Statistics for Linguistics Students

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Statistics for Linguistics Students

• Michaelmas 2004
• Week 1
• Bettina Braun

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Why calculating statistics?

• Describe and summarise the data
• E.g. examination results (out of 100)
• 22 98 40 45 16 31 77 78
• 55 45 61 91 87 45 54 66
• 75 87 88 49 64 76 58 61
• Average mark/Spread of scores/Lowest and highest
  marks?/Comparison with other results (e.g. from
  last years?)

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Population vs. Sample

• Population total universe of all possible
  observations.Populations can be finite or
  infinite, real or theoretical
• the IQ of all adult men in Britain
• The outcome of an infinite number of flips of a
  coin
• Descriptive statitics are called parameters

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Population vs. Sample (contd)

• Sample Subset of observations drawn from a given
  population
• The IQ scores of 100 adult men in Britain
• The outcome of 50 flips of a coin
• Descriptive statitics from a sample are called
  statistics
• Note In experimental research it is important to
  draw a representative, random sample that is not
  biased

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Histograms Frequency distribution of each event
Data Tutorial1.sav
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Central tendency mode and median

• Mode Most frequent mark (Note there may be
  multiple modes)
• Median score from the middle of the list when
  ordered from lowest to highest. Cuts data into
  halves (doesnt take account of values of all
  scores but only of the scores in middle
  position).

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Central tendency mean

• Mean sum of scores divided by the number of
  scores
• Note on notation Greek letters often used for
  population, roman letters used for statistic
  (properties of a sample)

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Comparing measures of central tendency

• Mode
• quick if we have frequency distribution
• Possible with categorical data
• Median
• Good estimate if we have abnormally large or
  small values (e.g. max aircraft speed of 450km/h,
  480km/h, 500km/h, 530km/h, 600km/h, and 1100km/h)
• Only influenced by values in the middle of
  ordered data
• Mean
• Every score is taken into account
• Some interesting properties ? Most widely used

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Types of variables

• Interval (scale) difference between consecutive
  numbers are of equal intervals (e.g. time, speed,
  distances). Precise measurements
• Ordinal assignments of ranks that represent
  position along some ordered dimension (e.g.
  ranking people wrt their speed, 1 fastest, 4
  slowest). No equal intervals
• Categorical (nominal) numerical categories,
  labels (e.g. brown 1, blue 2, green 3)
• Question on which type of data can we calculate
  a meaningful central tendency?

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Spread of distributions why?
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Spread of distributionsrange and quartiles

• Small spread often desirable as it indicates a
  high proportion of identical scores
• Large spread indicates large differences between
  individual scores
• Range difference between highest and lowest
  score rather crude measure
• Quartiles cuts the ordered data into quarters
  (second quartile median)

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Median, quartiles, and outliers

• Outlier (more than 1.5 box lengths above or below
  the box)
• Interquartile range
• Extreme value (more than 3 box lengths below or
  above the box)

Largest value which is not outlier
Upper quartileMedianLower quartile
Smallest value which is not outlier
tutorial1.sav simple bp, sep. var
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Spread of the population variance measures

• Variance sum of squared deviations from the mean
  
• Variance
• Standard deviation square root of variance

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Normal distribution (Gaussian distribution)

• Example IQ scores, mean100, sd16

Mean Median Mode
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Skewed distributions and measures of central
tendency
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Bimodal distributions
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Normal distribution (Gaussian distribution)

• Example IQ scores, mean100, sd16

Mean Median Mode
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z-scores

• Z-score deviation of given score from the mean
  in terms of standard deviations

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How likely is a given event?

• Example time to utter a particular sentence x
  3.45s and sd .84s
• Questions
• What proportion of the population of utterance
  times will fall below 3s?
• What proportion would lie between 3s and 4s?
• What is the time value below which we will find
  1 of the data?