Confidence Interval Calculator
Quickly compute a confidence interval for a population mean or proportion. Enter your sample statistics, choose a confidence level (90 %, 95 %, or 99 %), and get the lower bound, upper bound, and margin of error instantly.
Confidence Interval Calculator
Inputs
Enter sample data and confidence level to calculate the confidence interval.
Understanding Confidence Intervals
A confidence interval (CI) is a range of values that is constructed from sample data to estimate an unknown population parameter, such as a mean or a proportion. Instead of reporting a single point estimate, a CI communicates the precision of the estimate by providing an upper and lower bound along with a stated level of confidence.
The Core Formula
CI = x̄ ± z* × (σ / √n)- x̄ — the sample mean, your best single estimate of the population mean.
- z* — the critical z-value corresponding to your chosen confidence level.
- σ — the population standard deviation (or the sample standard deviation s when n is large).
- n — the sample size; appears under the square root, so its effect is non-linear.
Interpreting a Confidence Interval
A 95 % confidence interval does not mean there is a 95 % probability that the true parameter falls inside the interval you just computed. The true value is fixed; it either lies inside or outside the interval. The correct interpretation is: if you repeated the sampling procedure many times, 95 % of the resulting intervals would contain the true parameter.
Confidence Levels and Z-Scores
The confidence level determines how wide the interval will be. Common choices are 90 %, 95 %, and 99 %. A higher confidence level uses a larger z-score, which stretches the interval further from the point estimate to capture the population parameter with greater certainty.
- 90 % CI (z* = 1.645) — tighter interval, lower certainty. Useful when some risk of missing the parameter is acceptable.
- 95 % CI (z* = 1.960) — the most widely used level in academic research and industry reporting.
- 99 % CI (z* = 2.576) — broader interval, higher certainty. Preferred in medical trials and safety-critical applications.
Margin of Error
The margin of error (ME) is the “±” portion of the confidence interval formula: ME = z* × (σ / √n). It represents the maximum expected difference between the sample statistic and the true population parameter at the stated confidence level. Halving the margin of error requires quadrupling the sample size.
Factors Affecting CI Width
Three factors determine how wide or narrow a confidence interval will be:
- Sample size (n) — Larger samples produce smaller standard errors and therefore narrower intervals. The relationship is governed by √n, so doubling the sample cuts the standard error by about 29 %.
- Variability (σ) — When data points are spread out, the standard deviation is larger, and the interval widens.
- Confidence level — Increasing the confidence level from 95 % to 99 % raises the z-score from 1.960 to 2.576, widening the interval by roughly 31 %.
Z-Score Reference Table for Common Confidence Levels
The table below lists the critical z-values (z*) used in the confidence interval formula for the most frequently encountered confidence levels.
| Confidence Level | Z-Score (z*) | Typical Use Case |
|---|---|---|
| 80 % | 1.282 | Preliminary or exploratory research |
| 85 % | 1.440 | Internal business analytics |
| 90 % | 1.645 | Lower-stakes studies; tighter intervals |
| 95 % | 1.960 | Standard in most published research |
| 99 % | 2.576 | Medical and safety-critical research |
| 99.9 % | 3.291 | Six Sigma quality control processes |
How to Calculate a Confidence Interval
Follow these four steps for any confidence interval calculation: (1) identify the sample statistic (mean or proportion), (2) choose a confidence level and find the corresponding z-score, (3) compute the standard error, and (4) multiply the z-score by the standard error to get the margin of error, then add and subtract it from the point estimate.
Example 1: 95 % CI for a Mean (Exam Scores)
A university surveys 64 students. The sample mean exam score is 72, with a standard deviation of 16. Find the 95 % confidence interval for the true mean.
z* = 1.960 (for 95 %)
SE = σ / √n = 16 / √64 = 16 / 8 = 2.0
ME = 1.960 × 2.0 = 3.92
CI = 72 ± 3.92 = [68.08, 75.92]
Interpretation: We are 95 % confident the true mean score for all students is between 68.08 and 75.92.
Example 2: 99 % CI for a Proportion (Voter Poll)
A political poll surveys 1,000 voters. 54 % say they support a candidate. Find the 99 % confidence interval for the true proportion.
p̂ = 0.54, n = 1 000, z* = 2.576
SE = √(0.54 × 0.46 / 1000) = √0.0002484 ≈ 0.01576
ME = 2.576 × 0.01576 ≈ 0.0406 (4.06 %)
CI = 54 % ± 4.06 % = [49.94 %, 58.06 %]
Interpretation: We are 99 % confident the true support level is between 49.94 % and 58.06 %. Because the lower bound dips below 50 %, the result is not conclusive at 99 % confidence.
Example 3: 90 % CI for a Mean (Manufacturing)
A factory samples 49 widgets and measures a mean weight of 250 g with a standard deviation of 14 g. Compute the 90 % confidence interval for the true mean widget weight.
z* = 1.645 (for 90 %)
SE = 14 / √49 = 14 / 7 = 2.0
ME = 1.645 × 2.0 = 3.29
CI = 250 ± 3.29 = [246.71, 253.29]
Interpretation: We are 90 % confident the true mean weight of all widgets produced lies between 246.71 g and 253.29 g.
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