Confidence interval calculator
A 95% confidence interval is a procedure that captures the true mean 95% of the time — not a 95% chance that the true mean is in this particular interval.
Margin of error
1.611 467 1
- Lower bound
- 66.888 532 92
- Upper bound
- 70.111 467 08
- Standard error
- 0.822 192 19
- Interval width
- 3.222 934 2
- Margin as a share of the mean
- 2.353 %
Show the working
- Sample mean = 68.5
- Sample standard deviation = 5.2
- Sample size = 40
- Confidence level = 1.959 96
- Standard error = 5.2 / sqrt(40) = 0.822 192
- Margin of error = 1.959 96 * 0.822 192 = 1.611 47
- Lower bound = 68.5 - 1.611 47 = 66.8885
- Upper bound = 68.5 + 1.611 47 = 70.1115
- Interval width = 2 * 1.611 47 = 3.222 93
- Margin as a share of the mean = 68.5 != 0 ? 1.611 47 / abs(68.5) * 100 : 0 = 2.352 51
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- margin of error = z x sd / sqrt(n)
- interval = mean +/- margin
This uses the normal z value. For small samples with an unknown population variance the t distribution is correct and gives a wider interval.
Quadrupling the sample size halves the margin of error, which is why big surveys stop getting much better.