Collecting Data

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Unit 6

• Collecting Data

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My cabbages are better
Neighbour shows you a cabbage and claims it is
larger than any you could grow.

• So what? I grow tasty bug-free cabbages, not big
  ones.
• Anyone can grow one big cabbage, but all mine are
  above average.
• On my soil it takes real skill to grow cabbages
  at all. Anyone can grow cabbages on your soil.

Relevant
Representative
Unambiguous
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My cabbages are better
Neighbour shows you a cabbage and claims it is
larger than any you could grow.

• So what? I grow tasty bug-free cabbages, not big
  ones.

• Relevance.
• What do you measure and analyse?
• Practical problem, not specifically statistical

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My cabbages are better
Neighbour shows you a cabbage and claims it is
larger than any you could grow.

• Anyone can grow one big cabbage, but all mine are
  above average.

• Representative.
• Decide what objects your conclusions should refer
  to.
• Measure all or select a sample randomly.

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My cabbages are better
Neighbour shows you a cabbage and claims it is
larger than any you could grow.

• On my soil it takes real skill to grow cabbages
  at all. Anyone can grow cabbages on your soil.

• Unambiguous.
• Try to avoid other possible explanations for
  what you see.
• In an experiment, you can randomly allocate
  treatments to a pool of objects.

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Types of data collection

• Samples
• Observational look but dont touch!
• Representative objects from population
• Randomly select objects
• Avoids picking too many of one type
• Experiments
• Actively apply treatment to some objects
• Randomly select objects
• Randomly allocate treatments

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Sampling

• Why Sample?
• Measuring every object of interest is usually
  impractical or uneconomic.
• Sample must be representative.
• Advantages
• Cheaper
• Does not destroy the population
• Can make more careful observations
• Sampling variability can be estimated

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Sampling

• Selecting a Sample
• The reason for selecting an object must have
  nothing to do with the measurement to be made.
• Simple random sampling
• Every object has same chance of being selected.
• Ensures that selected objects are similar to
  rest.
• Statistical procedures guarantee this no basis
  for critics to argue.

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Sampling Terminology

• Population (Target Population)
• All of the objects we want to describe or draw
  conclusions about.
• Sample
• Objects selected from the population.
• Simple Random Sample
• A sample selected by a process that guarantees
  that every possible sample has the same chance of
  selection.

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Sampling Terminology

• Sampling Frame
• A list from which the sample is to be selected.
• Population Parameter
• A number, like the mean weight of cabbages per
  hectare, describing the whole population.
• Sample estimate
• A number calculated from a sample, which is
  intended to be close to the population parameter.

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Selecting a Sample

• Of sheep ( half a dozen)
• Select typical ones.
• First half dozen caught
• If tagged use random numbers
• Put through race select every 10th (if 60)

NO
NO
OK
OK

• Of Grass
• Choose typical areas
• Mark a grid on a drawing and select cells at
  random
• Make a quadrat and throw it haphazardly.

NO
OK
Maybe
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Selecting a Sample

• Number objects 1 to n
• Select random numbers
• Excel
• Minitab
• Calculator
• Random number tables
• Reject numbers outside 1 to n
• Reject duplicates

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Sampling from area

• Generate random x and y coordinates
• Design zig-zag path and sample
• every xxx paces
• at random numbers of paces over path
• Throw quadrats to fine-tune locally

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Notation

• N population size
• m population mean
• n sample size
• x1, x2, xi sample values
•   sample mean

is an estimate of ? is an estimate of
popn total
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Stratified Sampling

• Population may consist of 2 groups
• Large farms / Small farms
• Flat land / Hilly land
• Males / Females
• Maori / Pakeha
• Children / Young adults / Middle aged / Old

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Stratified Sampling

• Divide the population into strata (groups)
• For each stratum
• Separate simple random sample (SRS)
• Estimate mean in stratum
• Scale up to estimate total for stratum
• Add to get estimate of population total
• If required, divide by N to estimate overall popn
  mean

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Stratified sampling

• Provides separate estimates for strata
• More accurate than SRS for popn mean
• Deciding on strata to use
• You must know popn number in each stratum
• Best to use strata whose means are very different
• Sample size in strata?
• Proportional to popn size?
• Better ways? More for variable strata.

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Unambiguous causes from samples

• In survey of farms, those growing Variety A had
  higher mean yield than those growing Variety B
• Cannot conclude that Variety A is better

Farmers with poor soils chose to grow B because
it is more resistant to drought
An experiment is needed to remove ambiguity
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Experimental Design

• Uses a collection of objects experimental units
• Pots growing tomatoes
• Ideally all experimental units are identical
• Identical soil mix, light, variety of tomato,
• Different treatments applied to units
• Half the pots get additional fertiliser
• Only treatment differs, so differences are caused
  by treatment

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Randomisation

• In practice, the experimental units are not
  identical
• Cows are different
• It takes a long time to weigh a herd during which
  they still eat and excrete
• Try to make units as similar as possible
• Randomly allocate treatments to units
• Use random numbers to select cow for each
  treatment

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Replication

• More than 1 unit for each treatment
• Lets you assess natural variability of units
• Based on s.d. within each treatment
• Lets you assess whether difference between
  treatment means is more than random variation

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Blocking

• To improve the experiment, take account of known
  differences between units.
• Before experiment, split units into blocks.
• Like strata in sampling
• Blocks light levels in greenhouse (pot plants)
• Randomise treatments within each block.
• Same mix of varieties within each block

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Example

• Compare sizes of 4 varieties of cabbage
• Experimental units 24 positions in a plot
• Treatments 4 different varieties
• Randomisation Randomly pick positions for 6
  cabbages of each variety in plot.

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Cabbages (cont)

• Plot data
• Find the average cabbage weight for variety A, B,
  C and D
• Assess variability of the plants and the plots by
  looking at the variability for variety A, B, C
  and D

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Cabbages Blocking

• We may feel that there is a difference in
  fertility of the ground from left to right.
• Make each column a block and have one of each
  variety per block.

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Reasons for Blocking

• Allows for naturally occurring differences
  between the experimental units.
• Units are more similar within blocks
• The treatment means are each averaged over the
  same blocks.
• Treatment effects can be compared without the
  effects of the different blocks.
• Differences between the treatments can be found
  within each block.