Estimating Population Parameters
A poll cannot ask everyone, so it estimates a population value from a sample. The margin of error says how far the true value might be from the sample result.
A common approximation for the margin of error is , where is the sample size. Bigger samples give smaller margins.
Margin of error
The margin of error is the plus-or-minus band around a sample estimate. A poll result of 52% with a 5% margin means the true value is likely between 47% and 57%.
A quick estimate is : divide 1 by the square root of the sample size to get the margin as a decimal, then convert to a percent.
Bigger samples, tighter estimates
Because is under a square root, quadrupling the sample size only halves the margin of error. There are diminishing returns to larger samples.
So a sample of 400 gives a margin of , while 1600 gives .
Worked examples
Example 1: margin from sample size
A poll uses a random sample of 400 people. What is the margin of error, as a percent?
Answer:
Example 2: larger sample
What margin of error does a sample of 2500 give?
Answer:
Example 3: the interval around a result
A poll of 400 people finds 52% support. Between what two percents does the true value likely fall?
Answer: About to
Try one yourself
Common questions
What is the margin of error?
The plus-or-minus range around a sample estimate that likely contains the true population value.
How do I estimate it?
Use : divide 1 by the square root of the sample size, then convert to a percent.
Why do bigger samples help less and less?
The sample size is under a square root, so you must quadruple to cut the margin in half — diminishing returns.
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