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Standard normal: definition, properties, importance

Short answer

Standard normal is Z=(X-mu)/sigma with mean 0, variance 1, pdf (1/sqrt(2pi))e^(-z^2/2). It is symmetric about 0, total area 1, inflection at ±1, and one table serves all normals; it underlies z-tests and CLT.

Define standard normal distribution. Write down its properties and importance of this distribution. (Define Standard Normal Distribution with their respective probability density function and describe its properties.)

Definition

A normal random variable X with mean μ and standard deviation σ is converted to the standard normal variable Z = (X − μ)/σ. Z has mean 0 and variance 1, written Z ~ N(0, 1). Its pdf is

f(z) = (1/√(2π)) e^(−z²/2), −∞ < z < ∞.

Properties

  1. The curve is bell shaped and symmetric about z = 0.
  2. Mean = median = mode = 0; variance = standard deviation = 1.
  3. The total area under the curve is 1; P(Z < 0) = P(Z > 0) = 0.5.
  4. The curve is asymptotic to the z-axis; it never touches it.
  5. Points of inflection are at z = ±1.
  6. Areas: P(−1 < Z < 1) = 0.6827, P(−2 < Z < 2) = 0.9545, P(−3 < Z < 3) = 0.9973.
  7. Φ(−z) = 1 − Φ(z). Skewness is 0 and kurtosis is 3.
          .--'''--.
        .'    |    '.
      _/      |      \_
   __/        |        \__
  -3  -2  -1  0   1   2   3   z

Importance

  • A single table of areas for Z serves every normal distribution, whatever μ and σ are.
  • Probabilities for any N(μ, σ²) are found by standardising.
  • It is the basis for the normal approximations to the binomial and Poisson distributions.
  • The Central Limit Theorem makes the sample mean approximately standard normal after standardising, which is the basis of z-tests and confidence intervals.
  • It allows comparison of values from different scales, using z-scores.

✓ Answer checked by Claude Sonnet 5.5

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