Using population spread
STDEV.P divides by n; the sample variance here uses n-1.
Estimate a mean with a two-sided t interval
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Step-by-step guide · read-only ·
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Data & tools
Task requirements
Enter separate formulas in G2:G7. Read the numeric observations in A2:A21 and alpha in D2; only trailing unused sample cells are blank. G2 is n, G3 the sample mean, G4 the sample standard deviation, G5 the two-sided t margin at confidence level 1-alpha, and G6:G7 the lower/upper interval endpoints. Use sample spread, n-1 degrees of freedom and full computed precision.
Edit the sample report
Apply one numeric value to the actual worksheet. Observations and alpha retain their separate meanings. The complete sample design is validated before an edit replaces a valid input.
Applied value:
Follow the highlighted cells. Switch to Your turn to edit. Answer cells: G2, G3, G4, G5, G6, G7. Browser calculations may differ from your Excel version.
Selection & formula search
Practice stays on this device. No worksheet data is uploaded.
Notes & examples
About this lesson
Report the sample mean, sample variation, confidence margin and both endpoints instead of presenting the margin as the whole interval.
THE QUICK ANSWER01
Report the sample mean, sample variation, confidence margin and both endpoints instead of presenting the margin as the whole interval.
G2
=COUNT($A$2:$A$21)G3
=AVERAGE($A$2:$A$21)G4
=STDEV.S($A$2:$A$21)G5
=CONFIDENCE.T($D$2,STDEV.S($A$2:$A$21),COUNT($A$2:$A$21))G6
=G3-G5G7
=G3+G5For the example data
G2 8G3 10.5G4 1.6035674514745464G5 1.3406159386770387G6 9.159384061322962G7 11.840615938677038
Enter each formula in its labeled answer cell. These report formulas perform different jobs; do not fill the first formula down.
How it works
The design selects the calculation
Enter separate formulas in G2:G7. Read the numeric observations in A2:A21 and alpha in D2; only trailing unused sample cells are blank. G2 is n, G3 the sample mean, G4 the sample standard deviation, G5 the two-sided t margin at confidence level 1-alpha, and G6:G7 the lower/upper interval endpoints. Use sample spread, n-1 degrees of freedom and full computed precision.
Keep the model assumptions visible
The teaching inputs assume independent randomly sampled records and approximately normal outcomes (normal within-record differences for the paired task). The supplied numbers do not establish randomization, independence or distribution shape. A confidence procedure has a long-run coverage interpretation; p is not the probability that its null hypothesis is true. These are model-based sample calculations, not causal or practical-impact conclusions.
Declared numeric scope
Use 3–20 numeric observations per group, magnitudes at most 10000, positive relevant sample variance at least 1e-10, and alpha 0.01–0.20. Paired identities stay fixed and complete. Independent samples are contiguous with trailing blanks. Absolute paired/Welch t must be at most 20. Unsupported types, holes, zero spread and numerically ambiguous p-versus-alpha thresholds withhold grading instead of changing the data. Keep full precision; nonzero p values cannot pass as zero.
Read the full step-by-step walkthrough
- Identify the sampling model
Use independent observations from an approximately normal population for this small-sample teaching model. The sample does not prove random sampling or normality. The interval estimates the population mean, not the range of individual readings.
- Separate variation from standard error
STDEV.S estimates the spread of individual observations. Dividing by SQRT(n) gives the standard error of their mean. Do not divide the standard deviation by n.
- Keep alpha distinct from confidence
D2 is alpha, so 0.05 defines a 95 percent procedure. CONFIDENCE.T returns a half-width; T.INV.2T(alpha,n-1)*s/SQRT(n) gives the same margin.
- Construct both endpoints
Subtract the margin from the mean for the lower endpoint and add it for the upper endpoint. A negative lower endpoint is allowed for these signed quantities.
- Interpret uncertainty of the mean
The same sample gives a wider interval at smaller alpha. This is uncertainty of a population mean under the model, not the spread of individual future readings. Retain both endpoints and the point estimate.
Things to watch
Reporting only the margin
The interval is mean minus margin through mean plus margin.
Calling the interval an individual prediction
This calculation estimates a mean. It is not an interval for the next individual reading.
About this example
Original bounded source reports checked with real formulas and independent quadrature/incomplete-beta targets. Native recalculation and edit/save remain separate.
Content version 1. Practice checks compare calculated results across different datasets; they do not certify every possible formula.