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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; M=50, SD=28.6, 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,-

M SD

Incmid ƒ2543,- ƒ1481,-

Incmid/1000 kƒ2,543 kƒ1,481

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Linear transformation

• Say you want to know the deviation from the mean

• Subtract the mean (f2543,-) from INCMID

M SD

Incmid ƒ2543,- ƒ1481,-

Incmid-M ƒ0,- ƒ1481,-

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Recap: multiplication and addition and the number line

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Linear transformation

• Adding a constant (X’ = X+c)– M(X’) = M(X)+c

– SD(X’) = SD(X)

• Multiply with a constant (X’ = X*c)– 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: M=0, SD=1

• 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-)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 M=0 and SD=1.

• 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