47.648043
52.351957
2.351957
1.2
4.703914
1
0
47.648043
52.351957
2.351957
1.2
4.703914
1
0
A confidence interval (CI) gives you a practical range where the true population mean is likely to fall, based on a sample. Instead of showing just a single estimate (your sample mean), a CI adds uncertainty awareness — which is exactly what most real-world decisions need.
Use this Confidence Interval Calculator to get the lower bound, upper bound, margin of error, and standard error. You can also enable finite population correction (FPC) when you sample a meaningful portion of a fixed population.
For a sample mean, the classic confidence interval is:
CI = x̄ ± z* × (s / √n)
Where:
If you sample without replacement from a finite population and the sample is not tiny compared to the population, you can apply FPC:
FPC = √((N − n) / (N − 1))
Then the adjusted standard error becomes:
SEadj = (s / √n) × FPC
And the margin of error is:
MOE = z* × SEadj
A 95% confidence interval does not mean there is a 95% probability that the true mean is inside this specific interval. Instead, it means that if you repeated the same sampling process many times, about 95% of the intervals you build would contain the true population mean.
If your interval is wide, you have more uncertainty (often because n is small or variability s is large). If the interval is narrow, your estimate is more precise.
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Mean = 50, SD = 12, n = 100, 95% confidence. Margin of error ≈ 1.96 × (12/10) = 2.352, so CI ≈ [47.648, 52.352].
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Results
Keeping the same mean, SD, and n, but switching to 99% increases z* and widens the interval.
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If you sample a large portion of a fixed population, FPC shrinks the interval because you’re observing a bigger share of the population.
You typically need the sample mean (x̄), sample standard deviation (s), sample size (n), and your confidence level (e.g., 95%).
Higher confidence requires a larger critical value (z*), which increases the margin of error. So the interval must widen to capture the true mean more often.
The margin of error (MOE) is the “±” amount around the mean: MOE = z* × SE. Your CI is simply mean − MOE to mean + MOE.
Use FPC when you sample without replacement from a fixed population and your sample is not tiny compared to the population (a common rule of thumb is when n is more than about 5% of N).
This version uses z* critical values based on the standard normal distribution (the most common quick CI setup). If you want a strict t-based CI for small samples, the platform should support a t critical value lookup or t-inverse function.
The most direct way is increasing sample size (n), because standard error decreases with √n. Reducing variability also narrows the interval, but that depends on what you’re measuring.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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