Normal Distribution

1
Normal Distribution

• A particular family of distributions (bell
  curve)
• Where once you know the mean and the standard
  deviation
• you know the distribution
• Ae(x-ltxgt)2 gives a bell shaped curve
• Which many real world distributions approximate
• And which has characteristics that are known and
  useful
• About 68 within one stdev, 95 within two, 99.7
  within three
• If you know the mean IQ is 100 and the stdev is
  15, just how special is your IQ 150 kid?
• Z score table is the continuous version of that
  rule
• Z score is the number of standard deviations from
  the mean.
• Table tells you how likely it is that the Z score
  is no higher than that

2
Central Limit Theorem

• Population mean M, standard Deviation ?
• Take a sample of size N
• The average of the sample is an unbiased estimate
  of M
• The StDev calculated from the sample (dividing by
  N-1 instead of N) is an unbiased estimate of ?
• Suppose you repeated the experiment many times.
• Each time you get an average value
• The standard deviation of those averages is ? /?N
  
• So the bigger N, the closer the sample mean is to
  the population mean
• Why does this matter?
• To test the hypothesis that the population mean
  is 10
• You take a sample of size 16, calculate mean 8,
  ?2
• How likely is it that your sample mean would be
  that far off if the hypothesis is true?
• Compare the deviation (2) with the standard
  deviation
• Not of a sample of one but of the mean of a
  sample of 16
• ? /?16.5, so four standard deviations off.
  Unlikely.

3
Rents paid by law students at SCU

• Take a sample of 100
• First deduce standard deviation of the population
  from the sample
• Calculate the mean of the sample ltrentgt
• For each rent, calculate (rent - ltrentgt)2
• Add up and divide by 99 (why 99 not 100?)
• The square root is your estimate of the standard
  deviation of the population ?
• Which measures how much rents vary from student
  to student
• Then deduce the standard deviation of the mean
• Standard deviation of a sample of size n goes as
  ?/square root of n
• For samples of that size, thats how much their
  means would vary
• How likely is it that ltrentgt is at least that far
  from 1000?
• The distribution of means is approximately normal
• You know its standard deviation ?/10
• So ltrentgt-1000/(?/10) is z, consult the z table

4
The Calculation

• Hypothesis being tested average rent 1000
• Hypothetical numbers (from the book)
• Sample size 100
• ltrentgt950 Average of the sample
• ? 150 Standard deviation of the population
  (estimate)
• ?/?100 ?/10 15 Standard deviation of the
  mean
• So Z 50/15 3.33 standard deviations above
• Two tailed test why?
• Z table shows .995 below 3.33, .005 below -3.33
• So .99 between the two values
• So .01 probability that ltxgt at least that far
  from 1000 by chance

5
What does it mean?

• If the average rent for all students is 1000
• There is one chance in 100
• That a sample of 100 rents would have a mean
• At least 50 higher or lower
• Significance at .01--very strong result
• That does not mean either
• That the probability the rent is actually 1000
  is .01
• How high do you think it is?
• Or that the difference of the rent from 1000 is
  significant in the normal sense--i.e. large
• Suppose the population were San Jose, n10,000
• Z3.33 represents a mean how far from 1000?

6
Hypothesis Testing

• The basic logic of confidence results
• You have a null hypothesisthis coin is fair
• You have a samplesay the result of flipping the
  coin ten times. 7 heads.
• You want to decide whether the null hypothesis is
  true
• In the background there is an alternative
  hypothesis
• Which is relevant to how you test the null
  hypothesis
• For instancethis coin is not fair, but I don't
  know in which direction
• You ask If the null hypothesis is true, how
  likely is a result at least this far from what it
  predicts in the direction the alternative
  predicts
• For example, if the coin is fair
• How likely is it that the result of my experiment
  would be this far from 50/50?
• Suppose the answer is that if the coin is fair,
  the chance of being this far off 50/50 is less
  than .05 (i.e. 5)
• You then say that the null hypothesis is rejected
  at the .05 level

7
To Restate

• Confidence level tells you how strong this piece
  of evidence against the null hypothesis is
• but not how likely the null hypothesis is to be
  true
• analogously, it might be that a witness
  identification has only one chance in four of
  being wrong by chance
• but if you have a solid alibi, you still get
  acquitted
• "Statistically significant" doesn't mean
  "important" it means "unlikely to occur by
  chance"
• I take a random coin and flip it 10,000 times
• the result will prove it isn't a fair coin to a
  very high level of significance
• Even if it is "unfair" only by .501 vs .499
  probability

8
This is all sampling error

• Sampling error can be calculated, but..
• Other forms of error may be more important
• So "the margin of error is" may be misleading
• Consider DNA tests
• "The chance that the defendant's DNA would match
  this closely is less than one in a hundred
  million"
• May be a true statement about sampling error
• But there have been far more mistaken results
  than that number suggests
• Rates of human error are much higher than that
• As are rates of deliberate fraud
• Think of sampling error as a lower bound

9
Bayesian Statistics

• Consider again my coin flipping experiment
• Take a coin from my pocket, flip it twice
• Null hypothesis It's a fair coin
• Alternative It's double headed
• Get two heads
• Chance of evidence that strong for the
  alternative is .25
• We dont conclude it has that probability of
  being double headed
• Why?
• We start with a prior probability
• very few coins are double headed
• So the chance of drawing one and then getting
  heads twice
• Is much lower than the chance of drawing a fair
  coin and getting heads twice
• So the latter is what probably happened

10
Done formally

• Suppose one coin in 1000 is double headed
• Probability of pulling one from my pocket .001
• If it is double headed, prob of two heads 1
• So joint probability--that both happen--is .001
• 999 in 1000 coins are (approximately) fair
• P of pulling a fair coin from pocket .999
• If fair, p of two heads .25
• Joint probability is .25x.999.24975
• We know one of these two things happened
• Relative probability is .001/.24975aprox 1/250
• So odds about 250 to 1 that the coin is fair
• This is Bayesian statistics as opposed to
  classical statistics

11
Bayesian Statistics

• Tells you how to
• Start with a set of prior probabilities (.001,
  .999)
• Combine with the result of an experiment
• Deduce posterior probabilities (.004, .996)
• It doesn't tell you
• How to find your prior probabilities
• Those come from knowledge of the situation
• Modified by past experiments
• No prior, no posterior

12
How to Lie Part 2

• Report sampling error as if it was all error
• Report confidence result with meaning reversed
• The theory that the firm didn't discriminate
  against women
• Can be rejected at the .05 level
• So the odds are twenty to one that it did
• Report selected result
• This study found our product clearly worked
• And we aren't telling you about the other 19
  studies
• And this happens even without trying
• Academic version if you don't get results you
  can't publish
• Popular version the most striking result gets
  the press
• Both can cause unintentionally misleading
  results, but also
• Are incentives to deliberately distort results
• Since getting published and getting press may be
  the objectives

13
You can also just lie

• Statistics prove that
• 95 of quoted statistics are invented

Including this one
14
Multivariate statistics

• Each item (person, country, state, year) has two
  characteristics
• How are they related to each other?
• Why?
• Descriptive approach Scatterplot
• Approximate linear relationship. But note
• The plot might show you more complicated things,
  that calculating the correlation coefficient
  would miss.
• Humans come with very good pattern recognition
  built in.

15
Correlation Coefficient

• We have two characteristics, each associated with
  individuals in a population
• Height and weight of people
• Rainfall and average temperature of years
• Income and Lsat score
• Which could be parental income and student LSAT
  score or
• Entering LSAT and later income as a lawyer
• We want to know how the two are related
• When height is above average, is weight above
  average? (Probably)
• Do cool years have more rainfall?
• Correlation coefficient is a measure of how
  consistently
• When one variable is above its average, the other
  is above its (positive correlation)
• Or when one is above, the other is below
  (negative)
• 1 is perfect correlation--if you plot them they
  are on a straight line, slopes up
• -1 is perfect negative correlation--straight
  line, slopes down
• 0 is no correlation--but not necessarily no
  relationship.

16

• The first one you would get a positive
  correlation coefficientwhat would you miss?
• The second one, near zero correlation. But
• The scatter plot shows the pattern

17

• Summary
• The coefficient is from 1 to 1
• Sign tells you whether larger than average values
  of one variable imply larger than average values
  of the other () or smaller (-)
• The magnitude tells you how perfect the relation
  is, not the slope.
• Which of these has the higher correlation
  coefficient?
• This is the same point I made earlier about
  significance
• Statistically significant means we are sure the
  effect is there
• It says nothing about how large it is
• 550 heads/450 tails is much more significant
  evidence of unfairness than
• 3 heads/1 tail

18
Mathematical Definition

• For each value of the first variable, calculate
  how many standard deviations it is from the
  mean-- if greater than mean, - if less
• For each observation (person, state, ) multiply
  that figure for the first variable times that
  figure for the second
• Average over all observations
• (except you divide by n-1 instead of by n in
  averaging)
• for the same reason we did it earliersample
  slightly exaggerates the correlation for the
  population.
• I think
• Why this makes (some) sense
• If above average values of X occur for the same
  observation as above average values of Y, the
  product is positive
• If below go with below, the product is still
  positivenegative times negative is positive
• So if the two variables move together, get a
  positive correlation coefficient
• If they move in opposite directions, above
  average of one go with below average of the
  other, so times or times , which gives
  negative
• Average lots of negative numbers, get a negative
  correlation coefficient

19
Correlation need not be Causation

• It might be entirely due to some third variable
  that causes both
• Driving an expensive car has a negligible effect
  on life expectancyprobably negative if its a
  sports car
• But probably correlates with life expectancy.
  Why?
• Height has little effect on having children, but
  
• Number of children one has born is probably
  negatively correlated with height of adults
• Because?
• Or it might be partly due to such third factors,
  so you don't know how strong the causal effect is
• And third factors might push the other way,
  reducing, eliminating, or reversing the causation
• Death penalty and murder rates
• If factors that make murder rates high make death
  penalty more likely
• Either because high murder rates create pressure
  for death penalty
• Or because the social factors that make people
  more willing to kill illegally also make them
  more willing to kill legally.
• You might have a positive correlation masking a
  negative causation

20
And Causation may not lead to correlation
21
Causation, Correlation and Prediction

• Correlation can be used to predict
• "if the state has a death penalty, it probably
  has a high murder rate"
• doesn't depend on which causes which
• or whether there is a third factor causing both
• but if you have the causality wrong, you might
  get the prediction wrong
• because you are missing other relevant evidence
• taller adults are less likely to have born
  children than shorter

but taller females aren't.
You also might get the policy wrong Dying
correlates with being in the hospital. In order
not to die What if death penalty correlates
positively with murder rate?
22
Linear Regression

• instead of measuring how close to a line the
  points come (correlation coefficient)
• you try to estimate the line they come closest to
• which requires some definition of "close."
• You want to count both being too high and too low
  as errors
• So the difference between point and line wouldn't
  work
• Instead use the square of the differencepositive
  each way
• Find the line that minimizes the summed square
  deviation.

• Unlike the correlation coefficient, this one
  measures the size of the effect
• y ABx
• A is the interceptwhere the line crosses the
  vertical axis
• B is the slopehow much the line goes up for each
  unit it goes out

23
Goodness of Fit

• By convention, X (horizontal) is the independent
  variable, Y (vertical) the dependent YA BX
• Simplest "prediction" is that Y always equals its
  average value
• How much of the departure from that does the
  regression explain?
• TSS is the sum of squared residuals from the
  average

24

• So R2 is a measure of how much of the variance
  about the mean is explained by the regression
  line.
• Total variation minus variation unexplained by
  regression
• divided by total variation
• So R2 of 0 means the regression line does no
  better than just assigning the mean value to
  every point
• R2 of 1 means the regression explains all of the
  variance.
• Like correlation, this is a measure of goodness
  of fit
• In fact, R2 is the square
• of the correlation coefficient r
• And B, the slope, is a measure of the strength of
  the relationship.

25
Residuals

• If you plot the residuals from a
  regression--distance above or below the line
• It will show you which points don't fit the
  pattern
• In exploratory statistics, you might want to
  color points in ways reflecting other
  characteristics
• Men/women
• Blacks/whites
• Northern states/Southern states
• CEO's relatives/non-relatives
• And see if any such coloring explained the
  pattern
• In the book's example, Mary Starchway is both an
  outlier and an influential observation
• Outlier because her wage is much higher than
  anybody else's
• Influential observation because she is far off
  the experience/wage regression line
• Does the first necessarily imply the second?

26
Limitations of Linear Regression

• There might be a close relationship that isn't
  linear
• there are procedure analogous to linear
  regression for dealing with the first case
• Instead of plotting YABX you might plot
• YABXCX2 for example
• Giving something like that if Blt0 and Cgt0
• The second case strongly suggests that we need
  more than two variables
• Y is determined by X, and also by
• Whatever it is that distinguishes the two lines

27
Multiple Regression

• Suppose you believe the murder rate depends on
• The death penalty
• The fraction of the population that is males
  18-26
• This year's unemployment rate
• You could express that as Mab1Db2Fb3U
• Here M is the murder rate, by state
• D is the probability that a murderer will get the
  death penalty, by state
• F is the fraction of the state population that is
  male 18-26
• U is the state's unemployment rate
• The regression could be cross section All states
  in one year
• Or longitudinal One state in a series of years
• Or both

28
More Complicated Versions

• We could define D as
• The fraction of murderers who are executed, or
• Per capita number of executions per year, or
• Perhaps the murder rate depends on the square of
  D, or
• Perhaps D should be treated as a binary variable
  instead of continuous
• States with death penalty, D1
• States without, D0
• Perhaps murder rate in one year depends on
  current unemployment rate but last year's death
  penalty probability
• In which case you use current variables for
  everything else
• But a lagged variable for D
• Meaning that the value for NY in 1990 is the
  death penalty probability for NY in 1989

29
Running a regression means

• minimizing the sum of squared deviation from the
  regression's predictions
• Define as the value of M predicted by the
  regression
• i ab1Di b2Mi b3Ui
• Here i labels the particular observation (state
  and year in this example)
• We are looking for the values of a, b1, b2 and b3
  that minimize
• The sum of squared residuals, i.e. the sum of
  squared values of
• (Mi- i)
• summed over all i, which is to say over all
  states, or years, or

30
Running a regression means

• Minimizing the sum of squared deviation of the
  data from the regression's predictions
• Define as the value of M predicted by the
  regression
• i ab1Di b2Mi b3Ui
• Here i labels the particular observation (state
  and year in this example)
• We are looking for the values of a, b1, b2 and b3
  that minimize
• The sum of squared residuals, i.e. the sum of
  squared values of
• (Mi- i)
• summed over all i, which is to say over all
  states, or years, or

31
Significant Coefficients

• Regression results shows some coefficientgt0
• We want to know how sure we are it is true
• For instance, that whites get paid more than
  blacks
• Controlling for all other relevant factors
• We use a t test which is
• Analogous to the significance tests we have done
• Both in how it works and what it means
• t coefficient/its standard error
• I.e. how big it is relative to how uncertain
• Look up the corresponding confidence level
• On a t table--like a z table, but with one
  complication
• Degrees of freedom

32
Degrees of freedom

• Suppose I have only two data points
• (x1, y1) (x2,y2)
• And do a simple regression yabx
• How well will I fit the data?

• Perfectly
• You can always draw a straight line through two
  points
• The result generalizes
• With n parameters you can fit n data points
• Whatever the relation among them is
• So only fitting more points than that counts as
  evidence
• Which is what the degrees of freedom take account
  of

Give me enough parameters and Ill fit the
skyline of New York
33
Choosing Variables

• How do you decide what variables to include?
• From those that might be relevant and
• That you have data on
• One approach is trial and error
• Try each variable by itself, choose the one with
  the best R2
• Try adding each one, choose the one that
  increases R2 most
• Repeat
• There are computer programs that will do it for
  you
• Problem Out of all possible variables
• Some will fit your dependent variable well by
  chance
• And your procedure will find those ones
• So if you started with thirty candidate variables
• Getting a .05 result for one is not impressive

34
Problems or How to Cheat

• All the usual ways, such as
• Misstate the meaning of significance
• Use a biased sample
• Select which experiments to report
• Use unreliable data
• Plus some brand new ways
• Plaintiff claims aspartame causes cancer
• My regression found no significant relation
• Independent variables age, gender, use of diet
  drinks, aspartame consumption
• Defense claims his prostate medicine doesnt
  shorten life
• My regression shows a strong correlation
• Independent variables state of residency, race,
  use of prostate medicine
• Dependent variable Age at death

35
Collinearity problem

• Significance calculation is based on
• How much better you fit the data by adding this
  variable
• Which depends on what other variables are there
• Suppose you include both temperature F and
  temperature C
• How significant do you think either will be?
• t test is asking how many standard deviations out
  the coefficient is
• Which depends on how precisely you know the
  coefficient
• In my case, if you have one, the coefficient on
  the other could be anything
• Heating oil consumption A B(temp F) C(temp
  C)
• Do you see why?
• The same problem exists in less extreme cases
• Adding a variable that correlates closely with
  another
• Decreases the others significance, because
• The new one can explain most of the same
  variation.

36
Omitted Variable Problem

• You want to prove that X (prostate medicine)
  causes Y (shorter life)
• You leave out a variable that correlates with
  both
• Prostate medicine is only used by men
• Men have shorter life expectancies than women
• So dont include gender in your regression
• Your independent variable X
• Now seems to be predicting Y, because
• X predicts gender, which predicts Y

37
Furman v Georgia

• The case that (temporarily) abolished the death
  penalty
• Also a famous use of statistics
• Data on all capital cases
• Commonly said to have shown discrimination
  against blacks

• But black defendants had a lower probability of
  execution than white defendants!

• Control for race of victim
• black who killed a black more likely to be
  executed than
• White who killed a black. Ditto if victim was
  white. But
• Killer of a black much less likely to be executed
  than of a white
• And blacks mostly kill blacks, whites whites
• Which is why black killers less likely to be
  executed
• It was indeed evidence of racial discrimination
• Slight discrimination against black defendants,
  race of victim held constant
• Large discrimination against black victims, race
  of killer held constant

38
Significance and Standard of Proof

• Book discusses wage discrimination case
• Coefficient on the race effect nonzero but
• Not significant at .05 level
• Footnote suggests that since it is a civil case
• Perhaps .05 is too strong a requirement
• What should it be?
• Would .5 do it?
• Preponderance of the evidence
• Isnt that gt.5 probability?

39
Statistics and the Law School

• You want to raise the bar passage rate
• You have data on all students for the past ten
  years
• Information on them when they applied
• What courses they took, grades they got
• Bar exam outcomes
• How might you use it?
• What questions would you ask?
• How could statistics answer them?
• How could you use the information?

40
Who to admit

• Bar passage rate is the dependent variable
• Independent variables are what you knew about the
  student before admission
• LSAT score
• Undergraduate grades
• Undergraduate major
• Anything else?
• See which ones predict bar passage
• Alter your admission policies accordingly
• Any reasons why this might not work?
• Correlation is not causation
• Any reasons why changing independent variables
• Might not change dependent variable?

41
Class record

• Regress bar passage rate on
• What classes student took
• What grades he got on them
• Suppose you learn that
• Students who took class X were less likely to
  pass
• Students who took Y were more likely
• Would you raise bar passage rate by
• Abolishing class X
• Requiring class Y
• Suppose grades in class Z
• Predict bar passage rates
• Do well in Z, pass the bar, do badly, likely to
  fail
• Drop students who did badly in Z?
• In each case, why might it not work?

42
How about Professors?

• See how bar passage rate depends on
• Which courses the student took
• From which professor
• Take torts from Smith, pass the bar
• From Jones, fail the bar
• Fire Jones, raise Smiths pay or
• If Jones has tenure
• Have him teach something else
• More generally, rearrange who teaches what
• On the basis of regression coefficients showing
• The effect on bar passage rates

43
What do we need to know?

• In each of these cases
• To decide whether using the regression results
• Will let us improve outcomes
• Whether correlation is probably causation
• What additional information might we want?

• How were students assigned
• To courses and to professors
• Suppose X was a class failing students were
  assigned to
• Or Y a class with very selective admissions, or
• Smith a notorious hard grader who weak students
  avoided

44
ABA Fails Statistics

• ABA wants to include bar passage rate in deciding
  what law schools to certify
• What will the effect of doing this be?
• Why is it a mistake?

• Bar passage rate depends on at least two things
• Characteristics of the student
• Characteristics of the law school he went to
• Almost any school can get a student to pass the
  bar
• If he is sufficiently smart and hard working
• What matters is value added
• For a student with a given set of characteristics
• How likely is he to pass the bar if he goes to
  this school

To take account of bar passage, how should they
do it?
45
Use a Regression

• BPRabLsatc
• BPR Bar Passage Rate
• Lsat students Lsat score
• c represents other relevant student
  characteristics
• The higher a and b, the better the school
• Because the more likely to get a given student
• To pass the bar
• Some schools may do well with low Lsat students,
  some with high
• So report a, b, c
• And let the student calculate the probability
  that he will pass
• If he goes to that school
• Bar association could decide to certify any
  school
• That does relatively well for some
• Substantial group of students
• Including schools that are good for weak students

46
Statistics Exercises

• On the syllabus, for practice
• Do the calculations with numbers
• We will discuss them next class

47
Statistics You should know

• Ways of displaying and summarizing data
• Histogram, median, mean
• Some idea of what which are useful for
• What terms such as "significant" and "confidence
  interval" mean
• Testing a conjecture
• Null hypothesis/alternative hypothesis
• One tailed and two tailed tests
• Normal distribution, central limit theorem, z
• What a correlation coefficient shows
• What a regression result, single or multiple,
  means
• Coefficients and
• Measures of significance (R2, t)
• What can go wrong
• How statistical results can be presented to
  mislead
• How statistics can mislead, intentionally or not

48
You are not expected to

• Be able to do a regression
• If you ever need to, find the relevant software
• Or get a statistician to do it for you
• Prove things
• Give precise definitions
• Of correlation coefficient
• Least squares fit
• R2
• But you should understand about what they mean
• You need to understand enough
• Not to be fooled
• To know what questions to ask
• And about what the answers mean

49
First problem

• A friend, visiting SCU, comments on how young the
  law students look, and conjectures that their
  average age is only 24. You disagree, and assure
  him that it is older than that. To see which of
  you is right, you ask four students in one of
  your classes how old they are, and use what you
  have learned in this class to analyze their
  answers
• Aside from the small sample size, what possible
  problems are there with this procedure?
• 24, 26, 24, 30
• What is the mean age of the sample? (26)
• Standard deviation of the population (estimate)
• (24-26)2(26-26)2 (24-26)2 (30-26)2/(n-1)24/3
  8
• Take the square root ?82.8
• Standard deviation of the mean of 4 observations
  2.8/ ?41.4
• Z(26-24)/1.41.4
• One tailed or two tailed?
• How likely this far off by chance? .08