Data Science (Incl AI/ML): Inferential Stats

Notes

Most naturally occurring phenomena follow a normal distribution i.e. a bell curve
The bell curve has many interesting features
The mean & mode coincide
The 1-2-3 rule

The probability of any value lying within 1 standard deviation (sigma σ) from the mean (mu µ) is 68% (i.e µ+1 σ or µ-1 σs).
The probability of any value lying within 2 standard deviations (sigma) from the mean (mu) is 95% (i.e µ+2 σ or µ-2 σ).
The probability of any value lying within 3 standard deviations (sigma) from the mean (mu) is 99.7% (i.e µ+3 σ or µ-3 σ).

As a convention, a random variable's (X) value is specified in terms of its distance from the mean µ in units of std deviation σ i.e., (X-µ)/σ units - this is called Z, the standardised normal variable.
e.g if, say, µ is 35 and σ is 5, X value of 43.5 would be (43.5 - 35)/5 units = 1.7σ away from mean. Hence Z score is 1.7
(Ugly) Alternative to Z Score table
(Z)=1√2π∫Z−∞e−t22dt

or in Excel, NORM.S.DIST(z, TRUE/FALSE) where
TRUE means find the cumulative probability, FALSE means find the probability density.

Sampling

When dealing with large populations, it is more feasible to quantify the characteristics of the population by using "samples" of the population.
Working with the samples, we can approximate/extrapolate the characteristics of the original, full population
Sampling distributions play an important part in understanding the "margin of error" and our confidence to closeness to the population when inferring chracteristics.
A sampling distribution starts to approach/resemble a normal distribution at a sampling size (n) of 30.

The Central Limit Theorem (CLT)

For any kind of data (regardless of how it is distributed viz. normal, skewed, uniform etc.), the following properties hold true, provided a high number of samples has been taken :

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Data Science (Incl AI/ML)