How do I know which distribution to use?

Three discrete distributions

Proportional

Binomial

Poisson

Proportional

Binomial

Poisson

Given a number of categories Probability proportional to number of opportunitiesDays of the week, months of the year

Number of successes in n trialsHave to know n, p under the null hypothesisPunnett square, many p=0.5 examples

Number of events in interval of space or timen not fixed, not given pCar wrecks, flowers in a field

Proportional

Binomial

Poisson

Binomial with large n, small p convergesto the Poisson distribution

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

Binomial

Poisson

Generate expected values

Calculate

2 test statistic

Sample

Test statistic

Null hypothesis

Null distributioncompare

How unusual is this test statistic?

P < 0.05 P > 0.05

Reject Ho Fail to reject Ho

Sample

Chi-squaredTest statistic

Null hypothesis:Data fit a particular

Discrete distribution

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

Chi-squared goodness of fit test

Calculate expected values

The Normal Distribution

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Babies are “normal”

Babies are “normal”

Normal distribution

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Normal distribution

• A continuous probability distribution

• Describes a bell-shaped curve• Good approximation for many biological variables

Continuous Probability Distribution

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The normal distribution is very

common in nature

Human body temperature

Human birth weight Number of bristles on a Drosophila abdomen

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0.2

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0.4

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Discrete probabilitydistribution

Continuous probabilitydistribution

Poisson Normal

Pr[X=2] = 0.22 Pr[X=2] = ?

Probability that X is EXACTLY2 is very very small

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Discrete probabilitydistribution

Continuous probabilitydistribution

Poisson Normal

Pr[X=2] = 0.22 Pr[1.5≤X≤2.5] =

Area under the curve

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Discrete probabilitydistribution

Continuous probabilitydistribution

Poisson Normal

Pr[X=2] = 0.22 Pr[1.5≤X≤2.5] =

0.06

Normal distribution

€

f x( ) =1

2πσ 2e−x−μ( )

2

2σ 2

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Normal distribution

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Area under the curve:

€

Pr[a ≤ X ≤ b] = f [X]dXa

b

∫

€

Pr[a ≤ X ≤ b] = f [X]dXa

b

∫

=1

2πσ 2e−x−μ( )

2

2σ 2

a

b

∫ dX

But don’t worry about this for now!

A normal distribution is fully described by its mean and variance

Figure 10.3. The normal distribution with different values of themean and variance. (a) μ =5, σ =4; (b) μ= -3; σ=1/2.

A normal distribution is symmetric around

its mean

Y

ProbabilityDensity

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 μ-σ and μ+σ

• 95% of randomly-drawn observations fall between μ-2σ and μ+2σ

Standard normal distribution

• A normal distribution with:

• Mean of zero. (μ = 0)• Standard deviation of one. (σ = 1)

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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

x.x0 x.x1 x.x2 .x3 x.x4 x.x5 x.x6 x.x7 x.x8 x.x9

...

1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592

1.6 0.0548 0.0537 0.05262 0.05155 0.0505 0.04947 0.04846 0.04746 0.04648 0.04551

1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.0392 0.03836 0.03754 0.03673

1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938

1.9 0.02872 0.02807 0.02743 0.0268 0.02619 0.02559 0.025 0.02442 0.02385 0.0233

2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.0197 0.01923 0.01876 0.01831

2.1 0.01786 0.01743 0.017 0.01659 0.01618 0.01578 0.01539 0.015 0.01463 0.01426

2.2 0.0139 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.0116 0.0113 0.01101

Standard normal table: Z = 1.96

Pr[Z>1.96]=0.025

Normal Rules

• Pr[X < x] =1- Pr[X > x]

+ = 1 Pr[X<x] + Pr[X>x]=1

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

Normal Rules

• Pr[X > x] = Pr[X < -x]

• Pr[X < x] =1- 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 =Y − μ

σ

Z-score

€

Z =Y − μ

σ

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 =Y − μ

σ

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?

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Draw a rough sketch of the question

μ = 177.0cmσ = 7.1cmy = 180.3

Pr[Y > y]

€

Z =Y −μ

σ

Z =180.3−177.0

7.1Z = 0.46

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

0.5

0.30854

0.30503

0.30153

0.29806

0.2946 0.29116

0.28774

0.28434

0.28096

0.2776

Part of the standard normal table

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?