Modelling Variability

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Unit 7

• Modelling Variability

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Generalising from data
Weight gain of a litter of 40 pigs in 20 days

• A farmer is interested in specific pigs
• Researcher wants to generalise. What would you
  expect from other pigs with same diet?

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Generalising from data
Weights of 144 carrots

• No interest in specific carrots
• Researcher wants to generalise. What is
  distribution of this type of carrot in general?

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Randomness of data

• Repeat data collection
• Different data
• But similar data
• What do we mean by a generalisation?
• Any data set gives information about it

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Model for randomness of data

• Data Sample from underlying population
• Population may be real
• Herd sizes from sample of 20 dairy farms in
  region
• Popn is herd sizes of all dairy farms in region
• Usually population is hypothetical
• What we would get if infinite data set collected

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Describing population

• Smooth histogram

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The Normal Curve

• Many data sets are fairly symmetric, tailing off
  in the same way on both tails.
• Pig weight gains
• Skew data becomes more symmetric if averages or
  totals are examined.

Bunches of 4 carrots
Single carrots
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Examples with normal model
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Normal curves (distributions)

• Defined by two numbers (parameters)
• Mean (centre of distribution), ?
• Standard deviation (spread of distn), ?
• Shape all normal distns look as follows

??
??
?
?
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70-95-100 rule for normal distn

• Normal (?, ?)

??
??
?
?

• Exactly Easy approximation
• P(X within ? of ?) 0.683 approx 70
• P(X within 2? of ?) 0.954 approx 95
• P(X within 3? of ?) 0.997 approx 100

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70-95-100 rule-of-thumb
Any symmetric bell-shaped distn

• 70-95-100 rule-of-thumb
• P(X within s of x) is approx 70
• P(X within 2s of x) is approx 95
• P(X within 3s of x) is approx 100

s
s
s 5.09
Pigs
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Best-fitting normal distribution

• Best estimate of ? is the sample mean
• Best estimate of ? is the sample st devn

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Probabilities for normal distn

• Normal curve is a histogram
• Proportion of values (probability) equals area

• How likely is it that we will get a result
  between 25 28?

• 1/4 of area
• About 1 time in 4
• Probability 0.25

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Finding normal probabilities

• By eye
• Draw rough normal curve and guess area

• Exact
• Find z-scores for ends of interval (number of
  standard deviations from the mean)
• Look up areas to right of z-scores (tables)
• Add or subtract areas to give answer

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Using Standard Normal Tables

• Area (probability) of lower value than z for
  normal(?0, ?1) distribution
• Find probability of -
• z lt 1.30
• z lt -0.75
• z gt 1.23
• z gt -1.68
• Find the z value if the area is 0.6734

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Probabilities for Real Data

• For normal distns with ? ? 0 or ? ? 1
• z-value is number of st devns above mean
• Translate question into z-values
• Look up z values in the table as before.

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Examples

• Weight gains are approx normal
• Mean is 15 kg, standard deviation is 3 kg
• What proportion of animals -
• Gain less than 10 kg
• Gain less than 19 kg
• Gain more than 14 kg
• Gain more than 22 kg
• Gain between 13 and 20 kg

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Solution-

• Sketch distn
• From tables area to left is
• Probability is

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Weight gains of 40 pigs in 20 days

• 14.7 23.6 15.1 12.9 17.4 25.4 5.3 10.7
  17.4 16.0
• 14.2 13.8 4.9 13.4 8.5 10.7 23.6 19.6
  8.5 13.4
• 17.4 15.1 14.7 14.7 14.7 17.4 16.0 14.2
  14.2 13.4
• 7.6 9.8 8.9 8.5 23.6 9.3 23.6
  11.1 17.8 9.3

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Pig weight gains
s 5.09

• mean - 2s mean - s mean mean s mean 2s
  4.08 9.17 14.26 19.35 24.44

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Probability and Distributions

• Random Phenomenon
• A phenomenon is called random if individual
  outcomes are uncertain but there is none the less
  a regular distribution in a large number of
  repetitions.
• Probability
• The probability of an event is its relative
  frequency in a great many repetitions of the
  random phenomenon.

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Central Limit Theorem

• Averages or totals are closer to a normal distn
  than individual values

• Grass grubs 200 cores (100mm diameter)

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Central Limit Theorem

• Agricultural data is often totals from many
  individuals, so normal distribution is often
  reasonable model.
• We often summarise data with means
• Normal distribution for sample means