Formula or algorithm and variables.
interval = x̄ ± z × (s/√n)x̄ is sample mean, s is sample standard deviation, n is sample size, and z is 1.645, 1.960, or 2.576 for the selected level.
Formula or algorithm and variables
interval = x̄ ± z × (s/√n). x̄ is sample mean, s is sample standard deviation, n is sample size, and z is 1.645, 1.960, or 2.576 for the selected level. For a mean interval with a normal approximation, margin of error is critical value × standard error, where standard error is sample standard deviation divided by √n. Start by identifying the mathematical object represented by each input rather than treating every number as interchangeable. Keep the chosen mode visible when a tool has more than one interpretation. The calculation preserves exact integer arithmetic where the model is discrete and uses controlled numeric rounding where the result is real-valued. This makes the path from the input to the displayed answer inspectable and gives you a repeatable way to check a hand calculation.
Worked example
A mean of 50, standard deviation 10, sample size 100, and 95% confidence use z = 1.96, giving an interval of 48.04 to 51.96. Reproduce the example by writing the formula, substituting the values, and retaining units or place-value labels through each step. For a list-based tool, sort or group the values before checking the result; for a base-conversion tool, expand each digit by its place value. When the result is a large integer, compare the exact string rather than a rounded approximation. A worked example is a verification aid, not a default recommendation, and it should be replaced with the values from the problem you are actually solving.
How to interpret the result
The interval estimates a range for a population mean under the selected model; it is not the probability that a fixed parameter moves after the interval is calculated. Read the primary result together with its supporting details and the selected mode. A mathematically correct result can still answer the wrong question if the inputs describe a different population, representation, unit, ordering rule, or sample space. Compare scenarios only when the definitions and assumptions remain the same. Use exact notation for factors, matrices, repeating decimals, and integer bases; use the displayed precision for real-number summaries and avoid implying more certainty than the inputs support.
Validation and edge cases
Standard deviation must be positive, sample size must be at least 2, and confidence must be 90%, 95%, or 99%. Blank fields, malformed tokens, non-finite values, incompatible dimensions, invalid divisors, impossible probability ranges, and out-of-domain operations are rejected explicitly. Boundary values such as zero, one, a prime, a singular matrix, an empty remainder cycle, or a one-term sequence are tested according to the selected operation. If the output seems surprising, first verify the mode, the input order, the representation width, and whether a quantity was intended to be a count or a measurement.
Limitations and responsible use
The calculator uses a z critical value and assumes an independent, representative sample with a reasonably normal sampling distribution. It does not correct bias or study design. This page is an explanatory calculator, not a substitute for a statistical design, numerical-analysis package, software type specification, laboratory model, or professional review. Results are processed locally in the browser and no entered values are sent to a calculation service. Keep the original inputs, mode, and assumptions with any copied result. For decisions involving health, finance, safety, experiments, or production systems, verify the model with current authoritative guidance and an independent method before acting.
Check assumptions and sources.
The calculator uses a z critical value and assumes an independent, representative sample with a reasonably normal sampling distribution. It does not correct bias or study design.
Common questions.
What does the Confidence Interval Calculator calculate?
Estimate a normal-approximation confidence interval for a sample mean with clear assumptions and margin of error.
Which formula or algorithm does it use?
interval = x̄ ± z × (s/√n). x̄ is sample mean, s is sample standard deviation, n is sample size, and z is 1.645, 1.960, or 2.576 for the selected level.
How should the result be interpreted?
The interval estimates a range for a population mean under the selected model; it is not the probability that a fixed parameter moves after the interval is calculated.
What limitations or edge cases matter?
Standard deviation must be positive, sample size must be at least 2, and confidence must be 90%, 95%, or 99%. The calculator uses a z critical value and assumes an independent, representative sample with a reasonably normal sampling distribution. It does not correct bias or study design.