Margin of Error
In short: The margin of error is the plus-or-minus amount added to and subtracted from a sample result to give an interval likely to contain the true value for the whole population.
Algebra 2StatisticsS-IC.B.4
What is Margin of Error?
The margin of error admits that a survey is only a sample. If a poll finds 55% support, the true figure for everyone is probably near 55%, but not exactly there, and this number says how near.
When \(\hat{p}\) is a sample proportion from \(n\) people, the 95% version is about \(1.96\sqrt{\frac{\hat{p}(1 – \hat{p})}{n}}\). The result is written \(\hat{p} \pm \text{margin}\). Larger samples give smaller margins, and the margin shrinks by half when the sample size is multiplied by 4.
How to spot it
- A news poll reports a percentage followed by the phrase plus or minus a few points.
- A question asks for an interval in which the true population value probably lies.
- A sample size is given, along with a request to say how precise the survey result is.
- You are asked what happens to precision if more people are surveyed.
The method
- Identify the sample proportion \(\hat{p}\) and the sample size \(n\).
- Compute \(\sqrt{\frac{\hat{p}(1 – \hat{p})}{n}}\), then multiply by 1.96 for 95% confidence.
- Convert to percentage points if the result is reported as a percent.
- Form the interval \(\hat{p} – \text{margin}\) to \(\hat{p} + \text{margin}\).
Margin of Error: solved example
Problem
Out of 1,024 voters surveyed in a city, 55% back a new bike lane. Work out the plus-or-minus amount at 95% confidence and the resulting range for all voters.
Solution
With \(\hat{p} = 0.55\) and \(n = 1024\): \(\sqrt{\frac{0.55 \times 0.45}{1024}} = \sqrt{0.0002417} \approx 0.01555\).
Multiply by 1.96: \(0.01555 \times 1.96 \approx 0.0305\), about 3.0 percentage points. The interval runs from about 52.0% to 58.0%.
Try it
Question 1
A sample mean is 48.2 with a margin of error of 1.9. Give the interval.
Show the answer
\(48.2 – 1.9 = 46.3\) and \(48.2 + 1.9 = 50.1\), so the interval is 46.3 to 50.1.
Question 2
A survey’s margin of error is 4 points. How big is it if four times as many people are surveyed?
Show the answer
The margin is divided by \(\sqrt{4} = 2\), so it becomes about 2 points.
Margin of Error vs. Standard Deviation
The standard deviation describes how spread out the individual values in a data set are. This margin describes how uncertain a sample summary is, and it shrinks as the sample grows, while the data’s own spread does not. A sample of 1,000 can have a small margin even if the raw values vary a lot.
Keep going
Lessons
Worksheets and quizzes
- Sampling, Surveys and Experiments – Worksheet
- Sampling, Surveys and Experiments – Quiz
- Confidence Intervals for a Proportion – Worksheet
- Confidence Intervals for a Proportion – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Statistics