Normalization (Statistics) Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Test scores: Raw score 85/100. Class mean=70, SD=10. Z-score normalize this score and explain what it means relative to classmates.

Solution

  1. 1
    Z-score formula: z=xโˆ’ฮผฯƒ=85โˆ’7010=1510=1.5z = \frac{x - \mu}{\sigma} = \frac{85 - 70}{10} = \frac{15}{10} = 1.5
  2. 2
    Interpretation: the score is 1.5 standard deviations above the class mean
  3. 3
    Percentile (approximately): P(Z<1.5)โ‰ˆ93.3%P(Z < 1.5) \approx 93.3\% โ€” scored better than about 93% of classmates
  4. 4
    Z-score normalization allows comparison across different tests with different scales

Answer

z=1.5z = 1.5; scored 1.5 SDs above mean; better than approximately 93% of classmates.
Z-score normalization (standardization) converts raw scores to a common scale with mean=0 and SD=1. This allows comparison across different tests, classes, or years. A z-score of 1.5 has the same relative meaning regardless of the original scale.

About Normalization (Statistics)

Normalization rescales data to a standard range or distribution โ€” such as [0,1][0,1] or zero mean and unit variance โ€” to make different variables comparable.

Learn more about Normalization (Statistics) โ†’

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