1.825742
3.578454
46.421546
53.578454
7.156908
7.1569
%
1.825742
3.578454
46.421546
53.578454
7.156908
7.1569
%
The Confidence Interval Calculator constructs an interval estimate around a sample mean that is likely to contain the true population mean. Unlike a point estimate that gives a single value, a confidence interval provides a range of plausible values along with a level of confidence, making it one of the most important tools in inferential statistics. Whether you are analyzing clinical trial results, survey data, or quality control measurements, confidence intervals communicate both the estimate and its uncertainty.
This calculator supports the three most common confidence levels: 90%, 95%, and 99%. A 95% confidence interval means that if you were to repeat the sampling process many times, approximately 95% of the constructed intervals would contain the true population mean. The width of the interval depends on the standard deviation, sample size, and chosen confidence level, giving researchers direct control over the precision of their estimates.
The confidence interval for a population mean is calculated as:
$$CI = \bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}$$
Where x̄ is the sample mean, z* is the critical z-value for the desired confidence level, σ is the standard deviation, and n is the sample size. The term $$\frac{\sigma}{\sqrt{n}}$$ is the standard error, which measures how precisely the sample mean estimates the population mean.
The critical z-values used are: 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%. The margin of error (ME) is the product of the critical value and the standard error. Increasing the sample size narrows the interval (higher precision), while increasing the confidence level widens it (greater certainty).
The confidence interval should be interpreted as: We are X% confident that the true population mean lies between the lower and upper bounds. A narrow interval indicates a precise estimate, while a wide interval suggests more uncertainty. If the interval for a treatment effect does not contain zero, the effect is statistically significant at the corresponding alpha level. For practical decision-making, consider whether the interval falls entirely within an acceptable range.
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Results
With 100 respondents, the 95% CI for average income is $50,432 to $53,568, a margin of error of $1,568.
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Results
A 99% CI for pH measurement with 25 samples gives a tight interval of 7.245 to 7.555.
A 95% confidence level means that if you repeated the experiment many times and constructed a confidence interval each time, approximately 95% of those intervals would contain the true population parameter. It does not mean there is a 95% probability that the true value lies in this particular interval; the true value is fixed, and the interval either contains it or does not.
You can narrow a confidence interval in three ways: increase the sample size (most practical), use a lower confidence level (e.g., 90% instead of 99%), or reduce variability in your measurements. Quadrupling the sample size halves the margin of error because the standard error involves the square root of n.
Use a z-interval when the population standard deviation is known or the sample size is large (n > 30). Use a t-interval when the population standard deviation is unknown and you are estimating it from a small sample. The t-distribution has heavier tails to account for the extra uncertainty in estimating the standard deviation.
This calculator is designed for means. For proportions, the formula differs: CI = p̂ ± z*·√(p̂(1-p̂)/n). While the structure is similar, the standard error calculation is different. We offer a separate sample size calculator that can handle proportions.
A higher confidence level requires a wider interval to be more certain about capturing the true parameter. The 99% interval uses a larger critical value (2.576 vs 1.96), so it extends further in both directions. Every 99% interval for a given dataset will contain the corresponding 95% interval.
This calculator assumes: (1) the data comes from a random sample, (2) the population is approximately normally distributed or the sample size is large enough for the Central Limit Theorem to apply (typically n ≥ 30), and (3) the standard deviation is known or accurately estimated. Violations of these assumptions may affect the coverage probability of the interval.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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