Math & Statistics
Confidence Interval Calculator
Calculate standard error, margin of error, and lower and upper bounds for a mean using the entered critical value multiplied by sample standard deviation divided by the square root of sample size. The guidance makes the z-style approximation and its sampling assumptions explicit.
Standard error-
Margin of error-
Lower confidence bound-
Upper confidence bound-
Decision view
Confidence interval around the sample mean
Confidence interval around the sample meanThe lower and upper bounds are joined on one measurement scale with the sample mean, margin of error, and standard error labeled explicitly.
Exact scenario comparisonSample size changes while all other entered assumptions remain constant.
| Sample size | Standard error | Margin of error | Lower confidence bound | Upper confidence bound |
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How to use Confidence Interval Calculator
- Enter the sample mean, sample standard deviation, and number of independent observations from the same analysis sample.
- Choose a critical value that matches the intended confidence level and method; 1.96 is the familiar two-sided 95% normal critical value, not a universal default.
- Report the interval together with the confidence procedure, sample size, sampling design, units, and any weighting or clustering that is not represented here.
Calculator guide
Understanding Confidence Interval Calculator
A confidence interval combines a point estimate with a margin intended to reflect sampling uncertainty under a stated method. This page constructs a symmetric z-style interval from an entered sample mean, sample standard deviation, sample size, and critical value.
Calculation method
How the calculation works
Divide sample standard deviation by the square root of sample size and multiply by the entered critical value to form a symmetric interval. Divide sample standard deviation by the square root of sample size and multiply by the entered critical value to form a symmetric interval.
Method checklist
Verify the interval before interpreting it
A numerically correct interval can still overstate certainty when the design and multiplier do not match the data.
A confidence interval is strongest when the calculation method and data-generating process are reported together.
Worked situations
Practical examples
- For mean 52, standard deviation 11, and n = 120, standard error is 11 divided by the square root of 120, approximately 1.004.
- Using critical value 1.96 gives a margin of about 1.968 and a symmetric interval from approximately 50.032 to 53.968.
- Quadrupling an otherwise comparable sample size halves the standard error because uncertainty changes with the square root of n, not linearly with n.
Better inputs
Useful tips
- Use the effective independent sample size when observations are clustered, repeated, weighted, or correlated.
- For a small sample with unknown population standard deviation, a t critical value with appropriate degrees of freedom is often more suitable than 1.96.
- Keep more precision in the standard error and margin than in the final published interval to avoid rounding drift.
Before relying on the result
Limitations and common mistakes
- The page uses the entered standard deviation and critical value without selecting a t distribution, degrees of freedom, or design-based variance estimator.
- It assumes a symmetric interval around the sample mean and does not handle transformations, bootstrap intervals, finite-population correction, weights, or clustering.
- Confidence level does not measure the probability that this already-calculated fixed interval contains the parameter under the usual frequentist interpretation.
Reference
Key terms
- Standard error
- Estimated standard deviation of the sample mean, calculated here as s divided by square root of n.
- Critical value
- Multiplier selected from the intended confidence procedure.
- Margin of error
- Critical value multiplied by standard error.
- Confidence interval
- The point estimate minus and plus the method's margin of error.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Does 95% confidence mean a 95% chance the true mean is inside these bounds?
In the usual frequentist interpretation, 95% refers to the long-run coverage of intervals produced by the procedure, not a probability assigned to one fixed interval after calculation.
Why does a larger sample narrow the interval?
The standard error divides the entered standard deviation by the square root of sample size, so more independent observations reduce modeled sampling uncertainty.
When should a t critical value be used?
It is commonly used for a mean when population standard deviation is unknown, particularly with smaller samples; the appropriate value depends on confidence level and degrees of freedom.
Can the lower bound be negative?
Yes. The symmetric arithmetic does not enforce natural limits. For bounded variables, consider a method designed for that scale or a justified transformation.