Ask 40 people and you get one answer. Ask a different 40 and you get another. The pattern in those answers is what makes the rest of statistics possible.
Find the standard error
A population has σ = 18. Samples of size 36 are taken. How much does the sample mean vary?
The standard error is σ/√n = 18√36 = 186 = 3. A forgets that averaging steadies things: the sample mean varies far less than a single observation does. D divides by n instead of √n, 1836 — the central slip of this lesson, and it always makes the spread look far too small.
Take a sample, work out its mean, and you get a number. Take another sample and you get a slightly different number. Those means have their own distribution — a centre, a spread and a shape — and that is what a sampling distribution is.
Both curves sit above the same μ. Averaging does not move the centre — it only narrows the spread, because an unusually high reading in a sample tends to be offset by an ordinary one.
μx̄ = μ σx̄ = σ / √n
| Sample size n | √n | Standard error, σ = 18 |
|---|---|---|
| 1 | 1 | 18 |
| 4 | 2 | 9 |
| 36 | 6 | 3 |
| 144 | 12 | 1.5 |
Going from n = 36 to n = 144 is four times the work for twice the precision, because the divisor is √n. That is a real cost, and it is why large surveys stop where they do. Dividing by n instead of √n would promise 144 people the accuracy they cannot deliver.
For a large enough sample — n ≥ 30 is the usual working rule — the distribution of the sample mean is approximately normal, whatever shape the population had. Skewed, lumpy, two-humped: it does not matter.
This is the hinge of the whole unit. Confidence intervals and hypothesis tests all assume a normal sampling distribution, and the CLT is what earns that assumption without anyone having to know the population’s shape.
z = x̄ − μσ/√n
σx̄ = 6 000√100 = 600
the standard error first
z = 29 200 − 28 000600 = 2.0
divide by 600, not 6 000
P(z > 2) = 1 − 0.9772
area to the right
≈ 2.3%
n = 100 ≥ 30, so the CLT applies
The only change from the previous lesson is the denominator. A single income ₱1 200 above the mean would be z = 0.2 and utterly unremarkable; a sample of 100 averaging that much above is z = 2.0 and quite surprising. Same gap, very different meaning.
Read the question for the word sample or average before you write anything. The two calculations differ only by a √n, and the marker cannot tell a misread from a miscalculation — both simply come out wrong.
Still σ/√n; only the direction and the unknown change.
A normal population has μ = 75, σ = 12. For n = 16, find P(x̄ < 70.5).
σx̄ = 12√16 = 3
population normal, so small n is fine
z = 70.5 − 753 = −1.5
below the mean, so negative
P = 0.0668
read the left area directly
With n under 30 we may still go ahead, but only because the population itself is normal.
σ = 10 and n = 50. Find the standard error.
σx̄ = 10√50 = 105√2
√50 = √25 × √2
= √2 ≈ 1.41
2√2 = √2
Most real sample sizes are not perfect squares; simplify the surd or use a calculator, but still divide by √n.
σ = 15. How large a sample keeps the standard error at 2 or less?
15√n ≤ 2, so √n ≥ 7.5
the unknown is now n
n ≥ 7.5² = 56.25
square both sides
n = 57
always round a sample size up
Rounding down to 56 would leave the error just above 2 (15√56 ≈ 2.004).
Expect the standard error from σ and n, what happens to it when n changes, a statement of the Central Limit Theorem, and a probability that a sample mean lies above or below a value. Find σ/√n first, then z = x̄ − μσ/√n and the table. The 10-second check: the standard error must be smaller than σ for any n above 1, and four times the sample halves it.
Find the standard error
A population has σ = 40. Samples of size 64 are taken.
σ/√n = 40√64 = 408 = 5. A divides by n, 4064 — far too small, and it would claim a precision 64 observations cannot buy. C is √n itself, computed and then written down in place of the answer. B is the population SD, which is what one observation varies by, not an average of 64.
Hint: take the square root of n first and write it down on its own line, then divide. Half the errors in this topic are the square root being skipped in the rush.
