Enter your sample size, sample mean, and standard deviation to calculate your 90% confidence interval. The calculator returns the lower and upper bounds of the interval, plus the margin of error and standard error — giving you a complete picture of where the true population mean is likely to fall.
Confidence Interval
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Lower Bound
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Upper Bound
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Margin of Error (M.O.E)
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Standard Error (S.E)
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Z-Score Used
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A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (such as the mean) with a specified probability. For example, a 90% confidence interval means that if you repeated the sampling process 100 times, approximately 90 of those intervals would contain the true population mean.
A 90% confidence interval means there is a 90% probability that the calculated interval contains the true population parameter. It does not mean there is a 90% chance the true value is within that specific range — rather, it describes the reliability of the estimation procedure used to construct the interval.
The z-score (critical value) for a 90% confidence interval is 1.6449. This value comes from the standard normal distribution, where 90% of the distribution falls within ±1.6449 standard deviations of the mean, leaving 5% in each tail.
To calculate a 90% confidence interval: (1) Find the standard error by dividing the standard deviation by the square root of the sample size. (2) Multiply the standard error by the z-score of 1.6449 to get the margin of error. (3) Add and subtract the margin of error from the sample mean to get the upper and lower bounds of the interval.
A 95% confidence interval is wider than a 90% confidence interval because it requires more certainty. The z-score for 95% (1.9600) is larger than for 90% (1.6449), resulting in a larger margin of error and therefore a broader interval. Higher confidence comes at the cost of precision.
The margin of error is the amount added to and subtracted from the sample mean to create the confidence interval. It equals the critical z-score multiplied by the standard error (σ / √n). A smaller sample size or higher confidence level will result in a larger margin of error.
Standard error (SE) measures how much the sample mean is expected to vary from the true population mean. It is calculated as the standard deviation divided by the square root of the sample size (σ / √n). The standard error is a key component of the confidence interval — a smaller SE leads to a narrower, more precise interval.
Yes. A larger sample size reduces the standard error, which in turn reduces the margin of error and produces a narrower confidence interval. This means your estimate of the population mean becomes more precise as your sample size increases, without needing to change the confidence level.