Normal distributions
A class of continuous probability distributions.
- defined by parameters mu and sigma
- possible values are real numbers
Normal Probability Density
if x is a value X~Normal(mu,sigma)
f(x) = 1/sigma*sqrt(2pi)e^-1/2(x-mu/sigma)^2
Infinite different normal distribution, no constraints on the values a normal distribution can take on.
the are under the curve will be equal to 1.
Describing Normal Distribution
Unimodal, “bell-shaped” distributions, symmetric around the mean.
Changing the standard deviation affects how variable the values in the distribution are.
changing the mean while keeping the SD we get the same the graph with the same shape and spread but is positioned much higher up or lower on the axis.
Normal Quantile Plots
A type of graph used to judge the ‘fit’ the Normal distribution to data.
There are lots of unimodal bellshaped distributions that are not normal, normally is defined by the function (equation).
You can’t just look at the histogram and determine its normal.
You have to ask yourself if the model fits the normal equation
Takes the normal function and shows you what your data should look like if it was normal and lets you compare it to your data
Empirical Rule of Thumb
68-95-99.7
Any normal distribution has these characteristic.
If you go + and - one standard deviation from the mean, the area enclosed under the curve will be 68%.
If you go 2 standard deviation above and below the mean you will enclose 95% of the central distribution.
Standardized Score (z-score)
describes the number of standard deviations a value is from the mean of the distribution.
Standard Score = Value-Mean/(Standard Deviation)
-magnitude describes how extreme the value is
- doesn’t tell you if the distribution is normal or not.
- sign (positive/negative) describes location relative to mean
-Sd score gives you a sense of how spread out the distribution of a unimodal function is. Tells you where a value lies, sd score closer to zero, closer to the mean.
Use of standardized scores
- sense of location and frequency of a value from a unimodal, symmetric distribution
- common scale for comparing values from different distribution
- Even though density is not equivalent to probability, we can look at a curve and say values closer to the mean are much more probable (frequent/likely to occur).
Benefit of Standard Normal Distribution
N(0,1)
we may know probability of the distribution but not the mean, we want to understand how far a value is based on its cumulative probability. We can use standardized scores to calculate quantiles and percentiles.
