Is the Empirical Rule the same as the 68-95-99.7 rule?

Yes; 'Empirical Rule' is the technical name, while '68-95-99.7 rule' is the nickname that describes the three main percentages. They refer to exactly the same concept.

Can I use the Empirical Rule even if my data isn't perfectly normal?

Yes, as long as your data is approximately normal (roughly bell-shaped, not heavily skewed). For data that is visibly non-normal, Chebyshev's inequality is safer and will give you bounds that definitely hold.

Does 'within 1\sigma' include the exact values at \mu - 1\sigma and \mu + 1\sigma?

Yes, 'within 1\sigma' means the closed interval from \mu - \sigma to \mu + \sigma inclusive, including both endpoints. In practice, for continuous data, whether you include the boundaries makes almost no difference to the percentage.

Can I use the Empirical Rule to find the probability that a single value falls in a certain range?

Yes, exactly. If your data follows a normal distribution, the percentages from the Empirical Rule tell you the probability; for instance, the probability that a randomly selected observation falls within 1\sigma of the mean is approximately 68%.

Empirical Rule

The Empirical Rule describes where approximately 68%, 95%, and 99.7% of normally distributed data fall, measured in standard deviations from the mean.

68\% \text{ within } 1\sigma, ; 95\% \text{ within } 2\sigma, ; 99.7\% \text{ within } 3\sigma

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What each symbol means

What Empirical Rule takes

Symbol	Meaning
$\sigma$	Sigma represents the standard deviation, a measure of spread (in the same units as your data); confusing it with the mean μ or variance σ² destroys the calculation.

When to use it

Reach for the Empirical Rule when you have normally distributed data and want to quickly estimate what fraction of values lie within a specified distance of the mean.

Level

Usually taught in: Statistics · Appears on: AP Statistics

Worked examples

1. Estimate data within 1 standard deviation

Problem

A dataset of measurements is approximately normally distributed with mean 50 and standard deviation 5. What percentage of data falls between 45 and 55?

$45 = 50 - 5$
The lower boundary is exactly one standard deviation below the mean.
$55 = 50 + 5$
The upper boundary is exactly one standard deviation above the mean.
$5 = 1 \cdot \sigma$
Both boundaries sit at distance $1\sigma$ from the center, so the range is $\mu \pm 1\sigma$ .

Answer: $68\%$

The interval [45, 55] is symmetric about the mean and spans exactly $\mu - 1\sigma$ to $\mu + 1\sigma$ . The Empirical Rule directly gives us 68% without further calculation.

2. Find percentage of data outside a range

Problem

Test scores are approximately normally distributed with mean 72 and standard deviation 8. What percentage of scores fall outside the range 56 to 88?

$56 = 72 - 16$
The lower bound is 16 points below the mean.
$88 = 72 + 16$
The upper bound is 16 points above the mean.
$16 \div 8 = 2$
Dividing by the standard deviation shows both bounds are $2\sigma$ from the mean.
$95\% \text{ within } 2\sigma$
The Empirical Rule says 95% of data falls within $2\sigma$ of the mean.
$100\% - 95\% = 5\%$
Subtract to find the percentage lying outside the given range.

Answer: $5\%$

This problem asks for the 'outside' percentage, which requires an extra step: first apply the rule to find what's inside (95%), then subtract from 100%. Many students forget this final subtraction.

3. Determine normal distribution percentages in a ticket pricing scenario

Problem

A concert venue sells general admission and VIP tickets. General admission prices are approximately normally distributed with mean $85 and standard deviation $3, while VIP prices have mean $150 and standard deviation $6. What percentage of general admission tickets should be priced between $79 and $91?

$\text{GA: } \mu = 85, \; \sigma = 3$
Focus on the general admission tickets, noting the mean and standard deviation.
$79 = 85 - 6$
Calculate how far the lower price is from the mean.
$91 = 85 + 6$
Calculate how far the upper price is from the mean.
$6 \div 3 = 2$
Dividing each distance by $\sigma = 3$ shows both boundaries are $2\sigma$ from the mean.
$95\%$
The Empirical Rule tells us that 95% of data falls within $2\sigma$ of the mean.

Answer: $95\%$

The problem mentions both ticket types to add context, but only GA prices matter for this question. Recognizing that [ $79,$ 91] = $[85 - 2(3), 85 + 2(3)]$ means applying the $2\sigma$ rule directly. The VIP information is a distractor.

Common mistakes

Where Empirical Rule usually goes wrong

The mistake	Why it is wrong	The fix
Treating the Empirical Rule percentages as exact rather than approximate.	The rule derives from observed behavior in real data, not from a strict formula; the actual percentages vary slightly depending on how close the distribution is to perfectly normal.	Remember the rule uses 'approximately': ~68%, ~95%, ~99.7%. For precise probabilities, use the normal distribution calculator or z-tables.
Applying the Empirical Rule to skewed or non-normal data.	The rule only works when data has a bell-shaped (normal) distribution; if data is skewed left, right, or has heavy tails, these percentages will be incorrect.	Always check that your data is approximately normal first (using a histogram, Q-Q plot, or normality test). If not normal, use Chebyshev's inequality instead, which works for any distribution shape.
Confusing 'within $1\sigma$ ' with 'a range of width $1\sigma$ '.	Students sometimes think the interval $[\mu - \sigma, \mu + \sigma]$ has width $\sigma$ , when it actually spans a width of $2\sigma$ .	Write it out carefully: from $\mu - 1\sigma$ to $\mu + 1\sigma$ is a distance of $(\mu + \sigma) - (\mu - \sigma) = 2\sigma$ , not $\sigma$ .

Tips and when to use something else

The Empirical Rule is a quick estimation tool: once you identify a mean and standard deviation for approximately normal data, you can instantly estimate what fraction of data lies in any $1\sigma$ , $2\sigma$ , or $3\sigma$ band without a calculator.
Use Chebyshev's inequality as a fallback when your data is not normally distributed; it gives broader (more conservative) bounds that work for any distribution shape.
The rule becomes unreliable beyond $3\sigma$ : roughly 99.7% of data falls within $3\sigma$ , leaving only 0.3% in both tails combined, so claims about $4\sigma$ or $5\sigma$ should be treated with skepticism unless you have a very large dataset.
Practice the rule with real-world data like SAT/ACT scores, heights, or weights—these tend to be nearly normal, making the 68-95-99.7 percentages obvious when you plot them.

Related formulas

Normal DistributionThe normal distribution describes the probability density of a continuous variable, used to model measurements and natural phenomena.Standard DeviationStandard deviation measures how spread out data points are from the average—use it to understand variability in a dataset.Z-ScoreThe Z-Score tells you how many standard deviations a data point is from the mean, revealing whether it is typical or unusual.Outlier RuleThe Outlier Rule identifies data values farther than 1.5 times the interquartile range from the quartiles—use it to flag unusual observations.Confidence Interval for a MeanEstimates the range where a population mean likely falls using sample data when the population standard deviation is unknown.

Frequently asked questions

Is the Empirical Rule the same as the 68-95-99.7 rule?: Yes; 'Empirical Rule' is the technical name, while '68-95-99.7 rule' is the nickname that describes the three main percentages. They refer to exactly the same concept.
Can I use the Empirical Rule even if my data isn't perfectly normal?: Yes, as long as your data is approximately normal (roughly bell-shaped, not heavily skewed). For data that is visibly non-normal, Chebyshev's inequality is safer and will give you bounds that definitely hold.
Does 'within $1\sigma$ ' include the exact values at $\mu - 1\sigma$ and $\mu + 1\sigma$ ?: Yes, 'within $1\sigma$ ' means the closed interval from $\mu - \sigma$ to $\mu + \sigma$ inclusive, including both endpoints. In practice, for continuous data, whether you include the boundaries makes almost no difference to the percentage.
Can I use the Empirical Rule to find the probability that a single value falls in a certain range?: Yes, exactly. If your data follows a normal distribution, the percentages from the Empirical Rule tell you the probability; for instance, the probability that a randomly selected observation falls within $1\sigma$ of the mean is approximately 68%.

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Reviewed 2026-09-18