Class Width Calculator

What class width controls in a histogram

A histogram splits the range of a data set into equal-sized bins and counts how many values fall in each one. The NIST/SEMATECH e-Handbook of Statistical Methods describes the common form as splitting the range into equal-sized bins, called classes, and counting the points in each bin. The class width is the width of a single bin along the horizontal axis.

That one number sets how much detail the chart shows. Narrow bins separate nearby values but leave gaps and a jagged outline when the sample is small. Wide bins smooth the shape but can hide a second peak or a cluster near one end. OpenStax's Introductory Statistics 2e suggests 5 to 15 classes for most data sets, which is why the class count is an input on this page rather than something the tool decides for you.

Three inputs drive the whole calculation: the smallest value, the largest value, and the number of classes you want. Everything else, including the rounding step that most people get wrong, follows from those three.

The class width formula and the round-up rule

The formula is class width equals the range divided by the number of classes, where the range is the largest value minus the smallest. OpenStax Introductory Statistics 2e, section 2.2, gives the same rule and adds the step that matters: round the result up. In its worked case, 74.05 minus 59.95 equals 14.1, and 14.1 divided by 8 equals 1.76, which the text rounds up to 2 units per bar.

Coverage is the reason for rounding up. A width of 1.76 leaves the eighth class ending at 74.03, just short of the largest value at 74.05, so that observation has nowhere to go. Rounding up also keeps data values off the shared edges between classes, which is the second reason OpenStax gives for the rule.

One exception is worth knowing. When the division lands on a whole number, the width can stay as it is. A range of 42 split into 6 classes gives exactly 7, and the last class then ends on the largest value. Some instructors still add a unit so a visible gap remains, and the width override box on this page covers that case.

Worked example: test scores from 52 to 98

Suppose 30 students scored between 52 and 98 on an exam and you want five classes. The range is 98 minus 52, which is 46 points. Divide 46 by 5 and you get 9.2 points. Round up and every class is 10 points wide.

Starting at 52, the classes cover 52 to 61, 62 to 71, 72 to 81, 82 to 91, and 92 to 101. The top score of 98 sits in the last class with three points of room to spare. Nothing falls on an edge, and no score is left outside the chart.

Each midpoint is the average of its two limits. For the first class, 52 plus 61 is 113, and 113 divided by 2 is 56.5. The midpoints then climb by 10, the same as the width, ending at 96.5.

A second case: the 59.95 to 74.05 data

The OpenStax example is a useful check because the numbers are not tidy. The smallest value is 59.95, the largest is 74.05, and the text asks for eight bars. The range is 14.1, the exact width is 1.76, and the rounded width is 2.

Eight classes of 2 units each run from 59.95 to 75.95, which clears the maximum by 1.9 units. That spare room at the top is normal. The last class almost always extends past the largest observation, and that is fine as long as every class uses the same width.

Enter these numbers with a width of 1.76 instead and the eight classes end at 74.03, which leaves 74.05 with no class. That failure is exactly what the round-up rule prevents, and it is why the rule is stated as a rule rather than a preference.

Choosing the number of classes

OpenStax gives two starting points: 5 to 15 classes for readability, and the square root of the number of observations as a rough count. With 150 values, the square root is about 12.25, which rounds to 12 classes.

Sturges' rule, published by Herbert Sturges in 1926, is the other common choice. The number of classes equals 1 plus 3.322 times the base-10 logarithm of the sample size. For 40 observations, the log of 40 is about 1.602, so 3.322 times 1.602 equals 5.32, and adding 1 gives 6.32 classes, which rounds up to 7.

NIST's Dataplot documentation lists further class-width rules, including David Scott's rule based on the sample standard deviation and an interquartile-range rule, for anyone who needs something beyond range over classes. Both simple rules above are starting points, not laws. Skewed data often need a few more classes to show the tail, and a small sample needs fewer so that no bar sits empty. Enter a sample size and the calculator shows both suggestions, then leaves the final choice to you.

Class limits, boundaries, and midpoints

Class limits are the smallest and largest values that belong in a class, written at the precision of the data. For whole-number test scores, the first class runs from a limit of 52 to a limit of 61.

Class boundaries close the gap between neighboring classes. They sit half a unit below the lower limit and half a unit above the upper limit, so the first class is bounded by 51.5 and 61.5, and the second by 61.5 and 71.5. That shared edge is what makes the bins continuous along the axis.

The midpoint, also called the class mark, is the average of the two limits. For 52 and 61 the midpoint is 56.5. Midpoints matter when you only have grouped data and need a mean: multiply each midpoint by its frequency, add the products, and divide by the total count.

Width stays the same whether you measure it with limits or boundaries. The 52 to 61 class is 10 units wide as limits and 10 units wide as boundaries, from 51.5 to 61.5.

Five mistakes that spoil a histogram

• Rounding down, or not rounding at all. A width of 9.2 across five classes puts the top score of 98 exactly on the final edge, where the count becomes a judgment call.
• Mixing widths. One 5-unit class beside four 10-unit classes makes the narrow bar look tall and distorts the shape.
• Starting the first class above the smallest value. The lowest observation then has no bin, and the chart silently drops it.
• Using too many classes for a small sample. Twelve classes for 20 values averages 1.67 observations per bar, so most bars sit empty and the shape is noise.
• Reporting a width without units. A width of 10 means nothing until you say 10 points, 10 dollars, or 10 minutes.

The first mistake is the one to watch, because it can pass a quick eyeball test. A chart with the largest value on the last edge looks complete until you try to count that value.

What the calculator returns

The defaults are the exam score case: 52, 98, and 5 classes. The tool returns the range, the exact width, the rounded width, and the list of class starts.

Change any input and the boundary list rebuilds. If the last class ends below the maximum, the results area says so, because that means the chosen width cannot cover the data. The override box accepts a width you supply yourself, which is useful when a course or a report requires one exact interval.