How big a sample do I need?

Error shrinks like 1/√n, so quadrupling n halves the error. Population size barely matters.

Does a bigger sample fix bias?

No — bias from a bad sampling method doesn’t shrink with more data.

Shows how close a sample average usually lands to the true average.

x̄ ~ N(μ, (σ/√n)²)

The Central Limit Theorem says the sample mean (the average of a sample) is roughly bell-shaped for large n.
The spread of the sample mean is the standard error, equal to the standard deviation over the square root of n. (e.g. 10 / sqrt(100) = 1)
Bigger samples shrink the standard error, so the mean is more precise. (e.g. n 25 -> SE 2; n 100 -> SE 1)
This holds even when the original data is not bell-shaped, as long as n is large.
The average of all those sample means equals the true population mean.

$\overset{ˉ}{X} \sim N (μ, \frac{σ ^{2}}{n}), SE = \frac{σ}{n}$

Ask 5 kids their height and average it — your answer wobbles. Ask 100 kids and the average barely moves. Bigger groups give steadier answers.

Heights have spread σ = 10 cm. If you survey 100 people, the sample average has a standard error of 10 ÷ √100 = 1 cm, so your average should land within a couple of cm of the true mean.
With the same σ = 10, averaging only 25 people gives a standard error of 10 ÷ 5 = 2 cm — twice as wobbly — showing why a bigger sample gives a steadier answer.
To halve your error you need four times the data: jumping from n = 25 to n = 100 shrinks the standard error from 2 cm to 1 cm, because the spread falls with the square root of n.

Heights with spread σ = 10. Averaging n = 25 visitors, the sample average usually lands within about 4 cm of the true mean.

How big a sample do I need?: Error shrinks like 1/√n, so quadrupling n halves the error. Population size barely matters.
Does a bigger sample fix bias?: No — bias from a bad sampling method doesn’t shrink with more data.