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The normal approximation to the binomial

Edexcel IAL Mathematics · WST02 Statistics 2

For large n with p not too far from 0.5, a binomial distribution is close to a normal curve. You approximate it by a normal distribution with the same mean and variance, using a continuity correction.

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The approximation

If X ~ B(n, p) with n large and p close to 0.5, then X can be approximated by Y ~ N(np, np(1 - p)).

Example: B(200, 0.3) ≈ N(60, 42). Write the distribution with its parameters: mark schemes award a mark for N(60, 42).

Continuity correction

X is discrete and Y is continuous, so each whole number x is represented by the interval x - 0.5 to x + 0.5.

DiscreteContinuous
P(X ≤ 55)P(Y < 55.5)
P(X < 55)P(Y < 54.5)
P(X ≥ 70)P(Y > 69.5)
P(X > 70)P(Y > 70.5)
P(X = 60)P(59.5 < Y < 60.5)
P(55 ≤ X ≤ 65)P(54.5 < Y < 65.5)

Method

1. State the approximating distribution, N(np, np(1 - p)).

2. Apply the continuity correction.

3. Standardise: z = (x - μ)/σ, remembering σ = sqrt(np(1 - p)).

4. Use the table (z to 2 d.p.) or a calculator.

Examiners report that omitting the continuity correction, or standardising with the wrong value, loses the final marks.

Worked example

X ~ B(200, 0.3). Use a normal approximation to find P(X ≤ 55).

  1. Y ~ N(60, 42), σ = sqrt(42) = 6.4807.
  2. P(X ≤ 55) ≈ P(Y < 55.5)
  3. z = (55.5 - 60)/6.4807 = -0.6944
  4. P(Z < -0.69) = 1 - Φ(0.69) = 0.2451

Answer: 0.245 (3 s.f.) using the table; 0.244 with a calculator. Both are accepted

Key idea: B(n, p) ≈ N(np, np(1 - p)) for large n, p near 0.5. Always use a continuity correction and σ = sqrt(variance).

Check yourself

  1. X ~ B(200, 0.3). Use a normal approximation to find P(X ≤ 55). Give your answer to 3 significant figures.
    Show answer

    0.244. P(Y < 55.5), z = -0.6944, probability 0.2437 (table 0.2451).

  2. X ~ B(200, 0.3). Use a normal approximation to find P(X ≥ 70). Give your answer to 3 significant figures.
    Show answer

    0.0713. P(Y > 69.5), z = 9.5/6.4807 = 1.4659, probability 0.0713 (table 0.0708).

  3. X ~ B(200, 0.3). Use a normal approximation to find P(55 ≤ X ≤ 65). Give your answer to 3 significant figures.
    Show answer

    0.604. P(54.5 < Y < 65.5), z = ±0.8487, probability 0.6039 (table 0.6046).

  4. X ~ B(200, 0.3). Use a normal approximation to find P(X = 60). Give your answer to 3 significant figures.
    Show answer

    0.0615. P(59.5 < Y < 60.5), z = ±0.0772, probability 0.0615 (table 0.0638).

  5. X ~ B(200, 0.3) is approximated by N(μ, σ2). Find σ2.
    Show answer

    42. np(1 - p) = 200 × 0.3 × 0.7 = 42.

  6. X ~ B(100, 0.45) is approximated by Y ~ N(45, 24.75). Which is the correct continuity correction for P(X < 40)?
    • P(Y < 40.5)
    • P(Y < 40)
    • P(Y < 39.5)
    • P(Y > 39.5)
    Show answer

    P(Y < 39.5). X < 40 means X ≤ 39, which becomes Y < 39.5.

  7. For which binomial distribution is a normal approximation most suitable?
    • B(120, 0.45)
    • B(12, 0.45)
    • B(120, 0.02)
    • B(500, 0.001)
    Show answer

    B(120, 0.45). n is large and p is close to 0.5.

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Last updated 8 November 2026. Spotted a mistake? Tell us.