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Analysis of Variance

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This level of statistical analysis of data follows the basic 'error propagation ... two assignable causes vary (e.g. steam flowrate, reflux ratio) ... – PowerPoint PPT presentation

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Title: Analysis of Variance


1
Analysis of Variance
  • (read together with document VARIANCE.doc)
  • This level of statistical analysis of data
    follows the basic error propagation and
    uncertainty analysis (discussed in
    ERR_PROP.ppt)

2
Secondary Treatment
  • analysis of variance (ANOVA) extract information
    about variability
  • Why?
  • how much of the variation is due to
  • assignable causes
  • random effects
  • need to quantify degree of confidence in cause of
    variation

3
Analysis of Variance
  • How?
  • Must do several measurements
  • t-statistic quantifies distance from mean for
    certain degree of confidence that data will be in
    interval

4
Normal Distribution
  • distribution of events for large samples

5
t-statistic
  • smaller numbers of samples

6
t-statistic (cont.)
  • uncertainty in mean decreases as number of
    measurements (e.g. series of temperature
    measurements)

7
Analysis of Variance (ANOVA)
  • from several repeats of an observation under same
    experimental conditions
  • but
  • variance may change with operating conditions
  • may not have time to do repeat experiments under
    all conditions
  • equipment may break down, so repeat observations
    may only be available for some conditions.

8
One-Way Analysis of Variance
  • one assignable cause varies at a time (e.g.
    flowrate)
  • no repeats give information on variance between
    operating conditions, not within an operating
    condition due to uncertainties / errors
  • no repeats mean no error/variance estimate

9
One-Way Analysis of Variance (cont.)
  • worked examples given (in VARIANCE.doc) for
  • measurement uncertainty
  • no measurement uncertainty
  • mean square (error) variance within conditions
    (estimate of measurement uncertainty)

10
One-Way Analysis of Variance (cont.)
  • measurement uncertainty is not the only component
    of error variance (also systematic errors,
    unassigned causes)

11
Two-Way Analysis of Variance
  • two assignable causes vary (e.g. steam flowrate,
    reflux ratio)
  • again, repeat experiments required to get
    estimate of variance

12
Two-Way Analysis of Variance (cont.)
  • worked examples given (in VARIANCE.doc) for
  • no repeats (no estimate of error variance)
  • repeats (estimate of error variance)
  • repeats with no uncertainty/error
  • one-way ANOVA of two-way situation still gives
    estimate of error variance
  • unbalanced ANOVA

13
Unbalanced ANOVA
  • if equipment breaks down, if time runs out, may
    only have some repeat information
  • e.g.

14
Unbalanced ANOVA (cont.)
  • estimate of error variance not as good as from
    complete, balanced, design

15
Relative and Absolute Errors
  • Absolute error e.g. 0.3 - 0.1oC
  • appropriate if error / uncertainty independent of
    operating condition
  • Relative error e.g 0.3 - 10 kg/s
  • error / uncertainty dependent on operating
    condition
  • can be analysed in same way as absolute errors by
    using logarithms

16
Concluding Remarks 1
  • always possible to estimate errors and
    uncertainties associated with measurements (e.g.
    0.5oC in thermometer with 1oC divisions)
  • can always assess how these errors combine

17
Concluding Remarks 2
  • analysis of variance (ANOVA) gives more detailed
    information
  • need some repeat experiments
  • can deal with partial information (e.g. if
    equipment breaks down)
  • absolute and relative errors (relative error
    analysis using logarithms)
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