The t penalty fades as degrees of freedom accumulateUpdates with every input
Math & Statistics
Confidence Interval Approximation Calculator
Compare normal-z and Student-t confidence interval approximations for a mean, quantify their margin and endpoint gap, and follow how the difference contracts as sample size grows.
Student-t interval -
Normal-z interval -
t margin -
z margin -
Margin gap -
Relative gap -
First n meeting target -
Critical-value gap -
APPROXIMATION CONVERGENCE
The t penalty fades as degrees of freedom accumulate
A critical-value curve and paired interval rails show both the mathematical convergence and its consequence on endpoints.
SAMPLE-SIZE CHECKPOINTS
t and z margins at selected sample sizes
The table reveals when the numerical gap becomes negligible relative to the reporting scale.
| n | df | t critical | z critical | t margin | z margin | Margin gap |
|---|
COMPARISON SETUP
Keep every ingredient except the critical distribution fixed
- Enter the same mean, sample SD, and n for both intervals.
- Choose the reporting confidence level once.
- Set a convergence horizon large enough to show the gap shrinking.
- Express the target gap on the outcome scale.
- Use Student t when population sigma is unknown, even if the z result looks similar.
LIMITING BEHAVIOR
Numerical similarity does not erase the method distinction
At small n, the t distribution has heavier tails and yields a wider interval. As df rises, the critical values converge.
The gap also shrinks because standard error falls with sqrt(n). Both mechanisms appear in the curve.
APPROXIMATION MODEL
Separate unknown-sigma uncertainty from large-sample convenience
Both intervals use the same estimated standard error. Their difference comes entirely from replacing the t critical value with its limiting normal-z value.
Detailed calculation process and general formulas
SE_n = s / sqrt(n)ME_t = t*_(n-1) x SE_nME_z = z* x SE_ngap_n = ME_t - ME_zt*_(df) -> z* as df -> infinitySymbols, meanings, and units
- t*
- Student t critical valueunitless
- z*
- standard normal critical valueunitless
- df
- degrees of freedomunitless
- gap_n
- difference between t and z marginsoutcome units
- SE_n
- standard error at sample size noutcome units
APPROXIMATION JUDGMENT
Decide whether a shortcut matters at the displayed precision
A tiny gap may be operationally irrelevant while the formal model remains t-based.
Current penalty
-The margin gap quantifies the extra width from Student t.
Relative penalty
-Compare the gap with the t margin rather than reading raw units alone.
Convergence threshold
-The first sample size meeting the entered gap target is reported.
Decision takeaway: Use the correct distribution; use the gap only to understand how costly the approximation would be.
Applied decisions
Approximation comparisons
Small pilot study
A 12-observation pilot has an unknown population SD.
What the result clarifies: The t penalty is visible and should not be ignored.
Large monitoring sample
A routine report has hundreds of observations.
What the result clarifies: The two displayed intervals may round identically even though Student t remains the formal calculation.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
This calculator compares two critical-value conventions; it does not decide whether a t interval is appropriate for the sampling design or distribution. Large n does not cure bias, dependence, confounding, or invalid measurements.
Confidence Interval Approximation Calculator | z versus t Convergence FAQ
When are t and z exactly equal?
At finite df they are not exactly equal; t approaches z in the limit.
Why does the margin gap shrink faster than the critical gap?
The critical gap shrinks while the standard error also falls as n grows.
Should I switch to z once the chart looks flat?
Not merely for that reason. Use the model justified by whether population sigma is known.