Probability and Statistics > Basic Graphs and Charts > 68 95 99.7 rule

## What is the 68 95 99.7 rule?

When you use a standard normal model in statistics:

- About 68% of values fall within one standard deviation of the mean.
- About 95% of the values fall within two standard deviations from the mean.
- Almost all of the values — about 99.7% — fall within three standard deviations from the mean.

These facts are what is called the 68 95 99.7 rule, sometimes called the *Empirical Rule*. That’s because the rule originally came from observations* (empirical means “based on observation”).*

The normal curve is the **most common type of data distribution**. In a normal distribution, all of the measurements are computed as distances from the mean. These measurements are reported in standard deviations.

The normal curve is a symmetric distribution, so the middle 68.2% can be divided in two. Zero to 1 standard deviations from the mean has 34.1% of the data. The opposite side is the same (0 to -1 standard deviations). Together, this area adds up to about 68% of the data.

### When to use the Rule

You can use the rule when you are told your data is normal, nearly normal, or if you have a unimodal distribution that is symmetric. If a question mentions a normal or nearly normal distribution, *and* you’re given standard deviations; That almost certainly means you can use the rule to approximate how many of your scores will fall within a certain number of standard deviations.

## Example Question

The weights of stray dogs at a particular pound average 70lbs with a mean of 70 lbs with a standard deviation of 2.5 lbs. Assuming the weights follow a normal-shaped distribution:

- What weight is 2 standard deviations below the mean?
- What weight is 1 standard deviation above the mean?
- The middle 68% of dogs weigh how much?

Answers:

- 2 standard deviations is 2 * 2.5 (5 lbs). So if a dog is 2.5 standard deviations below the mean they weigh 70 lbs – 5 lbs = 65 lbs.
- 1 standard deviation is 2.5 lbs, so a dog 1 standard deviation above the mean would weigh 70 lbs + 2.5 lbs = 72.5 lbs.
- The 68 95 99.7 Rule tells us that 68% of the weights should be within 1 standard deviation either side of the mean. 1 standard deviation above (given in the answer to question 2) is 72.5 lbs; 1 standard deviation below is 70 lbs – 2.5 lbs is 67.5 lbs. Therefore, 68% of dogs weigh between 67.5 and 72.5 lbs.

## History of the 68 95 99.7 Rule

he 68 95 99.7 rule was first coined by Abraham de Moivre in 1733, 75 years before the normal distribution model was published. De Moivre worked in the developing field of probability. Perhaps his biggest contribution to statistics was the 1756 edition of The Doctrine of Chances, containing his work on the approximation to the binomial distribution by the normal distribution in the case of a large number of trials.

De Moivre discovered the 68 95 99.7 rule with an experiment. You can do your own experiment by flipping 100 fair coins. Note:

- How many heads you would expect to see; these are “successes” in this binomial experiment.
- The standard deviation.
- The upper and lower limits for the number of heads you would get 68% of the time, 95% of the time and 99.7% of the time

**Reference**:

MÃ³nica Blanco and Marta Ginovart. How to introduce historically the normal distribution in engineering education: a classroom experiment. Retrieved December 28 2015. http://upcommons.upc.edu/bitstream/handle/2117/6483/howtointroduce.pdf

Hello,

I trying to understand the origins of the 68-95-99.7 rule. I understand its meaning (or at least think I understand it) but cannot find concrete information on when and through what procedure those specific numbers were generated. In other words, who says that the normal distribution does not follows the, lets say. 55-87-95.6 pattern?

Is the rule is the result of rigorous mathematical derivation? if yes, I would like to understand, in lay terms, how it was calculated, or, is the rule is the result of empirical observations taken from the field? If its the case, how can we know for sure that those observations indeed yielded the right numbers?

I am not trying to be annoying, just genuinely curious about the question of how those numbers came to be.

thank you up front

Vitali

Vitali,

I looked into this for you and added a little to the history section at the end. De Moivre generated the numbers with experimentation.

Stephanie

This was ridiculously unhelpful.

Hi, Hannah,

What information were you looking for?

Stephanie

How would you go about answering a question such as “Using the 68-95-99.7% rule, what is the approximate proportion of values of a Normally distributed variable that lie between 2 and 3 standard deviations below the mean?”

It’s asking for the percentage between 2 and 3 std dev. 99.7% of data is between -3 and 3, so 50% of that is between 0 and 3. Basically, you’re going to be doing a little subtraction to find that %.