Does the bell curve fit your data?
Many methods quietly assume roughly normal data. This tool makes the assumption visible: it draws your histogram and lays over it the normal curve that has your data's own mean and standard deviation. Where the curve threads the bars, the assumption looks reasonable. Where the bars pile up somewhere the curve is thin (a second hump, a long tail, a cliff at zero), the assumption is doing work it should not be asked to do.
The commute data: mean 28.15, SD 4.04. The fitted curve hugs the histogram closely; 65% of values fall within one SD against the theoretical 68.3%, and 95% within two against 95.4%. Skewness is near zero. Verdict for this dataset: normal-based summaries and intervals are reasonable. Change the data to incomes and the same picture would show the curve's left half spilling below zero while a long bar tail escapes right: the model failing in plain sight.
The 68-95-99.7 yardstick
For a perfect normal curve: 68.3% within one SD, 95.4% within two, 99.7% within three. The tool tabulates your data's own percentages next to those benchmarks. Small samples wobble around the benchmarks naturally (twenty points cannot hit 68.3 on the nose; each point is 5%). Look for gross misses and shape problems, not decimal fidelity.
Testing normality by eyeball alone on tiny samples, then either relaxing or panicking. Ten points from a perfect normal distribution often look lumpy; ten thousand points reveal every real deviation. Match the strictness to the stakes and the sample size, and use a formal test (Shapiro-Wilk and friends) when the decision matters.
When the fit fails, what then?
Skewed data often normalizes under a log transform (incomes, reaction times). Two humps mean two populations: split before modelling. Heavy tails mean medians and IQRs should lead the summary. The graph's job is not to pass or fail your data; it is to show you which summary language the data speaks.
Frequently asked questions
What curve is drawn over my histogram?
The normal distribution with your data's mean and sample standard deviation, scaled so its total area matches the histogram's bar area. It is the best-fitting bell for this data, not a generic one.
How close should the percentages be to 68-95-99.7?
With small samples, within several points either way by chance alone. Systematic patterns (both tails heavy, one side missing, percentages off in the same direction at 1 and 2 SD) are the signal, not small wobbles.
Does non-normal data forbid means and t tests?
Not outright. Means still describe; t procedures tolerate mild non-normality, especially with larger samples, thanks to the central limit theorem. Strong skew or heavy tails call for medians, transforms, or rank-based tests instead.