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### Course: Statistics and probability>Unit 4

Lesson 5: Normal distributions and the empirical rule

# Standard normal distribution and the empirical rule (from ck12.org)

Using the Empirical Rule with a standard normal distribution. Created by Sal Khan.

## Want to join the conversation?

• Could you do a proof of the empirical rule?
• Look at a table of z-scores (which comes later, for folks who aren't up to that yet).
P(-1 < X < 1) = 0.6826
P(-2 < X < 2) = 0.9544
P(-3 < X < 3) = 0.9973

The empirical rule is just a rough estimate of that.
• Why isn't the mean 50%? And why isn't the standard deviation 68%?
• The way I understand it, the answer choice would have to explicitly state "The percentage of data below/above the mean/standard deviation" as opposed to "The mean/standard deviation". In a standard normal distribution, the mean (µ) by itself is equal to 0, and the standard deviation (σ) is equal to 1. We know this because normal distributions are given in the form: N(mean, standard deviation) or N(µ,σ), and the form for Standard Normal Distribution is: N(0,1).
• If you get all the stuff together, and you have not left something out, then could there still be anything left over?
• If you get all the data together and plot it in a plane, then there could not be anything left over. For the smallest number will fit in the less then -3 x the standard deviation. The greatest number will be found in the area of +3 x the standard deviation.

Godspeed jtlvhpublic,

Wim
• In the answer above, highest number is taken as standard deviation. But std dev is part of whole unit one (when you are expressing in fractions), Then how come std deviation is largest number?
• I know it's the same, but isn't σ ^2=1 and not σ =1?
• If you square root both sides of σ ^2=1 , don't you get σ =1.
• A compact disc is designed to last an average of 4 years with a standard deviation of 0.8 years. What is the probability that a CD will last less than 3 years?

1.11%
10.56%
86.65%
100%
I got 69.435% when 0.8-0.10565. I tried 1-0.10565= 89.435 %. I don't get what's going on here. Help me, please.
• How can 84% of the data be less that one standard deviation?
• total area under any curve is 1 and in above example the standard equation is also 1 .But its evident that the standard deviation is way too smaller than the whole curve (just by visualizing the curve) hence the last part in which sal arranged the data under the curve in a ascending order is confusing .....can someone help??
• you are right, (d). standard deviation is supposed to equal 34%, or half of 68%.
(1 vote)
• I dont understand any of this! Help