Quantitative Aptitude · Theoretical Distributions

Choosing and Comparing Theoretical Distributions: Binomial, Poisson and Normal

Q: What is the main difference between binomial, Poisson and normal distributions?

Binomial and Poisson are discrete and count outcomes. Normal is continuous and measures values. Binomial has a fixed number of trials and parameters n and p. Poisson has one parameter m with mean equal to variance. Normal has parameters μ and σ and a symmetric bell shape.

Q: Which distribution should I use, binomial or Poisson?

Use binomial when the number of trials n is fixed and each trial has the same success probability p. Use Poisson when you count events over a time or space interval with a known average and no fixed upper limit. If n is large and p is small, Poisson with m = np can approximate the binomial.

Q: When can the binomial be approximated by the normal distribution?

When n is large and p is not close to 0 or 1, so that the binomial is nearly symmetric. A common rule of thumb is that np and nq are both at least about 5. Then use mean np and SD √(npq), and convert to Z.

Q: Is the Poisson distribution symmetric?

No. It is skewed to the right, especially when m is small. As m becomes large it looks more symmetric and gets closer to a normal curve.

Q: Why is the normal distribution called continuous?

Because the variable can take any value in a range, not just whole numbers. The probability is the area under the curve between two points, and the probability at a single exact point is zero.

Updated 1 October 2026 · Fact-checked

Binomial counts successes in a fixed number of independent trials with constant probability p. Poisson counts rare events in an interval, with mean equal to variance. Normal describes a continuous, symmetric, bell-shaped variable. To solve a question, identify the variable type, check n, p and the mean-variance clue, then pick the distribution or approximation.

Understand Choosing and Comparing Theoretical Distributions

A theoretical distribution is a ready-made probability model. Instead of collecting data, you assume a pattern and use a formula or table. The exam tests whether you can pick the right model and read its parameters.

Start with one split: discrete or continuous. Binomial and Poisson are discrete. They count things: 0, 1, 2, and so on. Normal is continuous. It measures things like height, weight or marks, which can take any value in a range.

The binomial needs four conditions: a fixed number of trials n, only two outcomes (success or failure), the same probability p in every trial, and independent trials. Its parameters are n and p. The Poisson counts how many times a rare event happens in a fixed time or space, with no fixed upper limit. Its only parameter is m, the average number of events. The normal has two parameters, the mean μ and the standard deviation σ.

Shape helps you decide. Binomial is symmetric only when p = 0.5. It is skewed to the right when p < 0.5 and to the left when p > 0.5. Poisson is skewed to the right, and it looks more symmetric as m grows. Normal is perfectly symmetric, with mean = median = mode.

The distributions are linked. When n is large and p is small, the binomial is close to Poisson with m = np. When n is large and p is not too close to 0 or 1, the binomial is close to normal with mean np and SD √(npq). Because the binomial is discrete and the normal is continuous, a continuity correction of ±0.5 is used for accuracy. Follow the question's instruction if it says to ignore it.

Key formulas to remember

Binomial probability: P(X = r) = nCr × p^r × q^(n−r), where q = 1 − p, r = 0, 1, ..., n; Use when n is fixed and each trial is independent with the same p.
Binomial mean and variance: Mean = np; Variance = npq; SD = √(npq); Variance is always less than the mean because q < 1.
Poisson probability: P(X = r) = e^(−m) × m^r ÷ r!, where r = 0, 1, 2, ...; m is the average number of occurrences. The values of r have no upper limit.
Poisson mean and variance: Mean = m; Variance = m; SD = √m; Mean equals variance. This is the key clue for Poisson.
Poisson as limit of binomial: m = np (use when n is large and p is small); A common rule of thumb is np less than about 5. It is an approximation, not an exact rule.
Normal distribution parameters: X ~ N(μ, σ²); mean = median = mode = μ; skewness = 0; The curve is a symmetric bell. The total area under it is 1.
Standard normal variate: Z = (X − μ) ÷ σ; Z has mean 0 and SD 1. Use it with the Z-table.
Area rule for normal curve: μ ± 1σ ≈ 68.27%; μ ± 2σ ≈ 95.45%; μ ± 3σ ≈ 99.73%; These are the standard approximate areas under the curve.
Normal approximation to binomial: X ≈ N(np, npq), so Z = (X − np) ÷ √(npq); Use when n is large and p is not near 0 or 1. A common check is that both np and nq are at least 5 or so. Apply the ±0.5 continuity correction if asked.

How to solve Choosing and Comparing Theoretical Distributions questions

Use this order for any question that asks you to choose, compare or approximate a distribution.

1Decide if the variable is counted (discrete) or measured (continuous). Measured means normal. Counted means binomial or Poisson.
2If counted, check for a fixed number of trials n with two outcomes and constant p. If yes, it is binomial.
3If counted with no fixed n, and the question gives an average rate per interval, it is Poisson with m equal to that average.
4Look for the mean-variance clue. Variance less than the mean points to binomial. Variance equal to the mean points to Poisson.
5If the question is binomial but n is large, check p. A small p with np small suggests the Poisson approximation. A moderate p with large n suggests the normal approximation.
6Write down the parameters: n and p; or m; or μ and σ. For the approximations, convert first: m = np, or μ = np and σ = √(npq).
7Compute using the right tool: the formula for binomial or Poisson, or Z = (X − μ) ÷ σ with the table for normal.
8Match your value to the options. Check that it is reasonable: a probability must lie between 0 and 1.

Quickest way: Clue-word shortcut for choosing the distribution

When to use it: Use this for 'which distribution' and 'identify the parameters' MCQs. These take under a minute and need almost no calculation.

Scan for the clue words. 'Fixed number of trials', 'n items', 'each with probability p' means binomial.
'Average of ... per hour/page/km', 'rare', 'no upper limit' means Poisson.
'Height', 'weight', 'marks', 'bell-shaped', 'symmetric', 'mean and SD given' means normal.
Test the numbers. If mean = variance, pick Poisson. If variance < mean, pick binomial and solve q = variance ÷ mean, then p = 1 − q, then n = mean ÷ p.
For a normal approximation, find np and √(npq) first. Then compute Z and see how many SDs away X is. Use the 68-95-99.7 rule to eliminate options.
Do not spend time on long binomial sums with large n. Switch to an approximation or skip it. Wrong answers cost 0.25 marks, so attempt a question when you can eliminate at least one option, and skip only if you have no basis for an answer.

Common mistakes in Choosing and Comparing Theoretical Distributions

Choosing Poisson just because the event is 'rare' even though n is fixed and small.
Students match keywords without checking whether n and p are both given.
Fix: If n and p are both stated and n is small, use binomial directly. Poisson is an approximation only when n is large and p is small.
Saying the variance of a binomial equals np.
It is confused with the Poisson, where mean and variance are equal.
Fix: Binomial variance is npq. Only Poisson has variance equal to the mean.
Using the normal distribution for a discrete count with small n.
Students remember that normal approximates binomial but forget the conditions.
Fix: Check that n is large and p is not near 0 or 1 before using it. Otherwise use the exact binomial formula.
Using σ² instead of σ in Z = (X − μ) ÷ σ, or using npq instead of √(npq).
Variance and SD are both given or computed, so they get mixed up.
Fix: Always take the square root of npq before dividing. Write 'SD = √(npq)' as a separate line.
Treating the Poisson as symmetric and the binomial as always skewed.
The shapes are learnt as loose pictures rather than rules.
Fix: Remember: binomial is symmetric only at p = 0.5. Poisson is skewed to the right. Normal is always symmetric.
Using the binomial's n, p and q as the parameters of a normal approximation to a Poisson, or using the original parameters without converting them to a normal mean and variance.
Students forget that the Poisson has no n, p or q, and that every approximation needs its own conversion.
Fix: Convert first. For a binomial, the normal has mean np and variance npq. For a Poisson, the normal has mean = variance = m.

Worked examples

Example 1

The number of customer calls received by a helpline in an hour averages 4. There is no fixed maximum number of calls. Which distribution best models the number of calls in an hour? (A) Binomial (B) Poisson (C) Normal (D) None of these

Show the solution

The variable is a count, so it is discrete. This rules out normal.
There is no fixed number of trials n. This rules out binomial.
The data gives an average rate per interval, 4 per hour, and the count has no upper limit.
This matches Poisson with m = 4.

Answer: (B) Poisson

Example 2

A binomial variable has mean 6 and variance 4.5. Find the number of trials n. (A) 12 (B) 18 (C) 24 (D) 30

Show the solution

Mean = np = 6 and variance = npq = 4.5.
Divide: q = npq ÷ np = 4.5 ÷ 6 = 0.75.
So p = 1 − 0.75 = 0.25.
n = 6 ÷ 0.25 = 24.
Check: npq = 24 × 0.25 × 0.75 = 4.5, which matches.

Answer: (C) 24

Example 3

A fair-minded quality inspector tests 400 items. Each is defective with probability 0.2 independently. Using the normal approximation, the Z value for X = 96 defectives (ignoring continuity correction) is: (A) 1 (B) 2 (C) 4 (D) 16

Show the solution

n = 400 is large and p = 0.2 is not near 0 or 1, so the normal approximation is suitable.
Mean = np = 400 × 0.2 = 80.
Variance = npq = 400 × 0.2 × 0.8 = 64, so SD = √64 = 8.
Z = (96 − 80) ÷ 8 = 16 ÷ 8 = 2.
Note that 16 is the difference, not Z. Dividing by SD gives 2.

Answer: (B) 2

Exam tips

Most questions test identification. Learn the clue words and the mean-variance relationships by heart. They give you marks in seconds.
Write the parameters first: n and p, or m, or μ and σ. Many options are built from common slips like using npq instead of √(npq).
In approximation questions, check the conditions in the question. If n is small, the exam usually expects the exact formula.
Remember the 68-95-99.7 areas. They let you eliminate options without opening the Z-table.
With 0.25 negative marking, attempt a question if you can eliminate at least one option. Skip only when you have no basis for an answer, and skip long exact binomial sums with large n.

Practice questions from Theoretical Distributions

Choosing and Comparing Theoretical Distributions: frequently asked questions

What is the main difference between binomial, Poisson and normal distributions?

Binomial and Poisson are discrete and count outcomes. Normal is continuous and measures values. Binomial has a fixed number of trials and parameters n and p. Poisson has one parameter m with mean equal to variance. Normal has parameters μ and σ and a symmetric bell shape.

Which distribution should I use, binomial or Poisson?

Use binomial when the number of trials n is fixed and each trial has the same success probability p. Use Poisson when you count events over a time or space interval with a known average and no fixed upper limit. If n is large and p is small, Poisson with m = np can approximate the binomial.

When can the binomial be approximated by the normal distribution?

When n is large and p is not close to 0 or 1, so that the binomial is nearly symmetric. A common rule of thumb is that np and nq are both at least about 5. Then use mean np and SD √(npq), and convert to Z.

Is the Poisson distribution symmetric?

No. It is skewed to the right, especially when m is small. As m becomes large it looks more symmetric and gets closer to a normal curve.

Why is the normal distribution called continuous?

Because the variable can take any value in a range, not just whole numbers. The probability is the area under the curve between two points, and the probability at a single exact point is zero.