Margin of Error Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Margin of Error.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The maximum expected difference between the sample statistic and the true population parameter; it is half the width of a confidence interval.

When a poll says 'the approval rating is 52%52\% with a margin of error of Β±3%\pm 3\%,' it means the true value is likely between 49%49\% and 55%55\%. The margin of error is the 'Β±\pm' partβ€”it tells you how much wiggle room to give the estimate. Larger samples and less variability shrink the margin of error.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The margin of error is half the width of a confidence interval β€” how far the estimate might be from the truth.

Common stuck point: The procedure for margin of error is the easy part; the trap is reporting the full interval width as the margin of error. Asking "Am I reporting how far an estimate might be from the truth as a single Β±\pm value (half the interval width)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I reporting how far an estimate might be from the truth as a single Β±\pm value (half the interval width)?

Worked Examples

Example 1

easy
A poll of n=400n=400 voters finds p^=0.55\hat{p}=0.55 supporting a candidate. Calculate the margin of error at 95% confidence and interpret the result.

Answer

Eβ‰ˆΒ±5%E \approx \pm 5\%; 95% CI: (50.1%,59.9%)(50.1\%, 59.9\%). Majority support is likely.

First step

1
SE=p^(1βˆ’p^)n=0.55Γ—0.45400=0.2475400=0.000619β‰ˆ0.0249SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.55 \times 0.45}{400}} = \sqrt{\frac{0.2475}{400}} = \sqrt{0.000619} \approx 0.0249

Full solution

  1. 2
    E=zβˆ—Γ—SE=1.96Γ—0.0249β‰ˆ0.049β‰ˆΒ±5%E = z^* \times SE = 1.96 \times 0.0249 \approx 0.049 \approx \pm 5\%
  2. 3
    Confidence interval: 0.55Β±0.049=(0.501,0.599)0.55 \pm 0.049 = (0.501, 0.599)
  3. 4
    Interpretation: we are 95% confident between 50.1% and 59.9% of voters support the candidate
Margin of error quantifies polling uncertainty. 'Candidate leads 55% Β± 5%' means the true support could be anywhere from 50% to 60%. A margin of error that doesn't include 50% still suggests the candidate leads, but we can't be certain.

Example 2

medium
A news report claims: 'The margin of error for this poll is Β±3%.' Explain three things a careful reader should know about this margin of error.

Example 3

medium
You need E≀3E \leq 3 for a 95% CI on a mean with Οƒ=20\sigma = 20. Find the smallest nn.

Example 4

hard
Show that p^(1βˆ’p^)\hat{p}(1-\hat{p}) is maximized when p^=0.5\hat{p}=0.5, and find its max value.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A 95% CI for average commute time is (28,36)(28, 36) minutes. What is the margin of error, and what does it tell us about the survey's precision?

Example 2

hard
Two polls: Poll A (n=100n=100): margin of error Β±10%. Poll B (n=1600n=1600): margin of error Β±2.5%. By what factor was n increased, and by what factor did E decrease? Verify using E=zβˆ—0.5Γ—0.5nE = z^* \sqrt{\frac{0.5 \times 0.5}{n}}.

Example 3

easy
A poll reports 52%52\% approval with a margin of error of Β±3%\pm 3\%. What interval of values does this imply?

Example 4

easy
A confidence interval is (40,60)(40, 60). What is the margin of error?

Example 5

easy
The margin of error for a mean is zβˆ—β‹…SEz^* \cdot \text{SE}. If zβˆ—=1.96z^* = 1.96 and SE =2.5= 2.5, find the margin.

Example 6

easy
True or false: a larger sample size produces a smaller margin of error (same confidence).

Example 7

easy
Does the margin of error account for non-sampling bias such as bad question wording?

Example 8

easy
To reduce the margin of error, should you increase or decrease the sample size?

Example 9

easy
A 95%95\% margin uses zβˆ—=1.96z^* = 1.96; a 90%90\% margin uses zβˆ—=1.645z^* = 1.645. With the same data, which margin is larger?

Example 10

easy
A proportion estimate is p^=0.6\hat{p} = 0.6 with margin 0.040.04. State the confidence interval.

Example 11

medium
A sample has Οƒ=20\sigma = 20, n=100n = 100. Find the 95%95\% margin of error (zβˆ—=1.96z^* = 1.96).

Example 12

medium
A poll has p^=0.5\hat{p} = 0.5, n=1000n = 1000. Find the 95%95\% margin (zβˆ—=1.96z^* = 1.96) using SE =p^(1βˆ’p^)/n= \sqrt{\hat{p}(1-\hat{p})/n}.

Example 13

medium
Currently margin =6= 6 with n=100n = 100. Keeping confidence fixed, what margin results from n=400n = 400?

Example 14

medium
Two candidates poll at 48%48\% and 52%52\%, each with margin Β±3%\pm 3\%. Can we declare a clear winner?

Example 15

medium
A margin of 4.94.9 comes from zβˆ—=1.96z^* = 1.96 and SE =2.5= 2.5. If confidence rises to 99%99\% (zβˆ—=2.576z^* = 2.576), what is the new margin?

Example 16

medium
A study needs a margin of at most 22 for a mean with Οƒ=10\sigma = 10 at 95%95\% (zβˆ—=1.96z^* = 1.96). Find the smallest nn.

Example 17

medium
A margin of error is zβˆ—β‹…Οƒ/nz^* \cdot \sigma/\sqrt{n}. With Οƒ=18\sigma = 18, n=36n = 36, zβˆ—=1.645z^* = 1.645 (90%90\%), find the margin.

Example 18

challenge
Quadrupling the sample size from nn to 4n4n changes the margin by what factor, and why is this 'diminishing returns'?

Example 19

challenge
A poll uses p^=0.5\hat{p} = 0.5 for a conservative margin. Why does p^=0.5\hat{p} = 0.5 maximize the margin, and what is the maximum value of p^(1βˆ’p^)\hat{p}(1-\hat{p})?

Example 20

challenge
A margin of 44 at n=50n = 50 must be cut to 11 at the same confidence and Οƒ\sigma. What sample size is required?

Example 21

medium
A sample has Οƒ=24\sigma = 24, n=64n = 64. Find the 95%95\% margin of error (zβˆ—=1.96z^* = 1.96).

Example 22

medium
A poll of n=625n = 625 has p^=0.4\hat{p} = 0.4. Find the 95%95\% margin (zβˆ—=1.96z^* = 1.96) using SE =p^(1βˆ’p^)/n= \sqrt{\hat{p}(1-\hat{p})/n}.

Example 23

easy
A 95% CI for a mean is (12,18)(12, 18). What is the margin of error?

Example 24

easy
A 95% CI is (45,55)(45, 55). State the point estimate and margin of error.

Example 25

easy
With Οƒ=15\sigma = 15, n=225n = 225, zβˆ—=1.96z^* = 1.96, find the margin of error.

Example 26

easy
To shrink the margin of error, which of these helps: bigger sample, smaller sample, or higher confidence?

Example 27

easy
A poll reports p^=0.62\hat{p}=0.62 with margin Β±0.03\pm 0.03. Give the 95% CI.

Example 28

medium
Find the 95% margin of error for p^=0.4\hat{p} = 0.4, n=100n = 100 (zβˆ—=1.96z^* = 1.96).

Example 29

medium
With p^=0.5\hat{p} = 0.5, n=400n = 400, zβˆ—=1.96z^* = 1.96, find the 95% margin of error.

Example 30

medium
With Οƒ=30\sigma = 30, n=100n = 100, zβˆ—=1.96z^* = 1.96, compute the 95% margin of error.

Example 31

medium
Currently E=4E = 4 at n=50n = 50. Find EE at n=200n = 200 (same zβˆ—z^* and Οƒ\sigma).

Example 32

medium
A poll has p^=0.3\hat{p}=0.3, n=100n=100, zβˆ—=1.96z^*=1.96. Find the margin.

Example 33

medium
Currently E=4.9%E=4.9\% at n=400n=400, p^=0.5\hat{p}=0.5, 95% confidence. What is the new margin at n=1600n=1600?

Example 34

hard
Two candidates poll at 51%51\% and 49%49\%, each with margin Β±3%\pm 3\%. Can you declare a winner?

Example 35

hard
To cut the margin by a factor of 33, by what factor must nn grow?

Example 36

hard
For a 99% margin with Οƒ=10\sigma=10, E≀2E\leq 2, find the minimum nn (zβˆ—=2.576z^*=2.576).

Example 37

hard
A CI is (0.30,0.42)(0.30, 0.42). What is the margin of error and the point estimate?

Example 38

hard
With Οƒ=16\sigma=16, n=64n=64, find the 95% margin (zβˆ—=1.96z^*=1.96).

Example 39

hard
A study triples its sample size from n=100n=100 to n=300n=300. By what factor does margin change?

Example 40

hard
A poll has p^=0.5\hat{p}=0.5, n=2500n=2500, zβˆ—=1.96z^*=1.96. Margin?

Example 41

challenge
A pollster wants E≀0.02E \leq 0.02 at 95% confidence with no prior estimate of pp. Find the smallest nn (zβˆ—=1.96z^*=1.96).

Example 42

challenge
A 95% CI for a mean is (xΛ‰βˆ’4,xΛ‰+4)(\bar{x} - 4, \bar{x} + 4). If you re-run at 99% confidence with the same data, find the new margin (z95βˆ—=1.96z^*_{95}=1.96, z99βˆ—=2.576z^*_{99}=2.576).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

confidence intervalstandard deviation