Class Width in Statistics: How to Calculate It, Round It and Read It From a Histogram

§ 01

What class width is

Every class in a frequency table or histogram is supposed to cover the same width unless the data specifically calls for otherwise — how wide a slice of the data each bar represents, in the same units the data itself is measured in. Comparing bar heights across a histogram only means something when every bar covers equal ground; a class twice as wide as its neighbor draws a bar with twice the area for the same count, and it collects more values just from covering more of the range. This page builds the frequency table and its classes; the rest of the statistics-basics toolkit, from z-scores to populations and unusual values, is on the z table and other statistics basics.

§ 02

How to calculate class width: the formula

The formula: (max - min) / number of classes, rounded up

Class width statistics questions all use one formula: $\text{class width}=$$\text{class width} = \dfrac{\text{max}-\text{min}}{\text{number of classes}}$, rounded up to a convenient value. Sturges' rule, $1+$$1+3.322\log_{10}n$, is one common way to pick the number of classes before you compute the width, if the assignment doesn't specify one.

A worked example from a raw data set

Twelve exam scores: 62, 65, 70, 71, 74, 78, 81, 85, 88, 91, 95, 99. Range: $99-62=37$. With 5 classes, the raw width is $37 \div 5 = 7.4$, rounded up to 8.

§ 03

How to determine the number of classes

How to determine class width starts one step earlier than the formula above: picking how many classes to use in the first place, since the formula needs that number before it can run. How to determine class width in statistics, when an assignment doesn't hand you the number of classes directly, means picking one of the two rules below rather than guessing.

Sturges' rule, worked

Sturges' rule is $k=$$1+$$k=1+3.322\log_{10}n$, where $n$ is the number of data points. For $n=50$: ${\log }_{10}50\approx$$\log_{10}50\approx1.699$, so $k\approx$$1+$$3.322(1.699)\approx$$1+$$5.644=$$k\approx1+3.322(1.699)\approx1+5.644=6.644$, rounded to the nearest whole number: 7 classes. Sturges' rule tends to suggest fewer classes for smaller data sets and more for larger ones, which keeps each class holding a reasonable number of data points either way.

The square-root rule, worked

The square-root rule is simpler: $k=\sqrt{n}$. For the same $n=50$: $\sqrt{50}\approx7.07$, rounded to 7 classes — close to Sturges' rule here, though the two rules can disagree on other data sets, since they're two different approximations, not two paths to one guaranteed answer. Either is an acceptable starting point unless an assignment specifies which one to use; both exist only to keep the number of classes reasonable, neither too few to show a shape nor so many that classes sit empty.

§ 04

Class width with decimals: always round up

Rounding down would leave the highest data point, 99, outside every class. How do you find the class width in statistics when the division doesn't come out even? Always round up: a class width of 7 (rounded down from 7.4) only reaches 62+7×5=97, one class boundary short of covering 99.

An exact division needs the same care. With whole-number data, a width that divides the range exactly ends the last class one unit below the maximum, so go up to the next whole number: 10 to 94 in 6 classes gives $84\div6=14$, but six classes of 14 stop at 93, so use 15 (practice item 1 below).

A data set with decimals

Suppose rainfall totals are recorded to one decimal place, from 1.2 to 8.9 inches, split into 6 classes. The range is $8.9-1.2=7.7$ and $7.7\div6\approx1.283$, so round up at the data's own precision to a class width of 1.3. The classes run 1.2–2.4, 2.5–3.7, 3.8–5.0, 5.1–6.3, 6.4–7.6 and 7.7–8.9, and the last one ends exactly at the maximum. The boundaries sit halfway between classes (1.15, 2.45, 3.75 and so on), still 1.3 apart.

§ 05

Class width of a histogram

A class width histogram makes the formula visible: instead of subtracting and dividing, just read two neighboring labels off the x-axis and subtract one from the other. This figure marks each bar at its lower class limit, although its axis title calls those marks class boundaries; under the ±0.5 convention in the table below, the true boundaries are 61.5, 69.5, 77.5 and so on. Either way, the marks are 8 apart. Every gap between consecutive marks on a correctly drawn class width histogram is identical, so uneven gaps flag a mistake in the histogram or in its table. This is also the fastest way to answer how to calculate the class width when you're handed a finished histogram instead of a raw data set: skip the formula entirely and read the axis.

§ 06

When class width is not uniform

Not every frequency table uses one width throughout. Household income brackets are a common example: $0–$24,999 and $25,000–$49,999 (width 25,000 each), then $50,000–$99,999 (width 50,000), then an open-ended $100,000+ class. Unequal widths group the data more evenly when most values cluster at one end and thin out at the other, so heavy skew is the usual reason class width is not uniform, though many skewed data sets are still tabulated with equal widths.

§ 07

Class width vs class boundaries vs class limits

The three terms describe the same frequency table from three different angles, and mixing them up is a common source of wrong answers on an otherwise correct calculation.

Class limits come straight from the data and can never overlap between classes; class boundaries exist specifically to remove the gap between one class's limit and the next class's limit, which is what makes subtracting them give a clean class width. Without that half-unit adjustment, subtracting one class's lower limit from the next class's lower limit still works for whole-number data — $70-62=8$ matches the width found earlier — but boundaries are the version that generalizes to decimal data too, where limits alone would leave a gap between classes.

§ 08

Practice: five data sets

1. Min 10, max 94, 6 classes — range $84$, $84\div6=14$ exactly, but six classes of 14 starting at 10 end at 93 and miss 94, so go up to the next whole number. Class width: 15.
2. Min 5, max 100, 8 classes — range $95$, $95\div8=11.875$. Class width: 12 (rounded up).
3. Min 0, max 49, 5 classes — range $49$, $49\div5=9.8$. Class width: 10 (rounded up).
4. Min 120, max 245, 7 classes — range $125$, $125÷$$7\approx$$125\div7\approx17.857$. Class width: 18 (rounded up).
5. Min 1, max 60, 10 classes — range $59$, $59\div10=5.9$. Class width: 6 (rounded up).

Once a frequency table like the ones above has counts instead of probabilities in its right-hand column, turning a frequency table into a probability distribution is the next step — divide each count by the total and the two checks on that page take over from here.

§ 09

When the frequency-table assignment is due

A single class width takes a minute with the formula above; a full frequency-distribution assignment with several data sets rarely does, so most students turn to homework help by subject for the rest of it, and GradeDraft's statistics desk handles the whole statistics assignment, worked start to finish.