Sampling distribution, interval endpoints, benchmark, and precision leverLive current inputs
Math & Statistics
Confidence Interval Solver Calculator
Construct a two-sided confidence interval for a population mean from a sample mean, sample standard deviation, sample size, and confidence level using a Student-t critical approximation.
Lower confidence limit -
Upper confidence limit -
Margin of error -
Standard error -
Approximate t critical -
Degrees of freedom -
Interval width -
Benchmark relation -
CONFIDENCE-INTERVAL ANATOMY
Sampling distribution, interval endpoints, benchmark, and precision lever
The main panel centers the t-based interval on the observed mean; the lower precision curve shows how margin changes with sample size while sd and confidence stay fixed.
PRECISION SCENARIO TABLE
How sample size and confidence level change the interval
Scenarios hold the sample mean and SD fixed, isolating the arithmetic effect of n and confidence without pretending new observations have been collected.
| Scenario | Sample size | Confidence | t critical | Margin | Interval |
|---|
STATISTICAL SETUP
Confirm the estimator and sampling design before reporting an interval
- Enter the arithmetic sample mean.
- Use the sample SD computed with n-1 degrees of freedom.
- Count independent observations represented by that SD.
- Choose confidence before comparing with the benchmark.
- Report the interval with units and the sampling context.
WHAT CONFIDENCE MEANS
The interval describes a repeated procedure, not a probability assigned to this fixed endpoint pair
Under the model assumptions, repeated samples and intervals constructed by the same method would cover the population mean at the stated long-run rate.
The interval does not contain a chosen percentage of individual observations. That would require a prediction or tolerance interval, not a confidence interval for the mean.
ONE-SAMPLE T INTERVAL
Estimate uncertainty in the mean, not the spread of individual observations
The standard error divides sample SD by square root of n. A two-sided t critical value expands that standard error to the selected central confidence level.
Detailed calculation process and general formulas
df=n-1SE=s/sqrt(n)alpha=1-confidenceME=t_(1-alpha/2,df)*SECI=xbar +/- MESymbols, meanings, and units
- xbar
- sample meanmeasurement unit
- s
- sample standard deviationmeasurement unit
- n
- independent sample sizeobservations
- df
- degrees of freedomcount
- SE
- estimated standard error of the meanmeasurement unit
- ME
- two-sided margin of errormeasurement unit
DECISION CONTEXT
Read precision separately from benchmark comparison
A narrow interval can still be biased by a poor sample, and a wide interval can still be honestly calculated.
Sampling precision
-Margin combines SD, n, and the t critical value.
Evidence against a benchmark
-The benchmark is located below, inside, or above the interval.
Design leverage
-The lower curve shows diminishing margin reductions as n grows.
Decision takeaway: Treat benchmark exclusion as model-based evidence, not as proof of practical importance or causal effect.
ASSUMPTION REVIEW
Questions the interval arithmetic cannot answer
- Are observations independent?
- Is the sample representative of the target population?
- Are severe outliers or skewness material at this n?
- Was the endpoint selected before seeing the data?
- Does measurement error dominate sampling error?
Applied decisions
Confidence intervals with different decision implications
Process mean
A quality team estimates an average measurement from 24 independent units.
What the result clarifies: The interval quantifies mean precision while the benchmark marker supplies specification context.
Small pilot sample
A preliminary study has the same SD but far fewer observations.
What the result clarifies: The t critical value and standard error both widen the interval.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This is a one-sample mean interval using a Student-t critical approximation. It assumes independent observations and a sampling distribution suitable for t inference. It is not a prediction interval, tolerance interval, causal test, or substitute for study-design review.
Confidence Interval Solver Calculator | One-Sample Mean with t Critical Value FAQ
Why use t instead of z?
The population SD is unknown and estimated by the sample SD, adding uncertainty that is reflected by the t distribution.
Does 95% confidence mean 95% of observations lie inside?
No. The interval targets the population mean, not individual observations.
Why does doubling n not halve the margin?
Standard error decreases with square root of n, so four times as many observations are needed to roughly halve it.
What if the benchmark is inside the interval?
The interval procedure does not distinguish the mean from that benchmark at the selected two-sided confidence level; practical interpretation still needs context.