chart

The histogram

How a histogram differs from a bar chart

How to lie with a chart

1. The truncated axis

2. Unequal histogram classes

3. The three-dimensional effect

4. Cherry-picking the scale and the range

5. A mean where the median belonged

Exercises

Practice

Practice

Common mistakes

Formula card

Frequently asked questions

How does a histogram differ from a bar chart?

How many classes should a histogram have?

When is a pie chart the right choice?

Does the vertical axis always have to start at zero?

Why is quoting the mean sometimes an abuse?

What is the three-dimensional effect on a chart?

How can I check quickly whether a chart is lying?

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The average person does not exist