How do I read a Poisson distribution table?

Find the column for your λ, the average number of events per interval, and go down to the row for k events. For λ = 2.5 and k = 3, the individual table gives 0.2138 and the cumulative table gives 0.7576.

What is the difference between P(X = k) and the cumulative Poisson table?

An individual table gives P(X = k), the probability of exactly k events. A cumulative table gives P(X ≤ k), the probability of k or fewer, which is the running total of the individual entries from 0 to k. Use cumulative for “at most” questions and ranges.

How do I find “at least” or “more than” probabilities?

Use the complement. P(X ≥ k) = 1 − P(X ≤ k − 1) and P(X > k) = 1 − P(X ≤ k). For λ = 2.5, P(X ≥ 4) = 1 − 0.7576 = 0.2424. For λ = 10, P(X ≥ 15) = 0.0835. The calculator shows all five probabilities at once.

What if my λ is not in the table?

The tables have columns for λ from 0.1 to 1 in steps of 0.1, 1.5 to 10 in steps of 0.5 and 11 to 20 in steps of 1. For any other λ up to 10,000, such as 0.35 or 100, use the calculator. For λ = 100 and k = 120 it gives P(X = 120) = 0.0056.

When can I use the Poisson instead of the binomial?

When n is large, p is small and you know the rate λ = np. Rules of thumb such as n ≥ 20 and p ≤ 0.05 are guidelines. For n = 100 and p = 0.03, P(X ≤ 2) is 0.4198 exactly and 0.4232 with the Poisson.

Why does each column stop before k reaches 40?

Each column ends at the first k where P(X ≤ k) is 0.9999 or more, because the entries after it are smaller than 0.0001 and add up to less than 0.0001. Blank cells mean the same thing as a very small value, and the calculator gives any tail exactly.

Poisson Distribution Table

Poisson distribution table for P(X = k) and cumulative P(X ≤ k), λ 0.1 to 20, plus a calculator for λ up to 10,000 and a probability chart. Print or download.

Individual P(X = k) and cumulative P(X ≤ k)Calculator for λ up to 10,000, with a range optionPNG, SVG and print on A4 or Letter, no sign-up

Poisson probability calculator

λ (mean)

k (events)

Find

Examples:

Plot these probabilities as a bar chart

Copy the k, P(X = k) rows for your λ, then paste them into the bar chart maker as Label, value rows. Its value labels show two decimals.

Open bar chart maker

P(X = 3)

0.2138

P(X ≤ 3)

0.7576

P(X ≥ 3)

0.4562

P(X < 3)

0.5438

P(X > 3)

0.2424

Mean λ

2.5

Variance λ

2.5

Standard deviation

1.58114

Chart style

Cumulative line

Poisson probability tables, λ = 0.1 to 20

Choose a block of λ, then read down to k and across to λ. The first tab gives the probability of exactly k events, the second the probability of k or fewer. Click any cell to highlight its row and column.

Mean number of events λ

Export style

Paper for print

Poisson probabilities P(X = k) for λ = 1.5 to 5, by k and λ
k \ λ	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
0
1
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3
4
5
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8
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10
11
12
13
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15

0.0000 means the probability is below 0.00005. It is not exactly zero.

Blank cells: each column stops at the first k where P(X ≤ k) is 0.9999 or more, so a blank cell lies below that point and the blank cells together hold less than 0.0001 of probability. For longer tails, or a λ that is not a column, use the calculator, which accepts any λ up to 10,000.

Values are rounded to four decimals. We compared every printed cell (1,394 in all) with SciPy (poisson.pmf and poisson.cdf) and none differs after rounding.

Selected cell

λ = 2.5, P(X = 3) = 0.2138

More digits: 0.2138. Largest k in this column: 10 of 15 in this block.

Free Poisson Distribution Table Charts

Every chart below is drawn from the same code as the tool above. View, download and print them free, no sign-up.

Individual

Poisson Distribution Table: P(X = k), λ 1.5 to 5

Exactly-k probabilities for eight values of λ. For λ = 2.5 and k = 3 the entry is 0.2138.

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Reference tables

How to read

Distribution

Printable versions

Counting trials with a fixed chance? Use the binomial distribution table. For large λ, look up normal areas in the z table, and draw a probability bar chart with the bar chart maker. Not sure which table to use? See statistics tables.

What a Poisson distribution table shows

A Poisson distribution table lists probabilities for a count of events in a fixed interval of time or space: customers arriving in an hour, bacterial colonies on a small area of a plate, mutations in a stretch of DNA. The only parameter is λ (lambda), the average number of events per interval.
The Poisson model fits when events happen one at a time, independently of each other, at a steady average rate, and the number of events in one interval does not change the chance in another. If events cluster, the counts vary more than the model expects and the Poisson is a poor fit. A useful check is that the mean and the variance of a Poisson count are both λ.
The probability of exactly k events is P(X = k) = e^(−λ) λ^k ÷ k!, with mean λ and variance λ. The rate scales with the interval: 2.5 arrivals per hour is λ = 5 for a two-hour window.

How to read the table in three steps

Step 1: find the column for your λ. Step 2: go down to the row for k. Step 3: read the cell. In the individual tab, λ = 2.5 and k = 3 gives 0.2138, so the chance of exactly 3 events is 0.2138. In the cumulative tab the same position gives 0.7576, the chance of 3 or fewer. That entry is the sum of the first four individual entries: 0.0821 + 0.2052 + 0.2565 + 0.2138 = 0.7576. The cumulative entry is the exact sum rounded once, so adding rounded entries can occasionally differ in the last digit.
A 0.0000 in the table is rounded, not exact. Each column also stops at the first k where P(X ≤ k) reaches 0.9999, and the blank cells below it together hold less than 0.0001 of probability. For a longer tail or a λ between the columns, use the calculator, which takes any λ up to 10,000.

At least, more than, fewer than

The cumulative tab answers “k or fewer”. The other questions are one subtraction away. P(X ≥ k) = 1 − P(X ≤ k − 1). P(X > k) = 1 − P(X ≤ k). P(X < k) = P(X ≤ k − 1). For λ = 2.5, the chance of at least 4 events is 1 − 0.7576 = 0.2424. For a range, P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). The calculator at the top gives all five single-k probabilities, and the range for a to b, in one go.

Worked examples: arrivals, colonies, mutations

Arrivals. A help desk gets 2.5 calls per hour on average. The chance of at most 1 call in an hour is 0.2873, and the chance of more than 5 is 0.0420.
Colonies. A small area of a plate has on average 0.3 colonies. The chance of finding none is 0.7408, the λ = 0.3, k = 0 entry.
Mutations. If samples carry on average 1.5 mutations in a region, the chance of 3 or more in one sample is 0.1912. For λ = 10 and at least 15 events, P(X ≥ 15) = 0.0835, a tail value the table gives by subtraction and the calculator gives directly.
Large λ is no problem for the calculator. For λ = 100 and k = 120, P(X ≤ 120) = 0.9773, P(X ≥ 120) = 0.0282, and P(80 ≤ X ≤ 120) = 0.9599.

Poisson approximation to the binomial

When n is large and p is small, a binomial count is close to Poisson with λ = np. Common rules of thumb ask for n ≥ 20 and p ≤ 0.05, and some books use n ≥ 100 with np ≤ 10. These are guidelines, not guarantees, and the approximation improves as n grows and p shrinks.
Example: n = 100 and p = 0.03 gives λ = 3. The exact binomial P(X ≤ 2) is 0.4198 and the Poisson value is 0.4232. If you can use the exact binomial table, do; use the Poisson when n is beyond the table or only the rate is known.

Normal approximation when λ is large

For large λ the Poisson distribution looks like a normal distribution with mean λ and standard deviation √λ. A common rule of thumb asks for λ ≥ 10, and some texts use 20 or more. Add a continuity correction because the Poisson is discrete: P(X ≤ k) is approximately Φ((k + 0.5 − λ) ÷ √λ).
Example: λ = 36 and P(X ≤ 40). The exact value is 0.7771. The normal approximation with the correction, Φ((40.5 − 36) ÷ 6), gives 0.7734. Without the correction, Φ((40 − 36) ÷ 6) gives 0.7475, which is further off. For a z-score lookup, use the z table.

Reporting a Poisson probability

State the model, the rate and the interval, then the probability. In APA style, a probability that cannot exceed 1 is written without a leading zero and usually to two or three decimals: “Arrivals per hour were modeled as Poisson with λ = 2.5; the probability of exactly 3 arrivals was P(X = 3) = .214.” Say how λ was estimated, and whether the counts look Poisson, for example by comparing the sample mean and variance.

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