ARCH 21266126

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ARCH 2126/6126

• Session 2a Measurement

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To recap, the purpose of statistics..

• To provide insight into situations and problems
  by means of numbers
• How is this provided?
• Data are available or are collected
• Data are organized, summarized, analysed and
  results presented
• Conclusions are drawn, in context
• Whole process is often guided by critical
  appraisal of similar work already done

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From variation to variables

• Variables that can be analysed numerically are of
  several different sorts
• Categorical/qualitative/nominal variables
• Ranked/ordered/ordinal variables
• Numerical/quantitative/metric variables
• Different kinds of variables allow different
  kinds of numerical analysis
• This applies to the method of description or
  measurement, not the basic property

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Data sets

• Usually data do not come singly they come in,
  or are collected in, sets
• We collect them because we want to test some idea
  fairly against them
• E.g. we might want to test whether the stone
  artefacts from one site differ in size from stone
  artefacts from another
• For this, we measure artefact sizes
  systematically consistently

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What belongs in a data-set?

• We have considered it prudent to adopt the years
  1919-1925, excluding the drought year of 1926, as
  a fair standard for the future Queensland Land
  Settlement Advisory Board, 1927
• The tacit assumption that drought is an
  exceptional visitation to the inland country has
  shaped and infected public thought and official
  policy alike Francis Ratcliffe, 1937

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Making a measurement

• A variable is a measured property of a case
  measuring assigns numbers representing each
  cases value for that variable
• Variables must be exactly defined measurements
  reliably carried out
• Some variables are relatively simple but still
  need explicit specification, e.g. length
• Some are more complex and/or depend on
  non-obvious definitions, e.g. unemployment

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Measurement is never perfectly accurate but

• Our measurement of scraper length is valid, to
  the extent that it measures what it is supposed
  to measure
• Our measurement is reliable, to the extent that
  repetitions of the same measurement give the same
  result
• Our measurement is unbiased, to the extent that
  it does not tend to under-state or over-state the
  true value of the variable

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Recording a measurement

• Rare important observations deserve recording as
  insight-giving anecdotes
• But in many fields the bread butter of research
  are common observations where the issue is
  varying frequency
• Importance of a recording system
• Unsystematic recording is likely to lead to
  omissions or inconsistencies
• Limits to the benefits of precision

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Recording technology

• Pen paper still have their place
• Complex technology has its traps its
  vulnerability, your dependence
• But early, direct or automatic data entry into
  computers can bring big benefits in efficient
  use of time labour error reduction
  cross-checks
• Importance of duplicates back-ups

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How much data to collect?

• Limits to the benefit from measuring variables to
  many significant figures
• Limits to the benefit from increasing sample size
  indefinitely
• Limits to the benefit from increasing number of
  variables how many will you analyse?
• Attention to limits can save lots of time
• Limits not fixed, but depend on the situation
  under study the ideas under test

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Spreadsheets (e.g. Excel) databases (e.g.
Access)

• End point of data collection is often a matrix or
  table a column for each variable, a row for each
  case
• Often convenient to enter these into a
  spreadsheet or database (linkable, searchable)
• These can store, check, transform, calculate,
  apply conditions, select, test statistically,
  output to statpack

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Study design experiment versus observation

• How do we define? Variously but element of
  control often the key
• For practical, ethical etc. reasons, experiments
  rare in our subjects
• But experimental design important
• Dependent variable response variable under study
• Independent variable explanatory variable or
  factor

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Contexts and confounds

• Treatment a combination of specific conditions
  (levels of experimental factors)
• Extraneous variables ones not being studied but
  which may influence dependent variable thus
  part of relevant context
• Effects of different (independent or extraneous)
  variables are said to be confounded if they
  cannot be distinguished
• Good study design requires data on context

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Observational studies the risks of confounding

• Well designed experiments minimize confounding by
  appropriate choice of variables, cases and
  treatments random sequence of treatments
  random allocation of cases to groups
• Observational surveys lack this control
• Groups may be self-selected
• Differences in groups may have causes other than
  the variables under study
• But much can be done despite limitations

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ARCH 2126/6126

• Session 2b Description

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Examples of presentation

• Even the simplest forms of stating findings
  percentages, averages and the simplest
  graphical presentations emphasize selected
  aspects
• This can be legitimate can also be misleading
  much depends on honesty clarity with which
  procedure is described
• What as a percentage of what?
• Does the graph have linear scales? A zero?
• Please bring in examples yourselves

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How can we see patterns inherent in our data-set?

• Start simple e.g. frequency tables
• Frequency or
• Relative frequency
• Value of mental arithmetic cross-checks do
  figures make sense?
• Note rounding errors
• Keep an eye on sample sizes do they change?

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Frequency absolute relative

• Frequency of any value of a variable is the
  number of times that value is found i.e. it is a
  count, a whole number
• Relative frequency of any value is its frequency,
  expressed as a proportion of all observations
  (often a percent)

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Rates and ratios

• Ratio the size of a number relative to another
  number
• Proportion a ratio in which the second number
  includes the first
• Percentage a proportion multiplied by 100
• Rate a ratio of the number of events to the
  number of cases at risk of experiencing that event

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Good to graph results but graphs can also mislead

• Graphical depictions include line graphs, bar
  charts, histograms, pie charts, stem--leaf
  plots, scatterplots
• Line graphs usually used to plot variable against
  time (on horizontal axis) show seasons trends
• Different scales give different impressions, e.g
  non-zero base to vertical axis, unequal units,
  log scale

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Bar charts and histograms

• Bar charts compare the values of different
  variables, often categorical
• Histograms display frequency or relative
  frequency distributions of one variable at a time
• Width of histogram bars has meaning
• Eyes respond to impressions of area symbols,
  unequal widths, pseudo-3D can give a misleading
  impression

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Scatterplots

• Show the distributions of two variables at once
  (i.e. bivariate data)
• If one variable is independent and one dependent,
  independent goes on horizontal axis
• Essence of any relationship between them is
  apparent visually ve or -ve, strong or weak,
  simple or complex
• This can affect future statistical testing

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Measures of central tendency

• The arithmetic mean (average) add all
  observations together, divide the total by the
  number of observations
• (Also geometric harmonic mean)
• The median arrange all observations in order,
  find the middle one or the mid-point of the
  middle two
• The mode find the commonest value

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Central tendencies and distributions

• In a normal distribution, graph is symmetrical
  mean, median mode are similar
• But distributions may be different, e.g may be
  skewed to left or right
• Mean is often convenient but is strongly affected
  by outliers
• To avoid this, can use median less affected by
  outliers and skews

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Measures of central tendency are useful but

• If average income is above the poverty line, is
  poverty abolished?
• If the average child is at a weight//age which is
  thought to indicate healthy growth, are all
  growing healthily?
• If first agriculturalists diffused into Europe at
  an average rate of 1 km / year, does that imply
  rate was constant?
• Variation is ubiquitous sample is not fully
  characterized by its mean/median

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So we also need measures of dispersion
(variability)

• Range maximum minimum outliers
• Percentiles median is 50th percentile we can
  also find 25th 75th percentile (dividing sample
  into quartiles) or 20th, 40th, 60th 80th
  (quintiles) or 3rd 97th percentiles etc.
• Interquartile range (75th percentile 25th) is
  more stable than range

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Box plots

• A simple box--whisker plot consists of a box
  (interquartile range) with a central line
  (median) and a further line each side of the box
  (to the extremes)
• More elaborate versions represent outliers (gt1½ x
  box length from box) by dots not joined to the
  main whisker, far outliers (gt3 x box length)

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Dispersion around the mean

• Variance and standard deviation
• Standard deviation ? variance
• A little more complex to calculate but has some
  very useful properties
• Main component of variance is sum of squares,
  i.e. subtract mean from each observation, square
  result, add them up then divide SoS by sample
  size - 1

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Means, standard deviations normal
distributions

• Normal curves are symmetric, bell-shaped, drop
  off quickly, few outliers
• Mean median mode
• There are many normal curves mean standard
  deviation specify shape
• In a normal curve, point where tails flatten
  out is 1 SD from mean
• SD/mean coefficient of variation

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Properties of normal curve

• 68 of observations fall within 1 SD of mean
• 95 fall within 2 SDs of mean
• 99.7 fall within 3 SDs of mean
• A transformation may help to make a distribution
  approximately normal
• A raw observation can be converted into a
  standardized (z) score, to find the probability
  of its occurrence, with mean 0 SD 1