A single estimate is almost certainly a little wrong. A confidence interval reports a range instead, with a stated level of confidence.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The form
Every interval is estimate plus or minus margin of error. The margin depends on the confidence level and the sample size.
The usual level
For 95% confidence the margin is about 2 standard errors, since 95% of a normal distribution lies within two.
Sample size
Quadrupling the sample size halves the margin of error. Precision is expensive, because it improves with the square root.
What 95% means
It describes the method: about 95% of intervals built this way capture the true value. It is not a probability about this one interval.
The common misreading
Saying "there is a 95% chance the true value is in this interval" treats the fixed parameter as random. The randomness is in the sampling.
A range, with a stated reliability
A confidence interval is an estimate plus and minus a margin of error. Reporting a range acknowledges the uncertainty that a single number pretends does not exist.
What the confidence level describes
95% confidence means 95% of intervals constructed this way capture the true value. It describes the method over many samples, not the chance that this particular interval is right.
Confidence against precision
A wider interval is more likely to be correct and less informative. A 100% confidence interval covering every possible value is useless. Choosing a level is a trade between the two.
Narrowing the interval costs data
Halving the margin of error requires four times the sample. That square-root relationship is why national polls stop at around a thousand respondents — the next improvement is not worth the cost.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Sana Build an Interval
600 of 1000 people surveyed said yes, and the margin of error is 3 percentage points.
- Step 1
The sample proportion is 600 ÷ 1000 = 0.6, or 60%.
Step 4: Your Turn
Practice makes it stick.
The Interval
Problem 1 of 2
Estimate 60% with margin 3. What is the lower end, in percent?
%
The Width
Problem 2 of 2
An interval runs from 42% to 50%. What is the margin of error, in percentage points?
Build the Range
1 of 8
Estimate 45% with margin 4. Upper end, in percent?
2 of 8
Estimate 30% with margin 5. Lower end?
3 of 8
Interval 38% to 46%. What is the estimate?
4 of 8
Interval 38% to 46%. Margin of error?
5 of 8
450 of 900 said yes. Sample percentage?
6 of 8
Quadrupling the sample size does what to the margin? 1 halves it, 2 quarters it.
7 of 8
Which interpretation of 95% confidence is correct?
8 of 8
Estimate 72% with margin 6. Lower end?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Estimate 55% with margin 3. What is the upper end, in percent?
Question 2 of 2
How does a larger sample change the interval?
What You Learned
- A confidence interval is an estimate plus or minus a margin of error.
- Larger samples give smaller margins, improving with the square root.
- The confidence level describes the method, not one particular interval.