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Statistics and sampling distributions

Edexcel IAL Mathematics · WST02 Statistics 2

A statistic is a quantity worked out from a sample. Because samples vary, a statistic is itself a random variable, with its own sampling distribution.

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Statistic

A statistic is a quantity calculated only from the sample values, with no unknown parameters. The sample mean X̄, the sample total, the median, the range and the largest value are all statistics. X̄ - μ is not a statistic if μ is unknown.

Sampling distribution

The sampling distribution of a statistic lists every value it can take, with the probability of each. To find it:

1. List all possible samples (or all combinations, then count the orders).

2. Work out the statistic for each sample.

3. Add the probabilities for each value.

A bag holds many counters: 70% show 1 and 30% show 2. For a sample of 3, the sample total S has P(S = 3) = 0.73 = 0.343, P(S = 4) = 3 × 0.72 × 0.3 = 0.441, and so on.

Value12
Proportion of the bag0.70.3

A large bag of counters labelled 1 or 2

Counting orders

Examiners report that the most common error is forgetting that a combination such as (1, 1, 2) can occur in 3 orders, so its probability is multiplied by 3. A combination of three different values, such as (1, 2, 5), occurs in 3! = 6 orders.

S3456
P(S = s)0.3430.4410.1890.027

Worked example

Using the bag above, find the sampling distribution of the median M of a sample of 3 counters.

  1. M = 1 when at least two counters show 1: 0.343 + 0.441 = 0.784.
  2. M = 2 when at least two show 2: 0.189 + 0.027 = 0.216.

Answer: P(M = 1) = 0.784, P(M = 2) = 0.216

Key idea: A statistic uses sample values only. Its sampling distribution lists all its values and their probabilities; count the orders.

Check yourself

  1. A sample of 3 counters is taken from the bag. Find P(S = 4), where S is the sample total.
    Show answer

    0.441. Samples (1, 1, 2) in 3 orders: 3 × 0.72 × 0.3 = 0.441.

  2. A sample of 3 counters is taken from the bag. Find P(S = 5).
    Show answer

    0.189. Two 2s and one 1, in 3 orders: 3 × 0.7 × 0.32 = 0.189.

  3. A sample of 3 counters is taken from the bag. Find P(M = 2), where M is the sample median.
    Show answer

    0.216. M = 2 needs at least two 2s: 0.189 + 0.027 = 0.216.

  4. A sample of 3 counters is taken from the bag. Find the probability that the sample range is 0.
    Show answer

    0.37. All three the same: 0.343 + 0.027 = 0.37.

  5. A sample of 3 counters is taken from the bag. Find E(S).
    Show answer

    3.9. E(one counter) = 1 × 0.7 + 2 × 0.3 = 1.3, so E(S) = 3.9.

  6. A sample of three coins from a large bag gives the values 1, 2 and 5 in some order. In how many different orders can this combination occur?
    Show answer

    6. Three different values: 3! = 6 orders.

  7. X1, X2, ..., Xn is a random sample from a population with unknown mean μ. Which is a statistic?
    • X̄ - μ
    • The largest value in the sample
    • μ + 2
    • The population variance
    Show answer

    The largest value in the sample. It uses only the sample values.

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