Standard deviation graph

What the graph shows

Three things are drawn on top of each other, and each answers a different question.

• The bars are your data. The range is split into equal-width bins — Sturges' rule, ⌈log₂ n + 1⌉, sets how many — and each bar's height is the count of values that landed in that bin.
• The curve is the normal distribution with the same mean and standard deviation as your data, scaled so that its height in any bin is the number of values you would expect there if the data were perfectly normal. It is not fitted to the bars; it is fitted to the two summary numbers.
• The bands are centred on the mean and reach one, two and three standard deviations either side. The mean itself is the vertical line.

The preloaded example is 30 exam scores. Their mean is 75.73 and their sample standard deviation is 8.58, so the inner band runs from 67.15 to 84.32. Sturges gives six bins of width 5.83, holding 3, 4, 8, 7, 5 and 3 scores — a shape the curve follows closely, which is what a healthy fit looks like.

How to read standard deviation from a graph

On a smooth bell curve the standard deviation is visible as a distance. Starting at the peak, the curve falls away ever more steeply until it reaches a point where it begins to flatten out again. That is an inflection point, and there is one on each side. They sit at exactly μ − σ and μ + σ, so the horizontal gap from the peak to either inflection point is the standard deviation. A wide, low curve has a large σ; a tall, narrow one has a small σ. The peak height is 1 ÷ (σ√(2π)), so doubling the standard deviation halves the peak.

A histogram has no inflection points to read, but its range gives a serviceable estimate. Values from a normal distribution rarely stray beyond a couple of standard deviations, so the full width of the data spans roughly four standard deviations for a sample of about 30, and roughly six once you have several hundred values, because a larger sample has more chances to produce something far out in the tail.

σ ≈ range ÷ 4 (n around 30) σ ≈ range ÷ 6 (n in the hundreds)

For the example data the range is 93 − 58 = 35, and 35 ÷ 4 = 8.75 against the true 8.58. Treat these as rough guides for sanity-checking a number, not as methods for producing one. They fail on skewed data, where the range is dominated by one long tail, and on any dataset with an outlier.

Comparing the bars with the curve

The mean and standard deviation summarise a dataset in two numbers, and the overlay is a check on how much those two numbers leave out. When the bars hug the curve, they leave out little. When they do not, the pattern of the mismatch tells you what is going on.

• Skew. The tallest bar sits to one side of the mean line and a run of short bars trails off on the other. Right-skewed data — incomes, response times, anything bounded at zero — piles up left of the mean with a long tail to the right. The median will be noticeably lower than the mean.
• Two peaks. Two tall bars with a dip between them, and the curve peaking in the dip where hardly any data sits. Almost always two groups mixed together — two machines, two cohorts, two conditions — and the sensible move is to separate them and graph each.
• An outlier. A single short bar far from the rest, with empty bins between it and the main body. Its effect is on the curve rather than the bars: it drags the mean towards it and inflates the standard deviation, so the curve becomes wider and flatter than the bulk of the data deserves.
• Heavy tails. The centre bar overshoots the curve's peak and the outermost bars overshoot its tails, with a shortfall in between. More extreme values than a normal model predicts, which matters if you are using σ to set limits.

The 68–95–99.7 check

For a normal distribution, 68.3% of values lie within one standard deviation of the mean, 95.4% within two and 99.7% within three. The three "within ±k SD" cells in the results report what your data actually does, so you can compare without reading the chart by eye. The example gives 20 of 30 within one SD (66.7%), 28 of 30 within two (93.3%) and all 30 within three. The normal expectation for 30 values is about 20.5, 28.6 and 29.9, so the match is as close as a sample this size can give.

Expect the ±1 cell to wander. It is the most informative of the three, but with 30 values each observation is worth 3.3 percentage points, and a perfectly normal sample will land anywhere from the low 60s to the high 70s. The ±3 cell is nearly useless at this size — 99.7% of 30 is 29.9, so one value outside is already more than the rule predicts, and none outside proves nothing. Both cells become sharper as n grows. The empirical rule calculator turns any mean and SD into the exact band edges.

When the normal overlay misleads

The curve is drawn from the mean and standard deviation whether or not those numbers are a sensible summary. On a strongly skewed dataset they are not: the mean is pulled towards the tail, the standard deviation is stretched by it, and the curve ends up centred on a value that is not typical, with a lower band that may extend below values the data cannot take, such as zero. The bars will show this plainly, and that is the point of drawing them.

If that is what you see, switch summaries. The median gives a centre that the tail cannot drag, and the interquartile range gives a spread measured on the middle half of the data, with Tukey's fences for flagging outliers on a box plot. Report those instead of mean and SD, or transform the data — a log transform often makes a right-skewed variable close to normal — and graph the transformed values here.

Making a standard deviation chart in Excel

Excel 2016 and later will draw the histogram directly: select the column, then Insert → Statistic chart → Histogram, and set the bin width in the axis options. Get the summary numbers with =AVERAGE(range) and =STDEV.S(range). To add the curve, make a column of x values across the data range and next to it =NORM.DIST(x, mean, sd, FALSE) * n * width, which scales the density to the bar heights, then plot that column as a line. The Excel guide goes through the functions and the sample-versus-population choice.

Note that "standard deviation chart" also has a second meaning in Excel: a column chart of group means with error bars set to one standard deviation. That is a comparison of groups, not a picture of a distribution, and it is built from the Error Bars option on a chart element rather than from a histogram.

Finally, this page and the bell curve generator draw the same curve from opposite directions. The generator takes a mean and a standard deviation and draws the ideal shape, which is what you want when the parameters are given — a test with a known scale, a specification limit, a textbook problem. This page starts from raw data and lets you judge whether the ideal shape applies at all.