Enter your sample size, sample mean, and standard deviation to calculate your 99% confidence interval. You'll get the lower bound, upper bound, margin of error, and standard error — showing the range where the true population mean is expected to fall 99% of the time.
99% Confidence Interval
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Lower Bound
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Upper Bound
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Margin of Error (ME)
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Standard Error (SE)
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Critical Value (Z* or t*)
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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 a mean) with a specified level of confidence. For a 99% confidence interval, if you were to repeat the sampling process many times, 99% of the calculated intervals would contain the true population parameter.
A 99% confidence interval means that the procedure used to construct the interval will capture the true population parameter in 99 out of 100 repeated samples. It does NOT mean there is a 99% probability that the true value lies within any single computed interval — the true value is either in the interval or it isn't.
The Z-score (critical value) for a 99% confidence interval is 2.576. This value comes from the standard normal distribution, where 99% of the area falls within ±2.576 standard deviations from the mean. When using a t-distribution (small samples with unknown population SD), the critical value is slightly larger and depends on the degrees of freedom.
To find a 99% confidence interval: (1) Calculate the standard error: SE = σ / √n. (2) Multiply by the critical value: ME = Z* × SE, where Z* = 2.576 for a Z-interval. (3) The interval is x̄ ± ME, giving you Lower = x̄ − ME and Upper = x̄ + ME. If the population SD is unknown, use the sample SD with a t critical value instead.
The margin of error (ME) equals the critical value multiplied by the standard error: ME = Z* × (σ / √n). For a 99% confidence level using a Z-interval, Z* = 2.576. A larger sample size reduces the standard error and therefore narrows the margin of error.
Use a Z-interval when the population standard deviation (σ) is known, or when the sample size is large (n ≥ 30) and you're using the sample SD as an approximation. Use a t-interval when the population SD is unknown and the sample size is small (n < 30), as the t-distribution accounts for additional uncertainty from estimating σ with s.
A higher confidence level requires a larger critical value, which increases the margin of error and widens the interval. The Z* for 95% is 1.96, while for 99% it is 2.576. This trade-off means greater confidence comes at the cost of precision — you are more certain the interval contains the true value, but the range itself is broader.
The required sample size depends on your desired margin of error and standard deviation: n = (Z* × σ / E)². For a 99% confidence level (Z* = 2.576), if your SD is 10 and you want a margin of error of 2, you need approximately n = (2.576 × 10 / 2)² ≈ 166 observations. Larger samples yield narrower, more precise intervals.