Histogram maker

What is a histogram?

A histogram splits a range of numbers into equal slices, called bins, and draws a bar for each one whose height is how many values landed inside it. The bars sit side by side with no gaps, because the bins cover a continuous range with nothing missing between them.

It answers a question a list of numbers cannot: where do the values pile up. Twenty exam marks between 58 and 97 tell you almost nothing as a list. Grouped into bins of 10, you can see at a glance whether most of the class sat in the 70s or whether the marks split into a strong group and a struggling one.

The bars only make sense if the bins are the same width. A bar twice as wide covering twice the range will look twice as important even when it is not, which is the one rule most hand-drawn histograms break.

How to make a histogram

Four steps, whether you are doing it by hand or checking what software gave you:

1. Find the smallest and largest value. The distance between them is the range.
2. Decide how many bins you want. Five to 15 is the usual band, and the rules below give you a starting number.
3. Divide the range by the bin count to get a bin width, then round it to something readable such as 5 or 10.
4. Count how many values fall in each bin and draw a bar that tall.

Worked example

Twenty heights in centimetres, from 158 up to 185. The range is 27.

Sturges suggests 6 bins for 20 values, and 27 divided by 6 is 4.5. Rounding the width up to 5 and starting at 155 gives six bins: 155 to 160, 160 to 165, and so on up to 185.

• 155 to 160: 1 value
• 160 to 165: 1
• 165 to 170: 4
• 170 to 175: 7
• 175 to 180: 4
• 180 to 185: 3

The counts add back to 20, which is the check worth doing every time. The tallest bar is 170 to 175, the mean is 172.1 and the median is 172. Mean and median sitting almost on top of each other, with the tall bar in the middle, is what an even distribution looks like. Click “Heights” in the calculator above to see it drawn.

How many bins should a histogram have?

There is no correct answer, only answers that are easier or harder to read. Too few bins and two groups get merged into one bar. Too many and every bar is 1 or 2 values tall, which is a list of numbers with extra steps.

The OpenStax Statistics textbook puts the practical band at five to 15 bars for clarity, and suggests the square root of the number of values as a quick starting point. Four other rules are in common use, and the calculator lets you switch between all five:

Scott and Freedman-Diaconis give you a bin width rather than a count, and the count falls out of dividing the range by that width. The 3.49 constant comes from David Scott’s 1979 Biometrika paper, which worked out the bin width that minimises error for normally distributed data. Both rules depend on spread, so they react to your standard deviation, and through it your variance, in a way the counting rules do not.

The honest way to use them is to try two or three and see whether the picture changes. If the shape survives Sturges, square root and Freedman-Diaconis, it is in the data. If it appears under one rule and vanishes under another, it was in the binning.

The frequency distribution table

Every histogram has a table underneath it, and for a lot of coursework the table is the part being marked. The calculator prints it with four columns beyond the bin itself.

• Frequency: how many values landed in the bin. These are the bar heights.
• Relative frequency: the count divided by the total. A bin holding 7 of 20 values has a relative frequency of 0.35.
• Cumulative frequency: a running total as you move up through the bins.
• Cumulative percent: the same running total as a percentage, which answers questions like what share of the class scored under 80.

A relative frequency histogram is the same chart with the y-axis switched from counts to proportions. The bars keep their shape and only the scale changes, which matters when you want to compare two groups of different sizes. Twelve out of 20 and sixty out of 100 are both 0.6, and only the relative version shows that. The Frequency, Relative and Percent buttons under the chart switch between the three.

What the shape of a histogram tells you

The outline of the bars is the reason to draw the thing. Five shapes cover most real data, and the calculator names the one it finds along with its reasoning. A histogram shows you that shape and a five number summary puts numbers on it, which is why the two are usually reported together.

Symmetric and bell shaped

One peak near the middle with the bars falling away evenly on both sides. Mean and median land in nearly the same place. Heights, measurement errors and exam marks from a large group tend to look like this. Overlay a standard deviation graph on a histogram with this shape and the two line up closely.

Right skewed

The peak sits on the left and a thin tail stretches right. Salaries are the standard case: twenty salaries running from 22,000 to 240,000 have a mean of 57,300 and a median of only 34,000, because a handful of large values drag the mean up while leaving the median alone. Whenever the mean sits well above the median, expect a right skewed histogram, and read our note on mean vs median before reporting the average.

Left skewed

The mirror image. The peak is on the right and the tail runs left, which pulls the mean below the median. Test scores on an easy exam do this, since most people cluster near the top and a few low marks stretch the bottom.

Bimodal

Two separate peaks with a dip between them. This almost always means two different groups are sitting in one data set: two machines, two age groups, two shifts. The fix is to split the data and draw a histogram for each, because a single mean describing both peaks describes neither.

Uniform

Every bar is about the same height, so no value is more common than any other. Dice rolls and random numbers look like this, because every outcome is equally likely. With a small sample it is easy to see a flat histogram that is really just noise, so the calculator only calls a shape uniform when there are at least five values per bin to judge it on.

Histogram vs bar graph

They look similar and answer different questions. A histogram shows one numeric variable split into ranges. A bar graph compares separate categories.

• A histogram has no gaps between the bars, because the bins run into each other along a number line. A bar graph has gaps, because the categories are not next to anything.
• Histogram bars cannot be reordered. Moving the 70 to 80 bin next to the 90 to 100 bin would break the axis. Bar graph bars can be sorted any way you like.
• The x-axis of a histogram is numeric and continuous. The x-axis of a bar graph holds labels such as country, brand or colour.
• Bin width changes a histogram. Nothing equivalent exists for a bar graph, where one bar is one category, or for a pie chart, where one slice is one share of the total.

Sales by month is a bar graph, because months are categories. The distribution of order values is a histogram, because order value is a number that can land anywhere in a range.

Making a histogram in Excel and Google Sheets

Excel has had a built-in histogram chart since 2016. Select your column, open Insert, choose the statistic chart icon and pick Histogram. Excel chooses the bins for you, and you change them by right-clicking the x-axis, opening Format Axis and setting either the bin width or the number of bins.

Google Sheets does it through Insert, then Chart, then Chart type, then Histogram chart. Bin width sits under the Customise tab in the Histogram section. Sheets offers fewer bin widths than Excel does, which is worth knowing before you fight it.

Both of them hide the frequency table. If you need the counts written out, or you want to see what a different binning rule would do, paste the same column into the calculator above and you get the chart and the table together.