CONFIDENCE.T in Google Sheets: T-Based Mean Intervals
Calculate a t-based confidence-interval half-width with CONFIDENCE.T in Google Sheets, using sample deviation, observation count and independently checked examples.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: =CONFIDENCE.T(0.05,12,36) has expected result 4.0602158565. It is the half-width of a two-sided 95% t-based mean interval for an observation count of 36 and supplied standard deviation of 12.
Syntax and choosing the inputs
=CONFIDENCE.T(alpha, standard_deviation, size)
The Google CONFIDENCE.T page documents alpha as one minus confidence. Although its argument descriptions say population deviation and population size, Google’s MARGINOFERROR page explicitly defines its range-based calculation using CONFIDENCE.T with STDEV(range) and COUNT(range). This supports using the sample deviation and numeric observation count for the usual mean interval.
alpha: total probability outside the two-sided interval; 0.05 means 95% confidence.standard_deviation: the supplied deviation; use the sample deviation for this mean-interval workflow.size: number of numeric observations, not degrees of freedom and not the total population headcount.
Supply all three arguments. Examples use 0 < alpha < 1, positive deviation and an integer count above 1. Exact native validation at other inputs is not established here.
Worked example from five measurements
Enter 8, 10, 12, 14 and 16 in A2:A6. These synthetic values have mean 12 and sample standard deviation approximately 3.1622776602.
In D2, enter:
=AVERAGE(A2:A6)
In D3, calculate the margin:
=CONFIDENCE.T(0.05,STDEV.S(A2:A6),COUNT(A2:A6))
Expected D3: 3.9264863230. Then calculate the endpoints:
=D2-D3
=D2+D3
Expected endpoints: 8.0735136770 and 15.9264863230. The function returns neither endpoint on its own; it returns the quantity added to and subtracted from the mean.
This tiny synthetic dataset illustrates the arithmetic. For real small samples, the distribution and sampling assumptions deserve particular scrutiny.
Independent check and a printed-example discrepancy
The standard mean-interval half-width is t_(1-alpha/2,n-1)*s/SQRT(n), as described in NIST’s confidence limits for the mean. For n = 5 and 95% confidence, the t critical value is approximately 2.7764451052. Multiplying it by SQRT(10)/SQRT(5) gives 3.9264863230.
Google’s CONFIDENCE.T table prints alpha 0.05, deviation 6.43, size 27 and result 2.56340262541175. Independent t arithmetic gives 2.5436232842 for those printed inputs. The printed result instead matches deviation 6.48. Google’s MARGINOFERROR example independently matches the standard t formula. This guide uses original examples and discloses that table inconsistency rather than presenting the mismatched row as an executed check.
Advanced application: compare precision at different sample sizes
For a supplied deviation of 12 and 95% confidence:
| Formula | Expected margin |
|---|---|
=CONFIDENCE.T(0.05,12,9) |
9.2240165408 |
=CONFIDENCE.T(0.05,12,36) |
4.0602158565 |
=CONFIDENCE.T(0.05,12,100) |
2.3810603419 |
Increasing the count changes both the square-root denominator and the t critical value through n minus 1 degrees of freedom. Consequently, quadrupling the count does not exactly halve a t margin, even when the supplied deviation stays fixed.
At n = 36, the corresponding normal-based margin is approximately 3.9199279691, slightly smaller than the t margin 4.0602158565. This comparison is mathematical; the choice between methods depends on what is known about variability and the sampling model, not on which interval looks more favorable.
Interpret the interval correctly
A 95% confidence procedure is designed to cover the fixed population mean in about 95% of repeated samples under its assumptions. It is not an interval intended to contain 95% of individual measurements. It also does not establish a 95% posterior probability for the fixed mean without an additional probabilistic framework.
The formula does not repair nonrandom sampling, correlated observations or severe model mismatch. When raw values contain missing measurements, ensure that the mean, deviation and count use the same set of observations. Replacing a missing measurement with zero changes the analysis rather than merely cleaning its display.
Errors, availability and verification
Common input mistakes are entering 0.95 instead of 0.05, using population count rather than sample count, supplying n minus 1 as size, and subtracting the margin for both endpoints. The upper endpoint uses addition.
The reviewed documentation does not enumerate all native error codes, zero-deviation handling, fractional sizes or coercion. Current rolling Google Sheets documents CONFIDENCE.T, but its original introduction date and first or last functioning numbered release were not established.
NOT_TESTED_IN_GOOGLE_SHEETS. The individual Google pages and NIST reference were inspected on October 17, 2026. Original outputs and published-example discrepancies were independently checked numerically. No native execution is claimed.