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## Statistics and probability

### Course: Statistics and probability>Unit 4

Lesson 5: Normal distributions and the empirical rule

# Normal distribution problems: Empirical rule

The empirical rule (also called the "68-95-99.7 rule") is a guideline for how data is distributed in a normal distribution. The rule states that (approximately): - 68% of the data points will fall within one standard deviation of the mean. - 95% of the data points will fall within two standard deviations of the mean. - 99.7% of the data points will fall within three standard deviations of the mean. Created by Sal Khan.

## Want to join the conversation?

• At , Sal draws what looks like an upside down capital letter 'A' to left of 68,95,99.7.
What does it mean?
• That was an awkwardly-drawn asterisk.
For the record, ∀ is a common mathematical symbol in logic that is shorthand for "for all", but that's not what Sal was drawing. :)
• I'm wondering: Why use the empirical rule? How can I remember those percentages? I'd love a video on this subject that connects it to the other topics in statistics and explains why to use it!
• You use the empirical rule because it allows you to quickly estimate probabilities when you're dealing with a normal distribution. People often create ranges using standard deviation, so knowing what percentage of cases fall within 1, 2 and 3 standard deviations can be useful.
• This is a bit frustrating.

I started with the "AP Statistics" course. Ran into some exercises/quizzes with terms that were never taught.

Had to switch to the "Statistics and Probability" course to learn about those terms. In that course ran into "standard deviation" term and had to switch to "High school statistics" to learn about it.

Now on this course we get "normal distribution", which was never taught...

Do I now have to go to ck12.org or another course on KA to learn about normal distributions?

Is there no way to do these courses in sequence?
• It's out of order but you may want to start with the normal distribution review. And then further on down theres a video called "Deep definition of the normal distribution" in the "More on normal distributions" section, and that is labeled an intro to the normal distribution. There's definitely some weirdness with the stats stuff though.
• Why is it called empirical(something based on observations rather than a fixed formula) rule? The 68-95-99.7% distribution can be calculated through the normal distribution formula as well. How exactly is this empirical?
• The empirical rule is named as such because it was originally based on observation. 18th century French mathematician Abraham de Moivre flipped fair coins and tried to understand the probability of obtaining a specific number of heads from 100 coin flips. He observed that as the number of flips increased, the distribution approached a curve (the normal distribution).
• So, am I right to think that % of the distribution between 1 and 2 standard deviations is 13.5%? 95-68=27 and 27/2=13.5
So if the question were: what % of babies born are between the weights 7.3g and 8.4g? The answer would be 13.5%?
• Thanks Dave :)
• How would the problem be different, if the question had not specified that the data was "normally distributed"?
• We can say almost nothing if we do not know how our data is distributed!
• At Sal said "If we go one more standard deviation then, you know where this is headed, 99.7% of a deviation in that range." But why isn't it 100%?
• The Normal curve doesn't ever hit 0, so technically any place that we chop it off, we'll be chopping off a little bit of the probability. It so happens that at +/- 3 standard deviations we've captures 99.7% of the area, and for many folks that is close enough to being "basically everything."
• Is there some part of the course missing?

We just learned about Density curves in the previous lesson.

I would expect to first learn about the Normal Distribution, and then to learn about the empirical rule for the normal distribution.
But instead, this video is about how to solve exercises about the empirical rule (which hasn't been introduced yet) for the normal distribution (which also has not been introduced yet) ...
• How do we know that the empirical rule actually works?
• These numerical values (68 - 95 - 99.7) come from the cumulative distribution function (CDF) of the normal distribution. For example, F(2) = 0.9772, or Pr(x ≤ μ + 2σ) = 0.9772. Note that this is not a symmetrical interval – this is merely the probability that an observation is less than μ + 2σ. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):
Pr(μ − 2σ ≤ x ≤ μ + 2σ) = F(2) − F(−2) = 0.9772 − (1 − 0.9772) = 0.9545 or 95.45%.
This is related to confidence interval as used in statistics: μ ± 2σ is approximately a 95%.

For μ ± σ and μ ± 3σ we can find the probability using the same method!