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How do I know which
distribution to use?
Three discrete distributions
Proportional
Poisson
Binomial
Proportional
Given a number of categories
Probability proportional to number of opportunities
Days of the week, months of the year
Binomial
Number of successes in n trials
Have to know n, p under the null hypothesis
Punnett square, many p=0.5 examples
Poisson
Number of events in interval of space or time
n not fixed, not given p
Car wrecks, flowers in a field
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
Binomial
Poisson
Generate
expected
values
Calculate
2 test statistic
Sample
Test statistic
Null hypothesis
compare
Null distribution
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
Chi-squared goodness of fit test
Null hypothesis:
Data fit a particular
Discrete distribution
Sample
Calculate expected values
Chi-squared
Test statistic
compare
Null distribution:
2 With
N-1-p d.f.
How unusual is this test statistic?
P < 0.05
Reject Ho
P > 0.05
Fail to reject Ho
The Normal Distribution
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Babies are “normal”
Babies are “normal”
Normal distribution
0.4
0.3
0.2
0.1
-2
-1
0
1
2
3
Normal distribution
• A continuous probability distribution
• Describes a bell-shaped curve
• Good approximation for many biological
variables
Continuous Probability
Distribution
0.4
0.3
0.2
0.1
-2
-1
0
1
2
3
The normal distribution is very
common in nature
Human body temperature
Human birth weight
Number of
bristles on a
Drosophila
abdomen
Discrete probability
distribution
Continuous probability
distribution
Poisson
Probability density
Probability
0.4
0.3
Normal
0.2
0.1
-2
Pr[X=2] = 0.22
-1
0
1
2
3
Pr[X=2] = ?
Probability that X is EXACTLY
2 is very very small
Discrete probability
distribution
Continuous probability
distribution
Poisson
Probability density
Probability
0.4
0.3
Normal
0.2
0.1
-2
Pr[X=2] = 0.22
-1
0
1
2
Pr[1.5≤X≤2.5] =
Area under the curve
3
Discrete probability
distribution
Continuous probability
distribution
Poisson
Probability density
Probability
0.4
0.3
Normal
0.2
0.1
-2
Pr[X=2] = 0.22
-1
0
Pr[1.5≤X≤2.5] =
0.06
1
2
3
Normal distribution
Probability density
0.4
0.3
0.2
0.1
-2
-1
0
1
2
3
x
f x  

1
2
2
e
x  2
2
2
Normal distribution
Probability density
0.4
0.3
0.2
0.1
-2
-1
0
1
2
3
x
Area under the curve:
Pr[a  X  b] 

b
a
f [X]dX
Pr[a  X  b] 


b
1
a
2

2
e

b
a
x  
2
2
f [X]dX
2
dX
But don’t worry about this for now!
A normal distribution is fully
described by its mean and
variance
Fig ure 10.3. The normal distribution with different values of the
mean and variance. (a)  =5,  =4; (b) = -3; =1/2.
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. ( = 0)
• Standard deviation of one. (= 1)
0.4
0.3
0.2
0.1
-2
-1
0
1
2
3
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
2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018
2.1 0.01786 0.01743
2.2
0.0139
0.017
0.025
0.0197
0.01659 0.01618 0.01578 0.01539
0.01355 0.01321 0.01287 0.01255 0.01222 0.01191
0.02442 0.02385
0.0233
0.01923 0.01876 0.01831
0.015
0.0116
0.01463 0.01426
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

Z-score
Z
Y

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

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.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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
 = 177.0cm
 = 7.1cm
y = 180.3
Pr[Y > y]
Z
Y 

180.3 177.0
Z
7.1
Z  0.46
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
0.5
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?