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# Empirical Rule

Suppose that you have a set of data that is approximately normally distributed. Suppose, also, that you know the standard deviation of the data set. Is there much you can discern about the data from this information? Well, as a matter of fact, there is quite a bit, thanks to the empirical rule.

The empirical rule can be used to judge the likelihood of certain values in a dataset, as well as to check for outliers in your data set and much more. What is the empirical rule, and how does it relate to normal distributions and standard deviations?

## Definition of the Empirical Rule

The Empirical rule goes by several names, Sometimes it is called the $$95 \%$$ rule, the three-sigma rule, or the $$68$$-$$95$$-$$99.7$$ rule.

It is usually called the empirical rule as it is a rule informed by many observations of data sets, not a logical or definitive mathematical proof.

The empirical rule is a statistical rule based on observations that show almost all data in a normal data distribution fall within three standard deviations of the mean.

Where do the other names come from? Well, there's even more that the empirical rule can tell you, and the clues are in the names. It's all about percentages, and standard deviation.

## Empirical Rule Percentages

As mentioned previously, one of the names for the empirical rule is the $$68$$-$$95$$-$$99.7$$ rule. This name is actually quite telling when we look at the empirical rule in full. It states

For a set of normally distributed data, approximately $$68\%$$ of observations fall within one standard deviation of the mean, approximately $$95\%$$ of observations fall within two standard deviations of the mean, and approximately $$99.7\%$$ of observations fall within three standard deviations of the mean.

$$68\%$$, $$95\%$$, $$99.7\%$$, get it?

If you remember those three percentages, then you can use them to infer all sorts of normally distributed data sets.

But wait a minute, it's also sometimes called the three-sigma rule, why on earth is that?

Well, the symbol for standard deviation is sigma, $$\sigma$$. It is sometimes called the three-sigma rule because it states that almost all observations fall within three sigmas of the mean.

It is a standard convention to consider any observations that lie outside of these three sigmas as outliers. This means that they are not typically expected observations, and aren't indicative of the overall trend. In some applications, the bar for what is considered an outlier might be explicitly stated to be something else, but three sigmas is a good rule of thumb.

Let's take a look at what all of this looks like when put into a graph.

## Empirical Rule Normal Distribution Graph

Take the following normal distribution with a mean of $$m$$ and a standard deviation of $$\sigma$$ as an example.

Fig. 1. Normal Distribution Curve.

It's possible to divide it up according to the empirical rule.

Fig. 2. The empirical rule.

This graphical representation really demonstrates the main takeaways we can make of the empirical rule. It's very clear to see that virtually all observations fall within three standard deviations of the mean. There may very occasionally be outliers, but these are exceedingly rare.

The biggest chunk is clearly the middle $$-\sigma$$ to $$\sigma$$, just as the empirical rule states.

You may be thinking, 'great this rule seems useful, I'm gonna use it all the time!' But beware, and be careful. The empirical rule only holds true for data that is normally distributed.

## Empirical Rule Examples

Let's take a look at some examples to see how we can put all of this into practice.

(1) The heights of all the female pupils in a class are measured. The data is found to be approximately normally distributed, with a mean height of $$5ft\,2$$ and a standard deviation of $$2\, in$$. There are $$12$$ female pupils in the class.

(a) Using the empirical rule, roughly how many of the pupils are between $$5ft\,2$$ and $$5ft\,4$$?

(b) Using the empirical rule, roughly how many of the pupils are between $$4ft\,8$$ and $$5ft$$?

(c) One pupil is a height of $$5ft\,9$$, can this pupil be considered an outlier?

Solution:

(a) $$5ft\,4$$ is the mean plus one standard deviation. The empirical rule states that $$68\%$$ of observations will fall within one standard deviation of the mean. Since the question is only concerned with the upper half of this interval, it will be $$34\%$$. Therefore

$0.34 \cdot 12 = 4.08$

The number of female pupils in the class with a height between $$5ft\,2$$ and $$5ft\,4$$ is $$4$$.

(b) $$4ft\,8$$ is the mean minus two standard deviations, and $$5ft$$ is the mean minus one standard deviation. According to the empirical rule, $$95\%$$ of observations fall within two standard deviations of the mean, and $$68\%$$ of observations fall within one standard deviation of the mean.

Since the question is only concerned with the lower halves of these intervals, they become $$47.5\%$$ and $$34\%$$ respectively. The interval we are looking for is the difference between these two.

$47.5\% - 34\% = 13.5\%$

Therefore

$0.135 \cdot 12 = 1.62$

The number of female pupils in the class with a height between $$4ft\,8$$ and $$5ft$$ is $$1$$.

(c) $$5ft\,9$$ is over $$3$$ standard deviations greater than the mean, therefore this pupil can be considered an outlier.

(2) An ecologist records the population of foxes in a forest every year for ten years. He finds that on average there are $$150$$ foxes living in the forest in a given year in that period, with a standard deviation of $$15$$ foxes. The data is roughly normally distributed.

(a) According to the empirical rule, what range of population size could be expected over the ten years?

(b) Which of the following would be considered outlying population values?

$100, \space 170, \space 110, \space 132$

(a) According to the empirical rule, any observation not within three standard deviations of the mean is usually considered an outlier. Therefore our range is

$\mu - 3\sigma < P < \mu + 3\sigma$

$150 - 3 \cdot 15 < P < 150+ 3 \cdot 15$

$150-45 < P < 150+45$

$105 < P < 195$

(b) $$100$$ is the only one not within three standard deviations of the mean, therefore it is the only outlier.

## Empirical Rule - Key takeaways

• The empirical rule states that for normally distributed data sets, $$68\%$$ of observations fall within one standard deviation of the mean, $$95\%$$ of observations fall within two standard deviations of the mean, and $$99.7\%$$ of observations fall within three standard deviations of the mean.
• It is also known as the $$68\%$$-$$95\%$$-$$99.7\%$$ rule, the three-sigma rule, and the $$95\%$$ rule.
• Usually, any observation not within three standard deviations of the mean can be considered an outlier.

The empirical rule does not have a formula but it does state that for normally distributed data sets, 68% of observations fall within one standard deviation of the mean, 95% of observations fall within two standard deviations of the mean, and 99.7% of observations fall within three standard deviations of the mean.

In its simplest terms, the empirical rule states that virtually all data in a normally distributed data set falls within three standard deviations of the mean.

According to the empirical rule, 95% of all observations in a normally distributed data set fall within two standard deviations of the mean.

The empirical rule can be used to judge the likelihood of certain values in a dataset, as well as to check for outliers in your data set.

If the average lifespan of a dog is 12 years (i.e mean) and the standard deviation of the mean is 2 years, and if you want to know the probability of the dog living more than 14 years, you will use the empirical rule.

## Final Empirical Rule Quiz

Question

What is the empirical rule?

The empirical rule is a rule in statistics that indicates where most observations lie in a normal distribution of a data set.

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Question

What are other names for the empirical rule?

The three-sigma rule, the $$95\%$$ rule and the $$68\%$$-$$95\%$$-$$99.7\%$$

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Question

Why is the empirical rule sometimes called the $$95\%$$ rule?

Because it states that approximately $$95\%$$ of observations fall within two standard deviations of the mean.

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Question

Why is the empirical rule sometimes called the $$68\%$$-$$95\%$$-$$99.6\%$$?

Because it states that approximately $$68\%$$ of observations fall within one standard deviation of the mean, approximately $$95\%$$ of observations fall within two standard deviations of the mean, and approximately $$99.7\%$$ of observations fall within three standard deviations of the mean.

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Question

Why is the empirical rule sometimes called the three-sigma rule?

Because virtually all observations fall within three standard deviations of the mean.

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Question

What is the standard deviation of a data set?

The standard deviation of a data set is a measure of the spread of the data set.

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Question

According to the empirical rule, what is a standard way of determining outliers?

Observations not within three standard deviations of the mean are usually considered outliers.

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Question

In a data set with a mean of $$90$$ and standard deviation of $$5$$, which of the following observations could be considered outliers?

$85, \space 60, \space 103$

$$60$$, as it is not within three standard deviations of the mean.

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Question

What percentage of observations fall between one and two standard deviations from the mean?

$95\% - 68\% = 27\%$

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Question

What percentage of observations fall between two and three standard deviations greater than the mean?

$\frac{95\%}{2} - \frac{68\%}{2} = 13.5\%$

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Question

Of a population of $$30$$, with a mean of $$35$$ and standard deviation of $$8$$, roughly how many observations will be between $$43$$ and $$35$$?

$$43$$ is one standard deviation greater than the mean. So there's $$32\%$$ chance an observation will fall in this interval.

$0.32 \cdot 30 = 9.6$

The answer is therefore $$9$$ observations.

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Question

Of a normally distributed population of $$1000$$, how many observations will be outliers according to the empirical rule?

$(1-0.997) \cdot 1000 = 3$

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Question

Why is the empirical rule called the empirical rule?

The empirical rule is named so because it was found through repeated similar observations of normally distributed data.

'Empirical' means drawn from observation or experience, not from pure logic or theory.

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Question

What type of data distribution does the empirical rule hold true for?

Normal Distribution

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Question

In a data set of $$100$$ values, with a mean value of $$1000$$, and a standard deviation of $$200$$, what is the largest value you could reasonably expect to find?

The empirical rule says that $$99.7\%$$ of observations will fall within three standard deviations of the mean, therefore

$\mu + 3\sigma = 1000 + 3 \cdot 200 = 1600$

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