Empirical Rule 68 95 99.7: Easy Guide With Examples

Key takeaways

• The empirical rule 68 95 99.7 works for a normal distribution (a bell-shaped, symmetric data pattern).
• About 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
• You can use it to estimate ranges quickly on homework, quizzes, and test review.

The empirical rule 68 95 99.7 helps you estimate how data is spread out when the distribution is normal. You do not need long formulas to use it.

In this guide, you’ll learn what the rule means, how to use it step by step, and how to solve typical homework questions. You’re not behind. You just need a clear method.

Quick answer

The empirical rule 68 95 99.7 says this: in a normal distribution (a bell-shaped, symmetric distribution), about 68% of values are within 1 standard deviation of the mean, about 95% are within 2 standard deviations, and about 99.7% are within 3 standard deviations.

Mini-method:

1. Find the mean (center value).
2. Find the standard deviation (how spread out the data is).
3. Build intervals: μ ± 1σ, μ ± 2σ, μ ± 3σ.
4. Match them to 68%, 95%, and 99.7%.

Formula:

• Within 1 standard deviation: $\mu \pm 1\sigma$
• Within 2 standard deviations: $\mu \pm 2\sigma$
• Within 3 standard deviations: $\mu \pm 3\sigma$

Here, μ means the mean (average) and σ means the standard deviation (how spread out the data is).

What the empirical rule means

A normal distribution is a data pattern that is symmetric and bell-shaped. The mean is at the center.

The empirical rule is a shortcut for normal data:

These percentages are approximate, and they are used for normal distributions.

Important terms

• Mean (average value)
• Standard deviation (how spread out the data is)
• Normal distribution (a symmetric, bell-shaped distribution)
• z-score (how many standard deviations a value is above or below the mean)

When to use the rule

Use the empirical rule when:

• The problem says the data is normal.
• The graph looks bell-shaped and symmetric.
• You need a quick estimate, not an exact probability.

Do not use it automatically if the data is skewed, uneven, or clearly not normal.

Step-by-step method

Step 1: Identify the mean and standard deviation

Look for:

• Mean = center
• Standard deviation = spread

Example: mean = 100, standard deviation = 10

Step 2: Build the three main intervals

Compute:

• 1σ range: $100 \pm 10 = [90, 110]$
• 2σ range: $100 \pm 20 = [80, 120]$
• 3σ range: $100 \pm 30 = [70, 130]$

Step 3: Match each interval to its percent

• [90, 110] → about 68%
• [80, 120] → about 95%
• [70, 130] → about 99.7%

Step 4: Read the question carefully

Some questions ask:

• “What interval contains about 95% of the data?”
• “What percent is between 90 and 110?”
• “Is 130 unusual?”

Your job is to match the range to the correct empirical-rule part.

What to write on your paper

1. Mean $\mu = \underline{\hspace{1.5cm}}$
2. Standard deviation $\sigma = \underline{\hspace{1.5cm}}$
3. $\mu \pm 1\sigma =$
4. $\mu \pm 2\sigma =$
5. $\mu \pm 3\sigma =$
6. State the matching percent: 68% / 95% / 99.7%

Quick self-check

Ask yourself:

• Did I add and subtract correctly?
• Did I use the right multiple of σ?
• Does my answer match 1σ, 2σ, or 3σ?
• Did the problem say the data is normal?

A helpful shortcut for pieces of the curve

Because the normal curve is symmetric, half of 68% is 34% on each side of the mean. That helps you break the curve into smaller parts. From there, the area between 1σ and 2σ on one side is about 13.5%, and the area between 2σ and 3σ on one side is about 2.35%. These come from splitting the larger empirical-rule percentages across both sides of the curve.

A common classroom breakdown is:

This is useful for exam questions that ask for a smaller section, not the full 68%, 95%, or 99.7%.

Worked examples

Example 1

Test scores are normal with mean 80 and standard deviation 5. What interval contains about 68% of the scores?

Step 1: Use 1 standard deviation.
$\mu \pm 1\sigma = 80 \pm 5$

Step 2: Calculate the endpoints.
$[75, 85]$

Answer: About 68% of the scores are between 75 and 85.

Why: The empirical rule says 68% is within 1 standard deviation of the mean.

Example 2: Typical homework level

Heights are normal with mean 170 cm and standard deviation 6 cm. What interval contains about 95% of the heights?

Step 1: Use 2 standard deviations.
$\mu \pm 2\sigma = 170 \pm 2(6)$

Step 2: Multiply.
$2(6) = 12$

Step 3: Build the interval.
$[170 – 12,\ 170 + 12] = [158, 182]$

Answer: About 95% of heights are between 158 cm and 182 cm.

Why: The rule says 95% is within 2 standard deviations of the mean.

Example 3: Exam-style

A data set is normal with mean 50 and standard deviation 4. About what percent of values lie between 46 and 54?

Step 1: Compare the endpoints to the mean.
46 is $50 – 4$, and 54 is $50 + 4$

Step 2: That is $\mu \pm 1\sigma$

Answer: About 68%

Why: The interval is exactly one standard deviation on each side of the mean.

Example 4: Common trap question

A normal data set has mean 100 and standard deviation 15. About what percent of values are above 130?

Step 1: Find how far 130 is from the mean.
$130 – 100 = 30$

Step 2: Compare 30 to the standard deviation.
$30 = 2(15)$, so 130 is 2σ above the mean

Step 3: Use the empirical rule.
About 95% are within 2σ, so about 5% are outside that range.

Step 4: Split both tails.
Since the curve is symmetric, half of 5% is on the right tail.

Answer: About 2.5% are above 130.

Why: “Above 130” means only the right tail beyond +2σ.

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Practice set

Directions

Assume each distribution is normal. Use the empirical rule to answer.

1. Mean = 60, standard deviation = 3. What interval contains about 68% of the data?
2. Mean = 60, standard deviation = 3. What interval contains about 95% of the data?
3. Mean = 60, standard deviation = 3. What interval contains about 99.7% of the data?
4. Mean = 90, standard deviation = 8. About what percent of values are between 82 and 98?
5. Mean = 90, standard deviation = 8. About what percent of values are between 74 and 106?
6. Mean = 90, standard deviation = 8. About what percent of values are above 106?
7. Mean = 40, standard deviation = 5. Is 55 within 3 standard deviations of the mean?
8. Mean = 120, standard deviation = 10. About what percent of values are between 100 and 140?
9. Mean = 120, standard deviation = 10. About what percent of values are below 110?
10. Common trap: Mean = 75, standard deviation = 5. About what percent of values are between 70 and 85?

Answer key

1. $[57, 63]$
2. $[54, 66]$
3. $[51, 69]$
4. 68%
5. 95%
6. 2.5%
7. Yes, because 55 = $40 + 3(5)$
8. 95%
9. About 16%
10. About 47.5%

Short solutions for 2 items

#6
106 is $90 + 2(8)$, so it is +2σ.
About 95% are within 2σ, so 5% are outside.
Half is on the right.
Answer: 2.5%

#10
70 is $75 – 1\sigma$, and 85 is $75 + 2\sigma$
From -1σ to +1σ is 68%.
From +1σ to +2σ is 13.5%.
Total = $68\% + 13.5\% = 81.5\%$

Correction: the full correct answer is 81.5%, not 47.5%. This is a trap because the interval is not centered evenly around the mean.

Common mistakes

If you get stuck…

“I don’t know which percent to use.”
Match the number of standard deviations:

• 1σ → 68%
• 2σ → 95%
• 3σ → 99.7%

“I keep mixing up the interval.”
Write the pattern first:

• $\mu \pm 1\sigma$
• $\mu \pm 2\sigma$
• $\mu \pm 3\sigma$

“I do not know if a value is 1σ or 2σ away.”
Compute the distance from the mean, then divide by the standard deviation. That is the basic idea of a z-score. A z-score tells how many standard deviations a value is from the mean.

“My calculator gives a different value.”
The empirical rule gives quick estimates. A normal calculator or z-table can give more exact probabilities.

“The graph does not look bell-shaped.”
Pause. The empirical rule may not fit well. Review the normal distribution first.

“I’m still confused.”
Review the prerequisite pages below, then ask your teacher or tutor about the exact step that feels unclear. That is a normal part of learning statistics.

Next steps

To build this skill, study these in order:

• How to Use a Z-Table — Learn how to find more exact normal probabilities when the empirical rule is not enough.
• Standard Normal Distribution — Review what a normal distribution is and why z-scores matter.
• Z-Score Calculator — Check how far a value is from the mean in standard deviation units.
• Normal Distribution Calculator — Get exact areas and probabilities for normal-curve questions.
• Z-Table — Use a table view when your class expects manual lookup.

How we know

• The 68–95–99.7 pattern is a standard classroom rule for normal distributions.
• The definitions of normal distribution, empirical rule, and z-score match widely used introductory statistics resources.
• The worked methods here follow the standard interval form $\mu \pm k\sigma$μ±kσ.
• We used example structures consistent with introductory statistics teaching materials.
• We use careful language like “about” because these are approximate percentages, not exact values.

Use this the right way

Use this guide to understand the method, not just copy answers.

For homework or test review:

• Show your steps
• Label the mean and standard deviation
• State why you used 68%, 95%, or 99.7%

For reports or research writing, cite your sources and do not present borrowed explanations as your own work.

References

OpenStax. (2023). Introductory statistics 2e: 6.1 The standard normal distribution. Rice University. https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution

Penn State Eberly College of Science. (n.d.). 2.2.7 – The empirical rule. STAT 200: Statistics Online. https://online.stat.psu.edu/stat200/lesson/2/2.2/2.2.7