Uncertainty Calculations of Pressure Calibration

Uncertainty means how Uncertain (or certain) we are.

No readings are complete without units and Uncertainty.

Types of Uncertainty

There are two types of Uncertainty

• Type A uncertainty
• Type B uncertainty

The Type A

The Type A evaluation of standard uncertainty is the method of evaluating the uncertainty by the statistical analysis of a series of observations.

Type A uncertainty is based on readings taken during calibration.

The Type B

 The Type B evaluation of standard uncertainty is the method of evaluating the uncertainty by means other than the statistical analysis of the repeat measurements.

Type B uncertainty is based on other factors than readings taken during calibration.

Uncertainty can be reported either in % value or absolute value.

% Value can be reported as % of reading (% rdg) or % of Full-Scale deflection (%FSD)

Uncertainty Components of Pressure Calibration to be considered but not limited to:

Type A

1. Repeatability
2. Reproducibility

Type B

1. Uncertainty due to Standard equipment
2. Accuracy of Standard equipment
3.  Resolution
4. Hysteresis
5. Zero Error

Uncertainty Calculations

For understanding Uncertainty Calculations, we will take an example of Pressure Gauge calibration

5 Point Calibration with 1 Up cycle and 1 Down cycle

Consider UUC pressure gauge of range 0 to 10 bar with resolution 0.1 bar calibrated with master pressure Gauge of range 0 to 40 bar with resolution 0.001 bar.

Uncertainty of master pressure Gauge is 0.01 bar with K=2. Accuracy is 0.1% rdg.

Cal Point	UUC Reading	Standard Reading (UP)	Standard Reading (DOWN)	Average
Bar	Bar	Bar	Bar	Bar
0	0.0	0.000	0.000	0.000
2	2.0	2.003	2.005	2.004
4	4.0	4.006	4.004	4.005
8	8.0	8.005	8.006	8.006
10	10.0	10.008	10.007	10.008

Type A Uncertainty

Type A uncertainty is the standard deviation of readings.

Where:

x:  reading

x̄: Mean (average)

n: no of readings i.e. 2 in the above case

Using excel in can be calculated using formula = STDEV ()

Uncertainty due to Standard Deviation (U_a)

(U_a)= S.D / sqrt (n)

Sqrt: square root

Cal Point	Type A	Ua
Bar	Bar	Bar
0	0	0
2	0.001414	0.001
4	0.001414	0.001
8	0.000707	0.0005
10	0.000707	0.0005

Type B Uncertainty

Uncertainty due to Standard equipment

Uncertainty of standard equipment is mention on the calibration certificate. Also, the K value is mention in the calibration certificate.

Uncertainty due to Standard equipment (U_b1) = Uncertainty of standard equipment/ k

Here, in our example Uncertainty of Standard is 0.01 bar and K value is 2.

As this is absolute value directly in Pressure units, it is the same for the whole range.

If Uncertainty is provided in % rdg, it will be different for different calibration points.

Cal Point	Uncertainty of Standard	Ub1
Bar	Bar	Bar
0	0.01	0.005
2	0.01	0.005
4	0.01	0.005
8	0.01	0.005
10	0.01	0.005

Uncertainty due to Accuracy of Standard Equipment

Accuracy is expressed in % rdg or % full scale or absolute readings direct in units

Uncertainty due to Accuracy of Standard equipment (U_b2)

U_b2 = Accuracy of standard equipment/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

In our example, accuracy is % rdg. Therefore accuracy will be different for every reading.

Cal Point	Accuracy of Standard	Accuracy of Standard	Ub2
Bar	Bar	Bar	Bar
0	0.1%*0	0	0
2	0.1%*2	0.002	0.001155
4	0.1%*4	0.004	0.002309
8	0.1%*8	0.008	0.004619
10	0.1%*10	0.01	0.005774

Uncertainty due to Resolution

If UUC is set according to calibration point, and Standard reading is changed, resolution of Standard is considered.

If the Standard Pressure gauge is set according to calibration point, and UUC readings are changed, the resolution of UUC is considered.

In or above example, UUC is set and Standard readings are changing,

Therefore, the Resolution of a standard pressure gauge is considered.

The resolution of the Standard pressure gauge is 0.001 bar

Uncertainty due to Resolution (U_b3) = ((Resolution of Changing pressure gauge/2)) / SQRT (3))

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal Point	Resolution	Ub3
Bar	Bar	Bar
0	0.001	0.000289
2	0.001	0.000289
4	0.001	0.000289
8	0.001	0.000289
10	0.001	0.000289

Uncertainty due to Hysteresis

Hysteresis is the difference between Down Cycle reading and UP cycle reading. 

For 2 up and 2 down cycles. Hysteresis is the maximum difference between Down Cycle reading and UP cycle reading. 

Uncertainty due to Hysteresis (U_b4) = (Down reading- up reading)/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal Point	Hysteresis	Ub4
Bar	Bar	Bar
0	0	0
2	0.002	0.001155
4	-0.002	-0.00115
8	0.001	0.000577
10	-0.001	-0.00058

Uncertainty due to Zero Error

The Reading is noted when the pressure of the instrument is completely released.

The Zero Error is calculated as follows:

Zero Error= l x_2,0 – x_1,0 l

I.e. Zero readings of down cycle – Zero readings of up cycle

For Multiple cycles. The maximum value is Zero error.

Uncertainty due to Zero Error (U_b5) = Zero error/ Sqrt (3)

Assuming rectangular distribution, the value obtained is divided by a square root of 3

Cal Point	Zero Error	Ub5
Bar	Bar	Bar
0	0.000	0.000
2	0.000	0.000
4	0.000	0.000
8	0.000	0.000
10	0.000	0.000

Combined Uncertainty

Combined uncertainty is not simple addition.

Step 1: All individual uncertainty U_a, U_b1, U_b2, U_b3, U_b4, U_b5  are squared.

Step 2: All individual uncertainty is added after squaring

Step 3:  Square root of the value obtained in Step 2

Cal Point	Ua	Ub1	Ub2	Ub3	Ub4	Ub5	Combine Uncertainty
Bar	Bar	Bar	Bar	Bar	Bar	Bar	Bar
0	0	0.005	0	0.000289	0	0.000	0.005
2	0.001	0.005	0.001155	0.000289	0.001155	0.000	0.005
4	0.001	0.005	0.002309	0.000289	-0.00115	0.000	0.006
8	0.0005	0.005	0.004619	0.000289	0.000577	0.000	0.007
10	0.0005	0.005	0.005774	0.000289	-0.00058	0.000	0.008

Combined Uncertainty is also called Standard Uncertainty

Cal Point	Combine Uncertainty	Combine Uncertainty
Bar	Bar	Bar
0	Sqrt (02+0.0052+02+0.002892+02+02)	0.005
2	Sqrt (0.0012 +0.0052  + 0.0011552  +0.0002892 +   0.0011552 +02)	0.005
4	Sqrt (0.0012 +0.0052  + 0.0023092  +0.0002892 +   0.0011552 +02 )	0.006
8	Sqrt (0.00052 +0.0052  + 0.0046192  +0.0002892 +   0.0005772 +02)	0.007
10	Sqrt (0.00052 +0.0052  + 0.0057742  +0.0002892 +   0.000582 +02)	0.008

Expanded Uncertainty

Standard Uncertainty is multiplied by the k factor to obtained Expanded Uncertainty.

Where k is the coverage factor corresponding to the effective degree of freedom.

k factor is generally 2.

Cal Point	Expanded Uncertainty
Bar	Bar
0	0.010
2	0.011
4	0.011
8	0.014
10	0.015

Expanded uncertainty is reported in the calibration certificate.

A similar type of calculation is done for 9 Point calibration with 2 up and 2 down cycles.

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Read Next:

• Calibration Organizations
• Selection of Master Instrument
• Importance of Unit Measurement
• Why Calibration is Important?
• Repeatability and Reproducibility