# The Standard Normal Distribution

ck12.org Exercise: Standard Normal Distribution and the Empirical Rule

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• Some characteristics of a normal distributions are listed below.

1. Normal distributions are symmetric around their mean.
2. The mean, median, and mode of a normal distribution are equal.
3. The area under the normal curve is equal to 1.0.
4. Normal distributions are denser in the center and less dense in the tails.
5. Normal distributions are defined by two parameters, the mean ($\mu$) and the standard deviation ($\sigma$).
6. 68% of the area of a normal distribution is within one standard deviation of the mean.
7. Approximately 95% of the area of a normal distribution is within two standard deviations of the mean.

• It has everything to do with standard deviation $\sigma$, in other words, how much your values are spread around the mean.

Say you have a machine that fills kilo-bags of sugar. The machine does not put exactly 1000 g in every bag. The standard deviation may be in the order of 10 g.
Then you can say: mean = $\mu = 1000$ and $\sigma = 10$ (gram)

The empirical rule (easily verified by your GC) now says:

50% will be underweight and 50% will be overweight, by varying amounts, of course.

68% or all you sugar-bags will weigh between:
$\mu - \sigma$ and $\mu + \sigma$ or between 990 and 1100 gram
(so 16% will weigh more and 16% will weigh less, as the normal distribution is completely symmetrical).

95% will be between $\mu - 2 \sigma$ and $\mu + 2 \sigma$
So 2,5% will be under 980 gram and 2.5% over 1020 gram.

In practice
In the case presented, you may not want to be that much under weight (a small overweight is not a problem). So most manufacturers set their machines to slightly overweight. Let's calculate this:
$\mu = 1010 , \sigma = 10$
Now the 68% is all more than 1000 gram ($1010 \pm 10$)
And only 2.5% is more than 10 gram underweight.

Challenge
Now find out what happens - and what you would have to do - if the standard deviation of your filling machine were greater or smaller.

Some calculators also have approximate values of this integral.

Generally, when I need to compute this, I will fire up an 'R' session on my laptop, and then use the pnorm() function.

#### Explanation:

'R' is a free, open-source software for statistics and data analysis. You can download it from http://www.r-project.org.

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