The normal distribution and the standard normal table

Shape and notation

X ~ N(μ, σ²) means X is normally distributed with mean μ and variance σ² (so standard deviation σ). The curve is bell-shaped and symmetrical about μ; mean = median = mode. The total area under the curve is 1, and areas give probabilities. For a continuous variable, P(X = a) = 0, so P(X < a) = P(X ≤ a).

About 68% of values lie within 1σ of the mean, 95% within 2σ and almost all (99.7%) within 3σ.

Standardising

Z = (X - μ)/σ turns any normal variable into the standard normal Z ~ N(0, 1). The table gives Φ(z) = P(Z < z) for z ≥ 0, with z to 2 decimal places and Φ(z) to 4 decimal places.

Example: Φ(1.25) = 0.8944.

Using symmetry

• P(Z > a) = 1 - Φ(a)
• P(Z < -a) = P(Z > a) = 1 - Φ(a)
• P(a < Z < b) = Φ(b) - Φ(a)

Always sketch the curve and shade the area you want; examiners note that forgetting to subtract from 1 is the most common error.

Values from the standard normal table (computed with the error function; they match the Pearson table to 4 d.p.)

Worked example

Find P(-1 < Z < 2).

1. Φ(2) = 0.9772
2. P(Z < -1) = 1 - Φ(1) = 1 - 0.8413 = 0.1587
3. 0.9772 - 0.1587

Answer: 0.8185

Check yourself

1. Find P(Z < 1.25). Give your answer to 4 decimal places.
   Show answer

   0.8944. Directly from the table: Φ(1.25) = 0.8944.

2. Find P(Z > 0.84). Give your answer to 4 decimal places.
   Show answer

   0.2005. 1 - Φ(0.84) = 1 - 0.7995 = 0.2005.

3. Find P(Z < -1.5). Give your answer to 4 decimal places.
   Show answer

   0.0668. By symmetry, P(Z < -1.5) = 1 - Φ(1.5) = 1 - 0.9332 = 0.0668.

4. Find P(-1 < Z < 2). Give your answer to 4 decimal places.
   Show answer

   0.8186. 0.9772 - 0.1587 = 0.8185.

5. Find P(-1.96 < Z < 1.96). Give your answer to 4 decimal places.
   Show answer

   0.9500. 2Φ(1.96) - 1 = 2(0.9750) - 1 = 0.9500.

6. X ~ N(50, 16). Find the z value for x = 56.
   Show answer

   1.5. σ = sqrt(16) = 4, so z = (56 - 50)/4 = 1.5.

7. X ~ N(30, 9). What is the standard deviation of X?
   • 9
   • 3
   • 81
   • 30
   Show answer

   3. The second number is the variance σ² = 9, so σ = 3.

8. For X ~ N(μ, σ²), which statement is true?
   • P(X < μ) = 0.5
   • P(X = μ) = 0.5
   • The mean is greater than the median
   • P(X < μ - σ) = P(X < μ + σ)
   Show answer

   P(X < μ) = 0.5. The distribution is symmetrical about μ, so half the area lies below μ.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1
• Pearson Edexcel IAL Statistics S1 (WST01) examiners' report, October 2024

Last updated 7 November 2026. Spotted a mistake? Tell us.