Normal distribution

Also known as a Gaussian distribution.

Defined by a mean and standard deviation.

Shows up in nature frequently. You can often assume a normal distribution.

Values may be described as normally distributed.

Normal distribution at Khan Academy.

It has a special probability density function.

Distribution goes to infinity, but some examples cannot be infinite (or negative infinity), but still can be pretty good approximation.

$$ P(x,\sigma,\mu) = \frac{1}{{\sigma \sqrt {2\pi } }} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

The normal distribution converges to the the binomial distribution when you increase samples.Normal distribution excel at Khan Academy.

Examples of normal distribution measurements at Khan Academy.

Empirical Rule - 68-95-99.7

In a normal distribution, with known mean and standard deviation, then,

Break up the chart into +/- 3,2,1 standard deviations.

68% of values will fall between 1 std dev below, and 1 std dev above the mean.

95% for 2 std deviations below and above the mean

99.7% for 3 std deviations below and above the mean

For exactly 95%, use +/- 1.96 standard deviations from the mean..

Standard Normal Distribution

A normal distribution where

mean = 0

std dev = 1

Gaussian distribution

Another name for the normal distribution.

Skew normality test

Use a Z-test. Calculate the Z statistic

$$ \text{Skew} = \frac{1}{N} \sum_{i=1}^{N} (\frac{x_i - \mu}{\sigma})^3 $$

$$ \text{SE} = \sqrt{\frac{6}{N}} $$

Kurtosis normality test

Use a Z-test. Calculate the Z statistic

$$ \text{Kurtosis} = \frac{1}{N} \sum_{i=1}^{N} (\frac{x_i - \mu}{\sigma})^4$$

$$ \text{SE} = \sqrt{\frac{24}{N}} $$

3 is the kurtosis of a normal distribution, so compare Kurtosis against expected value of 3 in your Z test.