Empirical Rule Graph Generator

Mean (μ)

Standard Deviation (σ)

68%

Within 1σ

μ ± σ

95%

Within 2σ

μ ± 2σ

99.7%

Within 3σ

μ ± 3σ

Use the tool at the top of this page to visualise the 68–95–99.7 rule as a single bell curve. Enter a mean and a standard deviation, click Generate Graph, and the 68%, 95% and 99.7% bands are shaded together, so you can see at a glance what counts as typical, what is unusual and what is extreme.

For the background, see What the empirical rule means, or go straight to the Empirical Rule Calculator for the main calculation tool.

What this graph shows

For data that follow a normal distribution, values form a symmetric, bell-shaped curve centred on the mean (μ). The empirical rule describes three intervals around that centre:

68% within 1σ: μ − σ to μ + σ.
95% within 2σ: μ − 2σ to μ + 2σ.
99.7% within 3σ: μ − 3σ to μ + 3σ.

The graph turns those intervals into shaded bands on a single curve. The peak sits at the mean, most values cluster near it, and the shaded area shrinks quickly into the tails: only about 0.3% of values fall outside μ ± 3σ. The diagram beside this text shows the same bands on a single bell curve.

How to read the shaded SD bands

Read the graph from the centre outward. The bands are nested and cumulative, not separate slices.

68% within 1σ: the central area, from μ − σ to μ + σ.
95% within 2σ: from μ − 2σ to μ + 2σ. The bands between 1σ and 2σ add about 27% to the central 68%.
99.7% within 3σ: from μ − 3σ to μ + 3σ. The bands between 2σ and 3σ add about 4.7% more.

About 0.3% of values fall outside μ ± 3σ, in the tails. These are rounded figures: the exact values are 68.27%, 95.45% and 99.73%, so the exact steps between bands are 27.18% and 4.28% rather than the rounded 27% and 4.7%. For a single value, note that μ + 1σ is about the 84.13th percentile — see the Empirical Rule Percentile Calculator.

How mean and SD move the bands

Two numbers control the whole graph, and they do different jobs. The mean (μ) sets the centre: raise it and the entire curve and all three bands slide to the right; lower it and they slide to the left, without changing their shape. The standard deviation (σ) sets the spread: a larger σ stretches the curve wider and flatter, with a lower peak, and pushes the band edges further from the mean, while a smaller σ squeezes everything closer to the centre. The percentages do not change — 68%, 95% and 99.7% still apply — but the values at μ ± 1σ, μ ± 2σ and μ ± 3σ move with μ and σ.

Try it in the tool at the top: keep the mean fixed and double the standard deviation, and you will see every band widen by the same factor. That is why a graph, rather than the three percentages alone, is so useful: it shows at a glance whether a value is typical, unusual or extreme for that particular distribution.

Worked examples

Example 1 — mean 100, standard deviation 15. This is the classic IQ-style case, and it is the example to test in the tool above. The arithmetic is mean ± 1σ, ± 2σ and ± 3σ:

68% within 1σ: 100 − 15 = 85 to 100 + 15 = 115.
95% within 2σ: 100 − 30 = 70 to 100 + 30 = 130.
99.7% within 3σ: 100 − 45 = 55 to 100 + 45 = 145.

So a value of 130 sits exactly at μ + 2σ, on the upper edge of the 95% band, and a value of 145 sits at μ + 3σ, on the upper edge of the 99.7% band.

Example 2 — mean 60, standard deviation 5. Enter 60 and 5 in the tool to verify this second case:

68% within 1σ: 60 − 5 = 55 to 60 + 5 = 65.
95% within 2σ: 60 − 10 = 50 to 60 + 10 = 70.
99.7% within 3σ: 60 − 15 = 45 to 60 + 15 = 75.

Here a value of 75 is μ + 3σ and a value of 45 is μ − 3σ. Both examples use the same bands; only the centre and the width change.

How this graph differs from the bell-curve generator and the z-score plotter

This page is the SD-interval graph: it always shows all three 68%, 95% and 99.7% bands together on one curve, so you can compare the intervals at a glance. The Bell Curve Generator is a more general curve builder — it draws a custom normal curve for any mean and standard deviation, with its own statistics and download, rather than focusing on the three SD intervals. The Z-Score Graph Plotter answers a different question: it plots one z-score (or a tail between two z-scores) on the standard normal curve and shades that tail, rather than showing the 68–95–99.7 bands for a mean and SD. If you want to shade a single chosen range instead, use the Standard Deviation Shading Tool.

Common misconceptions

Not every dataset follows this rule. It applies only to data that are approximately normal — roughly bell-shaped and symmetrical. Skewed data, two-peaked data and very small samples need a different approach; run a normality check first if you are unsure.
The percentages are areas, not guarantees. 68% means about 68% of values in a normal distribution fall in that band. It does not guarantee that exactly 68% of any particular sample will.
Standard deviation is not a probability. It measures spread. The probability comes from the area under the curve over a range.
Curve height is not probability. Height is density. For a continuous variable, the probability of one exact value is zero; probability is always an area over an interval.
Beyond 3σ is rare, not impossible. About 0.3% of values lie outside μ ± 3σ. Check such a value carefully, but do not automatically delete it as an error.
95% here is not a 95% confidence interval. The 95% band describes where 95% of values in the distribution fall. A confidence interval, explained in Empirical Rule vs. Confidence Interval, is a different idea about estimating a population value from a sample.

FAQs

What is an empirical rule graph generator?

It is the tool at the top of this page. Enter a mean and a standard deviation and it draws one bell curve with the 68%, 95% and 99.7% standard-deviation bands shaded together, so you can see the intervals at a glance.

What do I enter?

Only two numbers: the mean (μ) and the standard deviation (σ). Then click Generate Graph. The standard deviation must be greater than zero; a zero or negative value cannot define a spread.

How do I read the 68%, 95% and 99.7% bands?

Start at the mean in the centre and read outward. The 68% band is μ ± 1σ, the 95% band is μ ± 2σ and the 99.7% band is μ ± 3σ. Each percentage is the total area up to that band, not a separate slice.

Why are the bands nested, not separate slices?

Because each wider interval contains the narrower one. The 95% band includes the central 68% plus the two bands between 1σ and 2σ, which add about 27%. The 99.7% band adds a further 4.7% between 2σ and 3σ. Do not add 68%, 95% and 99.7% together.

Are 68%, 95% and 99.7% exact?

No, they are rounded for convenience. The more exact values for a normal distribution are 68.27% within 1σ, 95.45% within 2σ and 99.73% within 3σ. The tool and this page use the rounded figures, as is standard for the 68–95–99.7 rule.

Can I use this graph generator for non-normal data?

No. The 68–95–99.7 percentages hold only for approximately normal data. For skewed or otherwise non-normal data, use a different method, such as Chebyshev’s theorem, which gives weaker bounds that apply more broadly. If you are unsure, run a normality check first.

How is this different from the bell-curve generator and the z-score plotter?

This page always shows all three SD bands together for your mean and SD. The bell-curve generator builds a general custom normal curve, while the z-score plotter plots one z-score, or a tail between two z-scores, on the standard normal curve.

What does a value outside 3σ mean?

It is rare but not impossible: about 0.3% of values in a normal distribution fall outside μ ± 3σ. Check the measurement and the data first, but do not automatically delete it as an error.