Quantitative Analysis Week 3: Probability and the Normal Distribution

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Quantitative Analysis Week 3Probability and the
Normal Distribution

• Arani Kajenthira
• arani.kajenthira_at_linacre.ox.ac.uk

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Survey of 1st Year Geography Undergrads

• 53 of students find statistics painful
• 78 of students find statistics boring
• 85 of students sit next to friends from college
  in lecture
• 95 of students are unwilling to volunteer
  answers in lecture
• 99 of students are hungry at noon
• 100 of students need to understand statistics to
  do well in their exams

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Recap of Last Week Dispersion
1. The concept of dispersion and relation to
central tendency 2. Measures of dispersion a)
Range b) Quartile Deviation c) Standard
Deviation 3. Dispersion Diagrams 4. Skew and
Kurtosis 5. Coefficient of Variation
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Lecture Notes Online

• http//weblearn.ox.ac.uk/site/socsci/ouce/uhs/prel
  ims/09_stats/
• Must log in to WebLearn with your Herald username
  and password to see the material

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Summary of Week 3
1. Probability 2. The Normal Distribution a)
Basic properties b) Probability
characteristics 3. The Standard Normal
Distribution a) Standard deviates and z-score
4. Examples, use of the probability table 5.
Cumulative Probability
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1. Probability

• The basis for inferential statistics
• Inferential statistics are tools which allow us
  to make
• -- quantitative probabilistic statements
• -- quantitative predictive statements

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1. Probability

• Tossing a coin
• What is the probability of getting a tail?
• 1 in 2 or 1/2 or 0.5 or 50
• Throwing a dice
• What is the probability of throwing a six?
• 1 in 6 or 0.167
• These are a priori probabilities

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1. Probability

• Rainfall
• What is the probability of daily rainfall of
  125mm in Oxford?
• Politics
• What is the probability that people will vote
  Labour, Conservative, Lib-Dem, Green?
• Need data to analyse
• These are empirical or posterior probabilities

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2. The Normal Distribution
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Probability in Statistics

• Relative frequency histogram sum of the lengths
  of the bars 1
• Rescaled into a histogram of relative frequency
  density sum of the areas of the bars 1

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Probability in Statistics

• The relative frequency density with a small
  sample size can be represented with a bar chart

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Probability in Statistics

• As the sample size is increased the cells become
  finer but the relative frequencies remain stable

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Probability in Statistics

• Once the sample size is very large, the sample
  distribution begins to represent the population
  probability
• The relative frequency density therefore becomes
  a smooth probability density curve

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2. The Normal Distribution

• Established by Karl Friedrick Guass 1777-1855
• See Wonnacott and Wonnacott (1990) p. 641
  Appendix 4-4

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2. The Normal Distribution
many natural phenomena are approximately normally
distributed

• For example, errors made in measuring physical
  and economic phenomena often are normally
  distributed

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The normal distribution is symmetricalon either
side of the mean value
50 of samples in the data set lie above the mean
50 of samples in the data set lie below the mean
lowest value in sample set
highest value in sample set
There is a 100 probability that any data value
in sample lies within extremes
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Remember A probability of 100 A probability
of 1.0 The area under the normal probability
density curve is equal to 1.0
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There is a 68 probability of a measurement lying
within 1 standard deviation of the mean

• The full power of the standard deviation is
  realized when it is combined with the normal
  distribution curve

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There is a 95 probability of a measurement lying
within 2 standard deviations of the mean
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3. The Standard Normal Distribution

• A standard normal distribution is a normal
  distribution with
• Mean 0
• Standard deviation 1
• Any normal distribution can be transformed to a
  standard normal distribution

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3. The Standard Normal Distribution

• z is the number of standard deviations that x
  lies above/below the mean
• a value equal to the mean transforms into 0.0
• a value one standard deviation larger than the
  mean transforms into 1.0

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3a. Probability tables for the z-statistic

• The z-score is the standard normal deviation
  from the mean
• The z-tables provide the probability of an
  observation occurring ? z standard deviations
  from the mean
• Probability Area under the standard normal curve

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Example 1

• What is the probability that an individual value
  will lie less than 1.5 standard deviations above
  the mean for data that are normally distributed?

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Example 2

• What is the probability that an individual
  measurement will have a value less than 0.8
  standard deviations below the mean in data that
  are normally distributed?

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Example 2

• in the tables for the z-statistic we only have
  values for positive z!
• However, the normal distribution is
  symmetricalso

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Example 2

• We look up value for z 0.80 p 0.7881
• However, because z -0.80

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Example 3

• What is the probability of getting less than
  500mm rainfall in any one year in Edinburgh?

Mean 664 Std Dev 120
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Example 3
First, we express 500 mm in terms of standard
deviations from the mean
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Example 3
First, we express 500 mm in terms of standard
deviations from the mean
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Example 3
Then, draw a sketch the table gives us p2 but
we need p1
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Example 3
p1 1.0 p2 1.0 0.9147 0.0853
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5. Cumulative Probability
Cumulative Normal Density Curve
Normal Density Curve
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5. Cumulative Probability
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5. Cumulative Probability
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Summary of Week 3
1. Probability 2. The Normal Distribution a)
Basic properties b) Probability
characteristics 3. The Standard Normal
Distribution a) Standard deviates and z-score
4. Examples, use of the probability table 5.
Cumulative Probability
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Excel skills in Practical 3

• Formatting cells
• More functions NORMSDIST and NORMSINV
• Entering formulas