How do I know which distribution to use?

Three discrete distributions Proportional Poisson Binomial

Proportional Binomial Poisson Given a number of categories Probability proportional to number of opportunities Days of the week, months of the year Proportional Number of successes in n trials Have to know n, p under the null hypothesis Punnett square, many p=0.5 examples Binomial Number of events in interval of space or time n not fixed, not given p Car wrecks, flowers in a field Poisson

Proportional Binomial Binomial with large n, small p converges to the Poisson distribution Poisson

Examples: name that distribution Asteroids hitting the moon per year Babies born at night vs. during the day Number of males in classes with 25 students Number of snails in 1x1 m quadrats Number of wins out of 50 in rock-paper-scissors

Proportional Generate expected values Calculate 2 test statistic Binomial Poisson

Sample Null hypothesis Test statistic Null distribution compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

Discrete distribution Chi-squared goodness of fit test Null hypothesis: Data fit a particular Discrete distribution Sample Calculate expected values Chi-squared Test statistic Null distribution: 2 With N-1-p d.f. compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

The Normal Distribution

Babies are “normal”

Babies are “normal”

Normal distribution

Normal distribution A continuous probability distribution Describes a bell-shaped curve Good approximation for many biological variables

Continuous Probability Distribution

The normal distribution is very common in nature Human body temperature Human birth weight Number of bristles on a Drosophila abdomen

Continuous probability distribution Discrete probability distribution Continuous probability distribution Normal Probability Poisson Probability density Pr[X=2] = 0.22 Pr[X=2] = ? Probability that X is EXACTLY 2 is very very small

Continuous probability distribution Discrete probability distribution Continuous probability distribution Normal Probability Poisson Probability density Pr[1.5≤X≤2.5] = Area under the curve Pr[X=2] = 0.22

Continuous probability distribution Discrete probability distribution Continuous probability distribution Normal Probability Poisson Probability density Pr[1.5≤X≤2.5] = 0.06 Pr[X=2] = 0.22

Normal distribution Probability density x

Normal distribution Probability density x Area under the curve:

But don’t worry about this for now!

A normal distribution is fully described by its mean and variance

A normal distribution is symmetric around its mean Probability Density Y

About 2/3 of random draws from a normal distribution are within one standard deviation of the mean

About 95% of random draws from a normal distribution are within two standard deviations of the mean (Really, it’s 1.96 SD.)

Properties of a Normal Distribution Fully described by its mean and variance Symmetric around its mean Mean = median = mode 2/3 of randomly-drawn observations fall between - and + 95% of randomly-drawn observations fall between -2 and +2

Standard normal distribution A normal distribution with: Mean of zero. (m = 0) Standard deviation of one. (s = 1)

Standard normal table Gives the probability of getting a random draw from a standard normal distribution greater than a given value

Standard normal table: Z = 1.96

Standard normal table: Z = 1.96 Pr[Z>1.96]=0.025

Normal Rules Pr[X < x] =1- Pr[X > x] Pr[X<x] + Pr[X>x]=1 + = 1

Standard normal is symmetric, so... Pr[X > x] = Pr[X < -x]

Normal Rules Pr[X > x] = Pr[X < -x]

Sample standard normal calculations Pr[Z > 1.09] Pr[Z < -1.09] Pr[Z > -1.75] Pr[0.34 < Z < 2.52] Pr[-1.00 < Z < 1.00]

What about other normal distributions? All normal distributions are shaped alike, just with different means and variances

What about other normal distributions? All normal distributions are shaped alike, just with different means and variances Any normal distribution can be converted to a standard normal distribution, by Z-score

Z tells us how many standard deviations Y is from the mean The probability of getting a value greater than Y is the same as the probability of getting a value greater than Z from a standard normal distribution.

Z tells us how many standard deviations Y is from the mean Pr[Z > z] = Pr[Y > y]

Example: British spies MI5 says a man has to be shorter than 180.3 cm tall to be a spy. Mean height of British men is 177.0cm, with standard deviation 7.1cm, with a normal distribution. What proportion of British men are excluded from a career as a spy by this height criteria?

Draw a rough sketch of the question

m = 177.0cm s = 7.1cm y = 180.3 Pr[Y > y]

Part of the standard normal table x.x0 x.x1 x.x2 .x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9 0.0 0.5 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.4721 0.46812 0.46414 0.1 0.46017 0.4562 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858 0.42465 0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 0.3 0.38209 0.37828 0.37448 0.3707 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 0.4 0.34458 0.3409 0.33724 0.3336 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 0.30854 0.30503 0.30153 0.29806 0.2946 0.29116 0.28774 0.28434 0.28096 0.2776 Pr[Z > 0.46] = 0.32276, so Pr[Y > 180.3] = 0.32276

Sample problem MI5 says a woman has to be shorter than 172.7 cm tall to be a spy. The mean height of women in Britain is 163.3 cm, with standard deviation 6.4 cm. What fraction of women meet the height standard for application to MI5?