ConfidLab

Confidence Interval Calculator

The range that likely contains the true population mean.

Sample mean

Sample std dev

Sample size (n)

Confidence level (%)

95% confidence interval

[94.632, 105.368]

Margin of error: ±5.368

What you entered

Mean: 100Std dev: 15n: 30

1
Standard error
15 ÷ √30= 2.7386
2
Margin of error
1.96 × 2.7386= 5.3677
3
Interval
100 ± 5.3677= [94.6323, 105.3677]

Result

Interval: [94.632, 105.368]

We're 95% confident the true population mean lies between 94.632 and 105.368.

95% confidence interval for a mean of 100, SD 15

The margin of error is the critical z of 1.96 times the standard error. The interval is the mean plus and minus that margin.

Sample size	Margin of error	Lower bound	Upper bound	Interval width
10	9.297	90.70	109.30	18.59
25	5.880	94.12	105.88	11.76
30	5.368	94.63	105.37	10.74
50	4.158	95.84	104.16	8.32
100	2.940	97.06	102.94	5.88
400	1.470	98.53	101.47	2.94

Precision improves with the square root of the sample size, not with the sample size itself, which is the single most useful fact here: going from 100 to 400 observations quarters the sample cost but only halves the margin of error. That is why large surveys hit diminishing returns quickly. A 95% interval means that if you repeated the study many times, about 95% of the intervals constructed this way would contain the true mean - it is not a 95% probability that this particular interval does. Use a t critical value instead of 1.96 when the sample is small and the population SD is unknown.

What '95% confident' really means

A 95% confidence interval doesn't mean there's a 95% chance the true mean is in this specific range — it means that if you repeated the sampling process many times, about 95% of the intervals you'd construct would contain the true mean.

Why bigger samples narrow the interval

The margin of error shrinks with the square root of sample size — quadrupling your sample size only halves the margin of error, which is why polls need surprisingly large samples to get tight intervals.

Frequently asked questions

I surveyed 200 customers. Average satisfaction = 7.2, SD = 1.5. What's the 95% CI?

Margin of error = 1.96 × (1.5/√200) = 1.96 × 0.106 = 0.208. The 95% CI is 7.2 ± 0.208 = (6.99, 7.41). You can be 95% confident the true average satisfaction of ALL customers falls in this range.

What's the difference between 90%, 95%, and 99% confidence?

Higher confidence = wider interval. 90% CI uses z = 1.645, 95% uses z = 1.96, 99% uses z = 2.576. A 99% CI is about 32% wider than a 95% CI. You gain more confidence at the cost of precision — 95% is the most common compromise.

How is confidence interval different from margin of error?

Margin of error is the ± part. Confidence interval is the resulting range. This calculator works with sample means; the margin of error calculator works with proportions (like poll percentages). Use this one for averages, that one for percentages.

How many people do I need to survey?

That depends on your desired margin of error and confidence level. Use the sample size calculator to work backwards from precision goals. As a rough guide: n = 100 gives ±10% precision, n = 400 gives ±5%, n = 1,600 gives ±2.5% (for proportions at 95% confidence).