Use the population standard deviation σ for one observation. For the mean of n independent observations from the same population, use σ/√n. Both variables have the same mean, but the sample mean is less spread out when n is greater than one.
The 2027 A-Level H2 Mathematics 9758 syllabus covers the sampling distribution of the mean. This guide compares two probabilities at the same cutoff and shows why the wording determines the standard deviation.
- Name the random variable before using a normal calculation.
- Use σ for a single observation and σ/√n for the sample mean.
- In normal distribution notation, the second parameter is the variance.
- Check whether the normal model is exact or an approximation.
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Name what the question measures
Let X describe a single value. If the population is normal with mean μ and variance σ², then X has distribution N(μ, σ²). For n independent observations, their mean X̄ has distribution N(μ, σ²/n).
The standard deviation of X̄ is the square root of σ²/n, namely σ/√n. Dividing σ by n would make the spread too small. If a calculator asks for standard deviation, enter σ/√n rather than σ²/n.
A normal population gives an exact normal model for the mean of an independent sample. For a non-normal population, using a normal model for the sample mean depends on whether a central limit approximation is suitable.
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Worked example | One reading and a mean of 16
Individual readings are independent and normally distributed with mean 100 and standard deviation 12. Compare the chance that one reading exceeds 106 with the chance that the mean of 16 readings exceeds 106.
| Variable | Standard deviation | z at 106 | Upper-tail probability |
|---|---|---|---|
| One reading | 12 | 0.5 | 0.3085 |
| Mean of 16 readings | 3 | 2 | 0.02275 |
For one reading, z = (106 − 100)/12 = 0.5. For the mean, the standard deviation is 12/√16 = 3 and z = (106 − 100)/3 = 2.
Both questions use an upper tail. The second tail is smaller because 106 is two standard deviations above the sample-mean centre and only half a standard deviation above the individual-reading centre.
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Practise when the wording changes
In the same example, take the mean of four independent readings. Its standard deviation is 6, so the chance that the mean exceeds 106 is P(Z > 1) ≈ 0.1587. This lies between the earlier two probabilities.
The phrase ‘at least one of 16 readings exceeds 106’ asks a third question. One extreme reading can occur while the mean stays below the cutoff. Independence gives the complement calculation 1 − P(X ≤ 106)¹⁶.
Underline the noun in the question. A reading, the mean, and at least one reading describe three different events. Write the corresponding variable or event before using a calculator.
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Frequently asked questions
Does the mean of the sample mean change with n?
Its expected value is μ for every sample size in this setup. Its variance decreases as n increases.
Can I use σ/√n for dependent readings?
The formula used here assumes independent observations with a common variance. Dependence can change the variance of the mean.
Why does N(100, 9) use 9 in the example?
It uses the variance of the mean of 16 readings. The standard deviation is 3, so the variance is 9.
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