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The normal approximation to the Poisson, and choosing an approximation

Edexcel IAL Mathematics · WST02 Statistics 2

A Poisson distribution with a large mean is close to a normal curve. With this and the two binomial approximations you need to pick the right one for each question.

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The approximation

If X ~ Po(λ) with λ large (a common guide is λ > 10), then X can be approximated by Y ~ N(λ, λ): same mean and same variance. Use a continuity correction exactly as for the binomial.

Example: Po(30) ≈ N(30, 30), so σ = sqrt(30) = 5.477.

Scale first

Change the interval before approximating. At 2.5 calls per minute, calls per hour are Po(150), which is approximated by N(150, 150).

Examiners note that using the exact Poisson when the question asks for an approximation scores no marks, and that a correct Po(56)-style model must still be given a continuity correction.

Choosing an approximation

Decide from the sizes of n, p or λ. If a question says 'use a suitable approximation', state which one and why, with its parameters.

SituationApproximation
B(n, p), n large, p smallPo(np)
B(n, p), n large, p near 0.5N(np, np(1 - p))
Po(λ), λ largeN(λ, λ)

Worked example

X ~ Po(30). Use a suitable approximation to find P(X ≤ 25).

  1. λ is large, so Y ~ N(30, 30).
  2. P(X ≤ 25) ≈ P(Y < 25.5)
  3. z = (25.5 - 30)/sqrt(30) = -0.8216

Answer: 0.206 (3 s.f.)

Key idea: Po(λ) ≈ N(λ, λ) for large λ, with a continuity correction. Scale λ to the interval before you approximate.

Check yourself

  1. X ~ Po(30). Use a normal approximation to find P(X ≤ 25). Give your answer to 3 significant figures.
    Show answer

    0.206. P(Y < 25.5), z = -0.8216, probability 0.2057 (table 0.2061).

  2. X ~ Po(30). Use a normal approximation to find P(X > 35). Give your answer to 3 significant figures.
    Show answer

    0.158. P(Y > 35.5), z = 1.0042, probability 0.1577 (table 0.1587).

  3. Calls arrive at a rate of 2.5 per minute. Use a suitable approximation to find the probability of more than 160 calls in an hour. Give your answer to 3 significant figures.
    Show answer

    0.196. Po(150) ≈ N(150, 150). P(Y > 160.5), z = 10.5/12.247 = 0.8573, probability 0.1956 (table 0.1949).

  4. X ~ Po(30). Use a normal approximation to find P(X = 30). Give your answer to 3 significant figures.
    Show answer

    0.0727. P(29.5 < Y < 30.5), z = ±0.0913, probability 0.0727 (table 0.0718).

  5. Which approximation should be used for B(400, 0.01)?
    • N(4, 3.96)
    • Po(4)
    • N(400, 0.01)
    • Po(400)
    Show answer

    Po(4). n is large and p is small.

  6. Which approximation should be used for B(80, 0.45)?
    • Po(36)
    • N(36, 36)
    • Po(80)
    • N(36, 19.8)
    Show answer

    N(36, 19.8). n is large and p is near 0.5: N(np, np(1 - p)) = N(36, 19.8).

  7. X ~ Po(45). Which approximation is suitable?
    • N(45, sqrt(45))
    • N(45, 45)
    • B(45, 0.5)
    • N(0, 1)
    Show answer

    N(45, 45). Mean and variance are both 45.

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