# Coverage Factor k by Effective Degrees of Freedom (t-Distribution Table)

> Source: https://netray.co/tools/coverage-factor-t-distribution-reference-for-uncertainty
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

Look up k for an expanded uncertainty once you know the effective degrees of freedom from the Welch-Satterthwaite formula.

## Category

quality

## Industry

Metrology

## Audience

- Metrologist
- Calibration technician

## Tool

**Kind:** worksheet

**Mode:** reference

**Columns:**

#### Effective degrees of freedom

**Key:** dof

**Type:** text

#### k at 68.27 percent

**Key:** k6827

**Type:** number

**Format:** decimal

#### k at 95.45 percent

**Key:** k9545

**Type:** number

**Format:** decimal

#### k at 99.73 percent

**Key:** k9973

**Type:** number

**Format:** decimal

**Rows:**

**Dof:** 1

**K6827:** 1.84

**K9545:** 13.97

**K9973:** 235.8

**Dof:** 2

**K6827:** 1.32

**K9545:** 4.53

**K9973:** 19.21

**Dof:** 3

**K6827:** 1.2

**K9545:** 3.31

**K9973:** 9.22

**Dof:** 4

**K6827:** 1.14

**K9545:** 2.87

**K9973:** 6.62

**Dof:** 5

**K6827:** 1.11

**K9545:** 2.65

**K9973:** 5.51

**Dof:** 6

**K6827:** 1.09

**K9545:** 2.52

**K9973:** 4.9

**Dof:** 7

**K6827:** 1.08

**K9545:** 2.43

**K9973:** 4.53

**Dof:** 8

**K6827:** 1.07

**K9545:** 2.37

**K9973:** 4.28

**Dof:** 10

**K6827:** 1.05

**K9545:** 2.28

**K9973:** 3.96

**Dof:** 20

**K6827:** 1.03

**K9545:** 2.13

**K9973:** 3.42

**Dof:** 50

**K6827:** 1.01

**K9545:** 2.05

**K9973:** 3.16

**Dof:** Infinity

**K6827:** 1

**K9545:** 2

**K9973:** 3

**Notes:** Values are from the t-distribution as tabulated in the GUM (JCGM 100), Annex G, Table G.2. Round the effective degrees of freedom down to the next lower tabulated value, do not interpolate upward.

## Content

**Intro:** An expanded uncertainty is the combined standard uncertainty multiplied by a coverage factor k. When the number of effective degrees of freedom is small, k = 2 understates the real coverage. This table gives the t-distribution value to use for 68.27, 95.45 and 99.73 percent.

**Sections:**

#### How to use the table

**Body:** First compute the effective degrees of freedom of your combined uncertainty with the Welch-Satterthwaite formula, veff = uc^4 / sum(ui^4 / vi). Type A components get n - 1 degrees of freedom; Type B components with well-founded limits are usually treated as infinite. Find the row at or below your veff and read the k in the column for the coverage you need. Multiply uc by that k to get the expanded uncertainty U.

#### Worked example

**Body:** A budget has a combined standard uncertainty of 0.012 mm, dominated by a repeatability term from 5 readings (4 degrees of freedom, 0.011 mm). Welch-Satterthwaite gives veff = 0.012^4 / (0.011^4 / 4), which is about 5.6. Use the row for 5, and k = 2.65 at 95.45 percent. U = 2.65 x 0.012 = 0.032 mm, not 0.024 mm from k = 2. That 33 percent difference matters for a pass/fail decision rule.

#### When k = 2 is acceptable

**Body:** Many laboratories quote k = 2 because their budget is dominated by Type B terms with infinite degrees of freedom, or because the repeatability data set is large, 20 or more readings. If a Type A term with few readings dominates, k = 2 is not defensible and accreditation bodies will question it. Document how you assessed veff.

#### Common mistakes

**Body:** Using the number of readings instead of n - 1, interpolating upward to a higher row, and applying k to a Type A term before combining are typical. Remember k is applied once, to the combined standard uncertainty. Also check that the coverage probability you quote matches the column you used.

**Cta:** If you want uncertainty data to flow from the lab into the equipment record and certificate in your ERP, we can scope that integration.

**Faq:**

#### Why is 95.45 percent used instead of 95?

**Answer:** With infinite degrees of freedom, k = 2 corresponds exactly to 95.45 percent for a normal distribution, which is why laboratories quote it as approximately 95 percent. k = 1.96 is exactly 95 percent. The GUM table is built on 95.45.

#### What is effective degrees of freedom?

**Answer:** It is a number that represents how reliable the combined standard uncertainty is, calculated from the contribution and degrees of freedom of each component. Low values mean the uncertainty estimate itself is uncertain, so a larger k is needed.

#### Can I interpolate between rows?

**Answer:** Use the next lower tabulated degrees of freedom rather than interpolating to a more favorable value. This keeps the coverage statement conservative.

#### Does ISO 17025 require this?

**Answer:** ISO/IEC 17025 requires an uncertainty evaluation, and accreditation bodies expect the coverage factor to be justified. Many accept k = 2 when the budget supports it, and expect the t-based factor when a small Type A sample dominates.

## Search terms

coverage factor k table, t distribution coverage factor 95.45, welch satterthwaite effective degrees of freedom, expanded uncertainty k=2, gum table g.2, coverage factor degrees of freedom, k factor calibration uncertainty, student t uncertainty

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