10-second primer
PIPrediction Interval
- What: Bounds the next observation from the same process.
- Use when: “Where will one more part/reading land?”
- Formula: x̄ ± t*·s·√(1+1/n) (two-sided)
- Note: Use t* (small n). For very large n, z is ok.
TITolerance Interval
- What: Bounds a chosen percent of the population.
- Use when: Make claims like “≥90% of all units lie in [L,U]”.
- Formula: x̄ ± k·s where k comes from a TI table/algorithm for your γ, n, and TI confidence.
Assumptions: roughly normal data, independent observations, and s is a reasonable estimate of σ.
Check a QQ plot or rely on process physics/CLT. Misusing normality ⇒ misleading intervals.
Inputs
α (for PI) = 0.050
Ready.
PIPrediction Interval — for the next observation
We compute x̄ ± t*·s·√(1+1/n). In Excel: t* = T.INV.2T(α, n-1).
Interpretation: If you take one more reading under the same conditions, it falls in [L,U] with ≈ stated confidence (normal model).
Common mistakes: (1) Using a CI when you actually want a PI. (2) Forgetting the √(1+1/n) factor. (3) Using z instead of t for small n.
TITolerance Interval — to cover a chosen % of the population
We use TI = x̄ ± k·s. Enter k from a TI table/software (n, γ, 1−α).
For a quick large-n intuition, the button suggests k ≈ z(1+γ)/2 (ignores TI confidence; not exact).
Reading: “With the stated confidence (from your table/software), at least γ% of the whole distribution lies in [L,U].”
Reminder: Exact k needs (γ, n, TI confidence). The quick k here is only a large-n intuition aid.
Reminder: Exact k needs (γ, n, TI confidence). The quick k here is only a large-n intuition aid.
Which one? (1-liners) & Excel cells
- CI: bounds a parameter (e.g., μ).
- PI: bounds a future single value (one random draw).
- TI: bounds a chosen fraction of the population (coverage).
PI (Excel, two-sided):
t* = T.INV.2T(α, N-1)
half = t* * S * SQRT(1 + 1/N)
PI = XBAR ± half
PI (z-approx large n): use NORM.S.INV(1−α/2) instead of t*
TI (normal, two-sided): TI = XBAR ± k*S (get k from TI table/software for your γ and confidence)
t* = T.INV.2T(α, N-1)
half = t* * S * SQRT(1 + 1/N)
PI = XBAR ± half
PI (z-approx large n): use NORM.S.INV(1−α/2) instead of t*
TI (normal, two-sided): TI = XBAR ± k*S (get k from TI table/software for your γ and confidence)