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MathidiaMathematics Academy
The normal distribution
Lesson 47 of 75
Easy practice

Standardisation

Objective. Transform values to and from standard normal scores.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Standardisation converts a value x from a normal distribution N(μ, σ²) into a z-score using z = (x−μ)/σ, placing it on the standard normal distribution N(0,1) so probabilities can be read from standard tables. A positive z means x is above the mean, and a negative z means it is below. For X ~ N(60, 100), standardising x = 75 gives z = (75−60)/10 = 1.5. Always divide by the standard deviation, not the variance, when standardising.

Worked example

Standardise the value x = 42 for the distribution X ~ N(50, 16).

42 … 50
Illustrated card showing 42 … 50
  1. 1Identify μ = 50 and σ² = 16, so σ = 4
  2. 2Apply z = (x−μ)/σ
  3. 3Substitute: z = (42−50)/4
  4. 4Simplify to z = −2

Answer: z = −2

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Common mistake. Students often divide by σ² instead of σ when standardising, for instance dividing by 100 instead of 10 for N(60,100). Always take the square root of the variance first to get σ before dividing.

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