S1 and S2 distributions cheat sheet

April 13, 2017 Robyn B

A quick guide to normal distribution, continuous uniform distribution and when to use binomial vs. Poisson, as well as a guide to approximations.

Normal distribution

- A way of modelling a trait among a population, e.g. heights in the UK.

- Can use it to calculate the probability of a certain trait being observed, e.g. likelihood of someone being up to 160cm tall. 

- It is bell shaped because it is more likely to have a value close to the mean, and less likely to have an extreme value, e.g. lots of people are around 5-6ft tall, but very few are 4ft or 7ft.

- The probability of X being up to a certain value is taken as the area under the curve up to that point, and these values can be looked up in the tables in your formula booklet.

- To look up the probability, you need to standardise the score, using Z = (X - mean)/standard deviation.

- Because it is a continuous scale, the probability of an exact trait being observed is equal to 0, e.g. probability of someone being exactly 158.973cm tall is taken to be 0, because it is so unlikely.

- The total area under the curve is 1.


Standard normal distribution

- The normal distribution is symmetrical, so with a standardised mean of 0: P(X< -1) is the same as P(X> 1)

Top tips for normal distribution questions:

1. Draw the distribution, labelling mean, standard deviation and the X value interested in

2. Shade the area you need to calculate

3. Calculate the Z values you need (you may be taking one away from another, or taking one away from 1)

4. Look up the probability associated with this value

5. Plug these values into the equation you made based on your shaded picture, and write out your full answer.


Continuous uniform distribution

- Used to model likelihood of events happening when there is a restricted range of possible outcomes, and it is equally likely for any of those values to occur.

- E(X) is the expected value and Var(X) is the variance, both of which have equation in your formula bookletImage title


Binomial vs. Poisson

- If there is a number of trials stated and two possible outcomes are being considered, use binomial.

- If there is an average or mean rate of something happening, use Poisson.


Approximation

- If n is large and/or if p is close to 0.5, use normal approximation to binomial so X ~ B(n, p) becomes Y~N(np,npq).

- If L is large, use normal approximation to Poisson so X ~ Po(L) becomes X ~ N(L,L)

- When doing this, you may need to use continuity correction because the probability of X being exactly equal to a value is 0 for normally distributed data. We can cover this next time if you like, but in the mean time here is a useful web page to help explain it: https://revisionmaths.com/advanced-level-maths-revision/statistics/normal-approximations

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