Key takeaway: The margin of error is the maximum expected distance, at a stated confidence level, between a sample statistic and the true population value, shrinking as sample size grows.
Grade 11 – Algebra 2S-IC.B.4Statistics and Data
Definition of Margin of Error
The margin of error puts a plus-or-minus band around a survey result. A poll finds that 62% of 850 students prefer a late start, but a different random group of 850 would give a slightly different percentage. The band says how much those percentages typically wobble.
Adding and subtracting it from the sample result gives an interval that probably contains the true value for everyone. A bigger sample gives a narrower band, while demanding higher confidence widens it. The word “error” does not mean a mistake was made; it names ordinary random variation between samples.
Formula and notation
For a sample proportion \(\hat{p}\) from \(n\) people, at 95% confidence:
\[\text{ME}\approx 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.\]
The interval is \(\hat{p}\pm\text{ME}\). The number 1.96 comes from the normal curve and corresponds to 95% confidence.
Key facts
- To halve the band, you must multiply the sample size by four.
- The band is largest when the sample proportion is near 50%.
- It covers random sampling variation only, not biased questions or poor sampling methods.
- Media reports often quote it in percentage points, such as ±3.3.
Margin of Error example
In a random poll of 850 students, 62% prefer a later school start. Find the 95% margin of error and the interval for the true percentage.
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Compute \(\hat{p}(1-\hat{p})=0.62\times 0.38=0.2356\).
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Divide by 850 and take the root: \(\sqrt{\frac{0.2356}{850}}\approx 0.01665\).
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Multiply by 1.96: \(\text{ME}\approx 0.0326\), about 3.3 percentage points.
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Interval: \(62\%\pm 3.3\%\) gives 58.7% to 65.3%.
The band is ±3.3 points; the true value is likely between 58.7% and 65.3%.
Common mistake
Watch out
Elias claims that doubling the sample from 850 to 1,700 will cut the band in half.
Why it is wrong
The sample size sits under a square root, so doubling it only divides the band by \(\sqrt{2}\approx 1.41\), giving about 2.3 points. To cut the band in half, the sample must be four times as large, 3,400.
Related terms and lessons
- Algebra 2 · Data, Normal Distributions and Statistical InferenceLessonWorksheetQuiz
- AP Statistics · Inference for ProportionsLessonWorksheetQuiz
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