Variance & Z scores

GY1421 Working With Geographic Information: Week 3 Summary

Transformations & Distributions

Data Transformations & Z-scores

• Importance of Normal Distribution: Descriptive statistics require normally distributed data. Transformations may be needed if data is skewed.

• Types of Distributions:

  • Symmetrical Distributions: Single peak, bell-shaped curve.

  • Multi-modal Distributions: Multiple peaks indicate different groups.

  • Skewed Distributions: Asymmetrical with long tails.

    • Positive skew: Tail to the right.

    • Negative skew: Tail to the left.

Transformation Techniques:

• For Moderately Positive Skew:

  • Square-Root: NEWX = SQRT(X)

  • Reflect & Square-Root: newx = sqrt(k - x)

• For Substantially Positive Skew: Logarithmic: NEWX = LG10(X). For zero: NEWX = LG10(X + C).

• For Moderately Negative Skew:

  • Square-Root: newx = lg10(x)

  • Reflect & Square-Root: NEWX = SQRT(K - X)

• For Substantially Negative Skew: Logarithmic: NEWX = LG10(K - X). Inverse: newx = 1/x.

Z-scores:

• Indicate the number of standard deviations from the mean.

  • Negative z-score: below mean.

  • Positive z-score: above mean.

• The Z-score distribution has mean = 0 and standard deviation = 1.

  • 34.1% within 1 SD on either side; 13.6% within 2 SDs.

• Z-scores can assess where data lies relative to a specific value using a standard normal probability table.