Histogram maker

What a histogram shows

A histogram takes one continuous variable, cuts its range into intervals of equal width, and draws a bar for each interval whose height is the number of values that fell inside it. The bars touch because the intervals do: there is no gap between "23 to 28" and "28 to 33". Read left to right, the picture tells you where the values cluster, how far they spread, and whether the spread is even on both sides of the centre.

That is what separates it from a bar chart. A bar chart compares categories — regions, products, months — and its bars can be shuffled into any order without losing anything. A histogram's bars are fixed to the number line, and their area carries the meaning. The sample above is 32 delivery times in minutes. The histogram maker puts them in 6 bins of width 5, from 18 to 48, with counts of 4, 13, 9, 3, 2 and 1. The bulk of deliveries take between 23 and 33 minutes; a thin tail runs out to the right.

How the bins are chosen

k = ⌈log₂ n + 1⌉ width = (max − min) / k

The number of bins comes from Sturges' rule, published in 1926 and still the default in R's hist() and in Excel. For the 32 delivery times, log₂ 32 is 5, so k = 6, and the width is (48 − 18) / 6 = 5 minutes. The first bin starts at the minimum and the last one ends at the maximum, closed on the right so the largest value is counted exactly once.

Sturges assumed the data is roughly normal and derived the count from the binomial distribution. That gives it two known weaknesses. It grows only logarithmically, so on a large sample — say 5,000 values, where it still offers just 14 bins — it oversmooths and can hide real structure. And on strongly skewed data it forces most values into one or two bars on the crowded side while the tail gets bins that hold almost nothing.

Two alternatives are worth knowing. The square-root rule, k = ⌈√n⌉, is what many spreadsheet tools use; it agrees with Sturges here (√32 rounds up to 6) but gives many more bins as n grows. The Freedman–Diaconis rule sets the width rather than the count, at 2 × IQR / ∛n, so a few extreme values cannot stretch the bins; on the sample data it asks for 9 bins of about 3.5 minutes. Neither is implemented here, and none of the three is right for every dataset. If the bars look like a comb or a block, the rule has misjudged your data, not the other way round.

Reading the shape

• Symmetric, one peak — the bars fall away evenly from the middle and mean and median almost coincide. A mean and standard deviation describe this well, and the normal curve will sit close to the bars.
• Right skew — a long tail to the right, mean above the median. Typical of delivery times, incomes, response times: anything with a hard floor and no ceiling. The sample data does this mildly, with a mean of 28.3 and a median of 27.
• Left skew — the mirror image, with a tail to the left. Common in scores on an easy test or ages at death.
• Bimodal — two peaks. Usually two populations mixed together: two machines, two shifts, two species. The mean lands between the humps, where hardly any values sit, and describes nothing.
• Outliers — an isolated bar far from the rest, often a single value. It pulls the mean and can inflate the standard deviation a great deal while leaving the median untouched.

The shape decides which summary to report. Symmetric and single-peaked: mean plus standard deviation. Skewed, bimodal or contaminated by outliers: median plus interquartile range, which the IQR calculator gives you along with the outliers flagged. The mean, median and mode calculator shows all three centres side by side, and the gap between mean and median is itself a quick measure of skew.

Standard deviation from a histogram

Sometimes the histogram is all you have — a figure in a report, with no raw data behind it. You can still recover a mean and standard deviation by treating each bar as if every value in it sat at the bar's midpoint. Multiply each midpoint by its frequency, sum, and divide by the total count to get the mean; then weight the squared deviations of the midpoints by the same frequencies for the variance.

x̄ ≈ Σ(f × m) / n s² ≈ Σ f (m − x̄)² / (n − 1)

On the sample data this gives a mean of 28.78 and a standard deviation of 6.04, against the true 28.31 and 6.11 from the raw values. The error comes from values not really being at their midpoints — the first bin's four values average 20.75, not 20.5 — and shrinks as bins get narrower. The grouped data standard deviation calculator does the whole computation from a table of intervals and frequencies, with the working shown.

The fitted normal curve

The curve drawn over the bars is the normal distribution with your data's mean and standard deviation, scaled so that its height at any point is the number of values a bin of that width would be expected to hold if the data were normal. It is a reference, not a fit in the least-squares sense: it uses only two numbers from your data and ignores the shape entirely.

That is what makes it useful. Where the bars and the curve agree, the mean and SD are doing their job, and tools that assume normality — the normal distribution calculator, z-scores, the 68–95–99.7 rule — will give sensible answers. Where they disagree, the mismatch is the diagnosis. On the delivery times the tallest bar, 23 to 28, holds 13 values where the curve expects about 9, the 33-to-38 bar holds 3 against an expected 5, and the last two bars hold more than the curve allows for. A crowded centre just left of the mean and a stretched tail to the right is what right skew looks like against a symmetric reference. The standard deviation graph draws the same histogram with the ±1, ±2 and ±3 SD bands shaded, so you can count how many of your values fall inside each.

Dot plots for small samples

With 16 values or fewer, Sturges' rule gives at most five bins, and a histogram turns into a handful of blocks that could be hiding anything. A dot plot — one dot per value, stacked where values repeat or nearly coincide — loses nothing, because there is nothing to bin. Every value stays visible, an outlier is unmistakable, and a gap in the data shows as a gap rather than being averaged into a bar. Most other calculators on this site draw one under the results for exactly that reason. Use this histogram maker when n is large enough that individual points would blur into a smear, and a dot plot when it is not.