Standardization of variables

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Standardization of variables

• Maarten Buis
• 5-12-2005

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Recap

• Central tendency
• Dispersion
• SPSS

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Standardization

• Is used to improve interpretability of variables.
• Some variables have a natural interpretable
  metric e.g. income, age, gender, country.
• Others, primarily ordinal variables, do not e.g.
  education, attitude items, intelligence.
• Standardizing these variables makes them more
  interpretable.

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Standardization

• Transforming the variable to a comparable metric
• known unit
• known mean
• known standard deviation
• known range
• Three ways of standardizing
• P-standardization (percentile scores)
• Z-standardization (z-scores)
• D-standardization (dichotomize a variable)

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When you should always standardize

• When averaging multiple variables, e.g. when
  creating a socioeconomic status variable out of
  income and education.
• When comparing the effects of variables with
  unequal units, e.g. does age or education have a
  larger effect on income?

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P-Standardization

• Every observation is assigned a number between 0
  and 100, indicating the percentage of observation
  beneath it.
• Can be read from the cumulative distribution
• In case of knots assign midpoints
• The median, quartiles, quintiles, and deciles are
  special cases of P-scores.

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P-standardization

• Turns the variable into a ranking, i.e. it turns
  the variable into a ordinal variable.
• It is a non-linear transformation relative
  distances change
• Results in a fixed mean, range, and standard
  deviation M50, SD28.6, This can change
  slightly due to knots
• A histogram of a P-standardized variable
  approximates a uniform distribution

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Linear transformation

• Say you want income in thousands of guilders
  instead of guilders.
• You divide INCMID by f1000,-

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Linear transformation

• Say you want to know the deviation from the mean
• Subtract the mean (f2543,-) from INCMID

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Recap multiplication and addition and the number
line
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Linear transformation

• Adding a constant (X Xc)
• M(X) M(X)c
• SD(X) SD(X)
• Multiply with a constant (X Xc)
• M(X) M(X)c
• SD(X) SD(X) c

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Z-standardization

• Z (X-M)/SD
• two steps
• center the variable (mean becomes zero)
• divide by the standard deviation (the unit
  becomes standard deviation)
• Results in fixed mean and standard deviation
  M0, SD1
• Not in a fixed range!
• Z-standardization is a linear transformation
  relative distances remain intact.

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Z-standardization

• Step 1 subtract the mean
• c -M(X)
• M(X) M(X)c
• M(X) M(X)-M(X)0
• SD(X)SD(X)

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Z-standardization

• Step 2 divide by the standard deviation
• c is 1/SD(X)
• M(Z) M(X) c
• M(Z) 0 1/SD(X) 0
• SD(Z) SD(X) c
• SD(Z) SD(X) 1/SD(X) 1

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Normal distribution

• Normal distribution Gauss curve Bell curve
• Formula (McCall p. 120)
• Note the (x-m)2 part
• apart from that all you have to remember is that
  the formula is complicated
• Normal distribution occurs when a large number of
  small random events cause the outcome e.g.
  measurement error

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Normal distribution

• Other examples the height of individuals,
  intelligence, attitude
• But the variables Education, Income and age in
  Eenzaam98 are not normally distributed

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Z-scores and the normal distribution

• Z-standardization will not result in a normally
  distributed variable
• Standardization in NOT the same as normalization
• We will not discuss normalization (but it does
  exist)
• But If the original distribution is normally
  distributed, than the z-standardized variable
  will have a standard normal distribution.

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Standard normal distribution

• Normal distribution with M0 and SD1.
• Table A in Appendix 2 of McCall
• Important numbers (to be remembered)
• 68 of the observations lie between 1 SD
• 90 of the observations lie between 1.64 SD
• 95 of the observations lie between 1.96 SD
• 99 of the observations lie between 2.58 SD

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Why bother?

• If you know
• That a variable is normally distributed
• the mean and standard deviation
• Than you know the percentage of observations
  above or below and observation
• These numbers are a good approximation, even if
  the variable is not exactly normally distributed

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P Z standardization

• Both give a distribution with fixed mean,
  standard deviation, and unit
• P-standardization also gives a fixed range
• Both are relative to the sample if you take
  observations out, than you have to re-compute the
  standardized variables

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P Z-standardization

• When interpreting Z-standardized variables one
  uses percentiles
• With P-standardization one decreases the scale of
  measurement to ordinal, BUT this improves
  interpretability.

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Student recap
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Do before Wednesday

• Read McCall chapter 5
• Understand Appendix 2, table A
• make exercises 5.7-5.28