For data that follows a normal distribution (the bell curve), three numbers cover almost everything you need to know about probability. The 68-95-99.7 rule — also called the empirical rule — connects standard deviation directly to the percentage of data near the mean.

The rule

For any normal distribution with mean μ and standard deviation σ:

  • About 68% of data falls within 1 SD of the mean (μ ± σ)
  • About 95% of data falls within 2 SDs (μ ± 2σ)
  • About 99.7% of data falls within 3 SDs (μ ± 3σ)

That's it. Memorize this and you can answer probability questions about normal data with mental math.

Worked example: SAT scores

For the class of 2025, SAT total scores had a mean of 1029 and a standard deviation of 235, according to the College Board's 2025 annual report. Treating the scores as roughly normal:

  • About 68% of test-takers score between 794 and 1264 (1029 ± 235).
  • About 95% score between 559 and 1499 (1029 ± 470).
  • The 3-SD band (1029 ± 705) would run past both ends of the 400–1600 score scale, so the 99.7% figure can't be applied literally here. Real scores stop at 400 and 1600.

A 1450 is (1450 − 1029) / 235 ≈ 1.79 SDs above the mean. Under a normal model, about 3.7% of test-takers would score higher, so a 1450 sits near the top 4%. For comparison, a score exactly 2 SDs above the mean would be in the top 2.5%: 95% of scores fall within ±2 SD, and the upper tail is half of the remaining 5%.

Worked example: adult heights

U.S. adult male heights ≈ mean 69 in, SD 3 in. CDC measurements for 2015–2018 put the average for men 20 and over at 69.0 inches, and their 5th to 95th percentile range, 64.1 to 73.8 inches, matches an SD of about 3 inches.

  • 68% of men are between 66" and 72" (5'6" to 6'0").
  • 95% are between 63" and 75" (5'3" to 6'3").
  • 99.7% are between 60" and 78" (5'0" to 6'6").

So a 7-foot man (84 in) is 5 SDs above the mean — under a normal model, about 1 in 3.5 million men. Now you know why NBA players stand out.

Why these specific percentages?

The percentages come from integrating the normal distribution's bell-shaped probability density function. The integral from μ−σ to μ+σ works out to ~0.6827; from μ−2σ to μ+2σ to ~0.9545; from μ−3σ to μ+3σ to ~0.9973.

The "68-95-99.7" labels are approximations rounded for memorability. The exact values are 68.27%, 95.45%, 99.73%. Close enough for almost any practical use.

Z-scores: how many SDs from the mean

A z-score measures how many SDs a value is from the mean:

z = (x − μ) / σ

If your test score is 1450 and the mean is 1029 with SD 235, your z-score is (1450 − 1029) / 235 ≈ 1.79. You're about 1.8 SDs above the mean.

Z-scores let you compare values across different distributions. A 90th-percentile score on the SAT (z ≈ 1.28) is statistically equivalent to a 90th-percentile score on the ACT (also z ≈ 1.28), even though the raw numbers are very different.

Tail probabilities

For more precise probabilities beyond 3 SDs, use a z-table or any statistics calculator. Some useful values:

  • z = 1.28: 90th percentile (top 10%)
  • z = 1.645: 95th percentile (top 5%)
  • z = 1.96: 97.5th percentile (top 2.5%) — used for 95% confidence intervals
  • z = 2.33: 99th percentile (top 1%)
  • z = 2.575: top 0.5% — used for 99% confidence intervals

What "normal" actually means

A normal distribution is the bell-shaped curve that arises from many independent factors adding together. Heights, test scores, blood pressure, machine measurement errors — all tend toward normal because each is the sum of many small effects.

This is the Central Limit Theorem: averages of large samples are approximately normal regardless of the underlying distribution. Hence the rule's wide applicability.

Where the rule fails

Real-world data isn't always normal. Watch out for:

  • Income: heavily right-skewed. The mean is much higher than the median because billionaires pull the tail. The 68-95-99.7 rule does not apply.
  • House prices: right-skewed and bimodal in many markets.
  • Stock returns: approximately normal in the middle but with fat tails — extreme moves happen more often than the rule predicts.
  • Binary outcomes: coin flips, yes/no surveys — these are binomial, not normal (though they approach normal in large samples).

Always check whether your data is approximately normal before applying the rule. A histogram and quick bell-shape sanity check is enough.

Quality control: the "six sigma" idea

Manufacturing aims for processes that are at least 6 SDs from any defect threshold. The "six sigma" methodology is named after this: drive variation low enough that 6 SDs fits comfortably between mean and any defect line. Its well-known target of 3.4 defects per million opportunities assumes the process mean drifts by 1.5 SDs over the long term, so it is really the tail beyond 4.5 SDs. Without that shift, the chance of landing beyond 6 SDs is about 0.002 per million, counting both tails.

On the same convention, a 3-sigma process produces about 66,807 defects per million opportunities (6.7%). Even the unshifted figure, one in 740 beyond 3 SDs on one side, means a major-airline operation pumping out millions of flights per year would have thousands of errors. Six-sigma is the aspiration.

Confidence intervals

The 68-95-99.7 rule underlies confidence intervals. A "95% confidence interval" is mean ± 1.96 × (standard error). The 1.96 is the exact z-score that captures 95% of the area under the normal curve; the rule's "2 SDs" is its rounded version.

Compute SDs from your data

Our standard deviation calculator computes the SD of any list of numbers. Pair that with the rule above to estimate where most of your data lies, and to spot potential outliers (anything beyond 2–3 SDs is statistically unusual under a normal model).

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