Scatter diagrams and correlation coefficients

1
Scatter diagrams and correlation coefficients

• Often its interesting to see how two variables
  are related. Examples
• GNP and infant mortality
• Profits of a firm and
• Returns from a share and
• Scatter diagrams
• Correlation coefficients
• Take care about cause and effect
• What do I need to understand about statistics?

2
Scatter diagrams

• For showing relationship between two number
  variables
• Easy with Excel select data and use chart
  wizard on toolbar
• Examples age and earnings, age and drink

3
Correlation coefficients

• Kendalls
• Consider all pairs of observations (of the two
  variables)
• Then Kendalls correlation coefficient
• proportion of SD observations minus proportion
  of RD observations
• (SD same direction RD reversed direction)
• If there are some equal observations, then some
  pairs will be neither SD or RD. Various ways of
  dealing with this (eg see SPSS manual)easiest is
  just to ignore these inconclusive pairs (see
  example )

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Kendalls correlationexample

• Data
• Person Height Shoe size
• A 160 cm 5
• B 180 cm 9
• C 180 cm 10
• No of SDs 2 (AB, AC)
• No of RDs 0
• BC is inconclusive
• Kendall Correlation 2/3 0/3 2/3 0.67
• (SPSS Kendalls tau_b 0.816. This is a more
  sophisticated way of dealing with equal
  observations.)

5
Correlation coefficients

• Pearsons formula is (not important for this
  course)
• This is the standard coefficient referred to as
  the correlation coefficient
• Work out using the correl function in Excel
• Answers similar to Kendalls but not identical

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Correlation coefficients (Kendalls and
Pearsonss)

• 1 for perfect positive correlation (uphill line)
• 0 for no correlation
• -1 for perfect negative correlation (downhill
  line)
• Positive correlation means variables go up and
  down together negative
• Correlation does not measure the slope, but how
  close the pattern is to a straight line
• Examples
• Age and Satunits
• Satunits and Sununits

7
Formulae, computers and understanding (1)

• You can usually get the answer (eg sd, correl,
  regression coefficient)
• with a computer
• Using the formula / method
• Computer is
• quicker and more accurate, but
• You may not understand what the answer means or
  how to use it . This can be serious!

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Formulae, computers and understanding (2)

• Sometimes the formula / method will help you
  understand what the answer means
• Eg percentiles, Kendall correlation coefficients
• Then its a good idea to do simple examples with
  formula/method to help you understand, then use a
  computer

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Formulae, computers and understanding (3)

• Sometimes the formula / method will not help you
  understand what the answer means
• Eg formulae for a regression coefficient, and
  normal distribution (later in the course)
• Here you need much more mathematical background
  to understand properly (especially the normal
  distribution)
• Then its a good idea to
• try to find an alternative approach which is
  easier to follow (regression), or
• Concentrate on understanding the formula/method
  in intuitive terms (normal distribution)

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What do I need to understand for the exam, the
assignment, etc?

• What the answer means, how it relates to the
  inputs, and how it can be used
• How to work it out with a computer (although in
  an exam you will not have a computer and will not
  be expected to remember details of computer
  menus, etc)
• In some cases, how to estimate a rough answer
• For easy methods only, how to work it out without
  a computer

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What might I need to follow published statistical
research?

• All the above
• Mathematical notation
• Sigma (summation) and Pi (product) notation
• Use of a bar above a symbol for mean (average)
• Subscripts Rjk etc
• Eg mean, Pearson correlation coefficient, page
  from a journal.

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Probabilities what they are and where they come
from

• A way of expressing uncertainty
• Probability of 1 means
• Probability of 0 means
• Probability of 0.5 means
• Probability of 0.1 means
• Make sure probabilities are realistic

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Where do probabilities come from?

• Equally likely assumptions
• Drawing balls from a bucket
• Winning the lottery jackpot
• Tossing a coin
• Empirical data
• Customer making a purchase
• Number of customers arriving in a morning
• Weather
• Subjective opinion
• New product selling more than 1000 units in the
  first year may be based on expert assessment
• Or a combination of the above

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Some examples

• Try to work out the following probabilities
• The probability of a card dealt from a well
  shuffled pack being an ace
• What if you did this 100 times. How many times
  would you expect an ace?
• The probability of a card dealt from a well
  shuffled pack being an ace or a spade
• The probability of a black card being a spade
• What is the principle you used to work these out?
  Are you sure the answers are right?

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Some more examples

• Estimate the following probabilities. How would
  you get a more accurate estimate?
• The probability of rain falling in Portsmouth
  tomorrow
• The probability of a randomly selected person in
  Portsmouth having the name Smith
• The probability of GlaxoSmithKline shares going
  up by more than 10 in the next month
• The probability of me (Michael Wood) living to
  the age of 100
• What are the principles you used to work these
  out? Are you sure the answers are right?

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Seminar exercises

• See handout