1. Introduction 

Struggling to calculate measurement uncertainty for analog pressure gauges? This guide explains the GUM method step-by-step using a real example. You’ll learn how to build an uncertainty budget, handle analog readability correctly, and calculate expanded uncertainty at 95% confidence (k = 2).

Measurement uncertainty tells us how reliable a measurement is. It quantifies the doubt that exists in our measurements.

Instead of just reporting a value, we report: 

Result = Measured Value ± Uncertainty 

This uncertainty represents a range where the true value is expected to lie, typically at 95% confidence (k = 2). 

2. What Makes Analog Gauges Different? 

Analog gauges are unique because: 

• They rely on human interpretation (reading a pointer) 

• There is no digital rounding 

• The biggest uncertainty often comes from readability (eye estimation) 

This is the most important concept to understand. 

3. Measurement Model 

For pressure calibration: 

E = I_UUC − R_STD 

Where: 

• E = error of the gauge 

• I_UUC = indication of the gauge (Unit Under Calibration) 

• R_STD = value from the reference standard 

All uncertainty contributors affect this result. 

4. Types of Uncertainty 

Type A (Statistical) 

Obtained from repeated measurements. 

Example: 

• Repeatability test 

Formula: 

u_A = s / √n 

Where: 

• s = standard deviation 

• n = number of readings 

Type B (Non-statistical) 

From other sources such as: 

• Calibration certificate 

• Instrument specifications 

• Experience or judgment 

5. Identify Uncertainty Sources 

For this tutorial, these are the identified sources for an analog pressure gauge: 

Type A 

• Repeatability 

Type B 

• Reference standard uncertainty 

• Reference standard drift 

• Reference standard resolution 

• UUC readability (analog) 

• Hysteresis 

6. Convert Each to Standard Uncertainty 

All values must be converted to standard uncertainty before combining. 

6.1 Reference Standard Uncertainty 

From certificate: 

u_ref = U / k 

Example: 

U = 0.0414 bar, k = 2 

u_ref = 0.0207 bar 

6.2 Drift 

Rectangular distribution: 

u_drift = a / √3 

Example: 

±0.034 bar 

u_drift = 0.0196 bar 

6.3 Resolution (Reference Standard) 

u_res = r / (2√3) 

Example: 

0.1 bar 

u_res = 0.0289 bar 

6.4 Readability (Analog Gauge) 

This is the most important part. 

Step 1: Define reading capability 

Example: 

• Smallest division = 0.5 bar – this is the resolution of the gauge

• The technician reads to 1/4 division, based on the agreed-upon capability to read 

So: 

a = 0.125 bar 

Step 2: Apply rectangular distribution 

u_readability = a / √3 

u = 0.0722 bar 

6.5 Hysteresis 

If negligible: 

u = 0 

Otherwise: 

u = (H/2) / √3 

7. Uncertainty Budget Table 

This is where we summarize everything.

Source	Type	Standard Uncertainty (bar)
Repeatability	A	0.0000
Reference Standard	B	0.0207
Drift	B	0.0196
Resolution	B	0.0289
Readability	B	0.0722
Hysteresis	B	0.0000

8. Combine Uncertainties 

Use Root Sum Square (RSS): 

u_c = √(Σu²) 

u_c = 0.0828 bar 

9. Expanded Uncertainty 

U = k × u_c 

U = 2 × 0.0828 = 0.17 bar 

10. Final Result 

>>>Result = x ± 0.17 bar (k = 2, 95% confidence) 

11. Key Learning Points 

• Analog gauges are limited by human reading 

• Always define the ± range (a) first 

• Then apply: 

u = a / √3 

• Do not blindly use formulas 

12. Common Mistakes 

• Treating analog like digital 

• Using √3 incorrectly 

• Forgetting to divide by k 

• Ignoring drift 

13. Practical Tips 

To reduce uncertainty: 

• Use finer scale gauges 

• Improve reading technique 

• Reduce the drift interval 

• Use a better reference standard 

14. Simple Step-by-Step Summary 

1. Define the measurement model 

2. Identify sources 

3. Classify Type A / Type B 

4. Convert to standard uncertainty 

5. Build budget 

6. Combine (RSS) 

7. Multiply by k = 2 

15. Conclusion 

Measurement uncertainty is not just a calculation — it is a model of reality. 

For analog pressure gauges, the most critical factor is how well the operator can read the scale. 

Once you understand that, the rest becomes systematic and repeatable. 

💡 Pro Tip

If your uncertainty is high, improving the reference standard may not help. Focus on improving readability (scale resolution or digital upgrade).

This article is part of our Pressure Calibration Hub
👉Pressure calibration

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📌 Frequently Asked Questions (FAQ)

1. What is measurement uncertainty in pressure gauge calibration?

Measurement uncertainty is the quantified doubt in a measurement result. In pressure gauge calibration, it defines the range within which the true pressure value is expected to lie, typically expressed as: $$\text{Result} = x \pm U$$Result=x±U

where $U$U is the expanded uncertainty at a defined confidence level (usually 95%, $k=2$k=2).

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2. How do you calculate uncertainty using the GUM method?

The GUM method follows these steps:

1. Define the measurement model
2. Identify all uncertainty sources
3. Classify them as Type A or Type B
4. Convert all to standard uncertainty
5. Combine using root sum square (RSS)
6. Multiply by coverage factor $k=2$k=2

This produces the expanded uncertainty.

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3. Why is readability important for analog pressure gauges?

Readability is critical because analog gauges rely on human interpretation of the pointer position. Unlike digital instruments, there is no rounding—only estimation.

This often becomes the largest contributor to uncertainty, especially when readings are taken at fractions of a scale division (e.g., 1/4 division).

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4. How do you calculate readability uncertainty for an analog gauge?

Readability uncertainty is calculated using a rectangular distribution:$$u = \frac{a}{\sqrt{3}}$$u=3​a​

Where:

• $a$a = estimated reading limit (e.g., 1/4 of the smallest division)

Example:
If smallest division = 0.5 bar and reading = 1/4 division:$$a = 0.125 \text{ bar}$$a=0.125 bar $$u = \frac{0.125}{\sqrt{3}} = 0.0722 \text{ bar}$$u=3​0.125​=0.0722 bar

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5. What is the difference between Type A and Type B uncertainty?

• Type A: evaluated using statistical analysis (e.g., repeatability from repeated measurements)
• Type B: evaluated using other information (e.g., calibration certificates, specifications, experience)

Both are converted into standard uncertainty and combined in the same way.

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6. Why do we divide by √3 in uncertainty calculations?

The factor √3 comes from the rectangular (uniform) distribution assumption, where all values within a range are equally likely.

The standard uncertainty is:$$u = \frac{a}{\sqrt{3}}$$u=3​a​

where $±a$±a is the assumed limit of error.

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7. What is expanded uncertainty and why use k = 2?

Expanded uncertainty is the final reported uncertainty after applying a coverage factor:$$U = k \cdot u_c$$U=k⋅uc​

Using $k = 2$k=2 corresponds to approximately 95% confidence, which is standard practice in calibration and ISO/IEC 17025.

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8. What is the biggest contributor to uncertainty in analog gauges?

In most cases, the largest contributor is:

👉 UUC readability (human estimation error)

This often dominates over:

• reference standard uncertainty
• drift
• resolution

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PRACTICAL IMPLEMENTATION

If you are performing in-house calibration:

You need:

• uncertainty calculations
• standard procedures
• datasheets

👉Check out this ready-made pressure gauge calibration procedure package, which consists of a procedure, measurement uncertainty calculator, a datasheet, and a calibration certificate in one package. visit this link >>buymeacoffee.com/edsponce/e/312258

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