Formula sheet

Engineering Measurements and Uncertainty Formula Sheet

Engineering physics formula sheet for measurement chains, calibration models, sensitivity, signal-to-noise ratio, sampling, quantization, bandwidth, first-order response, uncertainty budgets, guard bands, and decision limits.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects measurement-chain relationships used to convert sensor outputs into defensible engineering evidence. It focuses on calibration, signal scaling, noise, sampling, dynamic response, uncertainty budgets, guard bands, and decision limits. It does not replace device-specific physics, sensor datasheets, calibration standards, or validation in the installed system.

A measurement formula is useful only when the measurand, range, installation, environment, signal conditioning, sampling, calibration model, uncertainty contributors, and decision rule are stated.

Notation

Symbol	Meaning	Typical unit
$x$	measurand or input quantity	varies
$y$	sensor or instrument output	V, count, A, or display unit
$\hat{x}$	estimated measurand after calibration	varies
$b$	calibration intercept or offset	output unit
$m$	calibration slope	output unit per input unit
$S$	sensitivity	output unit per input unit
$R$	displayed or ADC resolution	varies
$N$	ADC bit count or sample count depending on context	dimensionless
$FS$	full-scale range	varies
$f_s$	sampling frequency	Hz
$f_c$	cutoff frequency	Hz
$\tau$	first-order time constant	s
$u_i$	standard uncertainty component	varies
$u_c$	combined standard uncertainty	varies
$U$	expanded uncertainty	varies
$k$	coverage factor	dimensionless
$L_L$ , $L_U$	lower and upper decision limits	varies

Linear Calibration Model

Linear calibration relation:

y=b+mx

Estimated measurand from output:

\displaystyle \hat{x}=\frac{y-b}{m}

Calibration residual at point $i$ :

r_i=y_i-(b+mx_i)

Root-mean-square calibration residual:

\displaystyle r_{RMS}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}r_i^2}

Sensitivity:

\displaystyle S=\frac{\Delta y}{\Delta x}

For a nonlinear calibration:

\displaystyle S=\frac{dy}{dx}

Use

A calibration curve is a model. It is valid only over the range, configuration, environment, installation, reference standard, and direction of approach used during calibration. Extrapolation is usually weak release evidence.

Offset, Gain Error, and Percent Error

Absolute error:

e=x_m-x_{ref}

Relative error:

\displaystyle e_r=\frac{x_m-x_{ref}}{x_{ref}}

Percent error:

e_\%=100e_r

Gain error for an indicated slope $m_i$ against reference slope $m_r$ :

\displaystyle e_g=\frac{m_i-m_r}{m_r}

Offset error:

e_b=b_i-b_r

Use

Offset and gain errors affect different decisions. Offset can dominate near zero. Gain error grows with measured value. A two-point check can miss nonlinearity, hysteresis, drift, and installation effects.

Resolution and Quantization

ADC code count:

C=2^N

Ideal ADC step:

\displaystyle q=\frac{FS}{2^N}

Quantization standard uncertainty for uniform rounding:

\displaystyle u_q=\frac{q}{\sqrt{12}}

Resolution-to-tolerance screen:

\displaystyle R\leq\frac{T}{10}

where $T$ is the tolerance or decision band.

Use

Resolution is not accuracy. A 16-bit converter can still support a poor measurement if the reference, amplifier, installation, noise, grounding, temperature, or calibration model is weak.

Signal-to-Noise Ratio and Averaging

Signal-to-noise ratio by amplitude:

\displaystyle SNR_{dB}=20\log_{10}\left(\frac{A_s}{A_n}\right)

Signal-to-noise ratio by power:

\displaystyle SNR_{dB}=10\log_{10}\left(\frac{P_s}{P_n}\right)

Sample mean:

\displaystyle \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i

Standard error of the mean for independent random samples:

\displaystyle u_{\bar{x}}=\frac{s}{\sqrt{n}}

Use

Averaging reduces independent random noise. It does not remove bias, drift, aliasing, saturation, thermal gradients, calibration error, or a wrong measurand. Do not average invalid data into credibility.

Sampling and Aliasing

Ideal minimum sampling condition:

f_s>2f_{max}

Nyquist frequency:

\displaystyle f_N=\frac{f_s}{2}

A simple alias frequency relation is:

f_{alias}=|f-nf_s|

where $n$ is the integer that brings the apparent frequency into the recorded band.

Samples collected over a record length $T_r$ :

N_s=f_sT_r

Frequency resolution of a simple record:

\displaystyle \Delta f=\frac{1}{T_r}

Use

The sampling theorem assumes band-limited signals and ideal reconstruction. Real measurement systems need anti-alias filtering, timing control, bandwidth margin, and enough record length for the statistic or event being reported.

Filtering and First-Order Dynamic Response

First-order low-pass cutoff:

\displaystyle f_c=\frac{1}{2\pi\tau}

First-order step response:

y(t)=y_\infty+(y_0-y_\infty)e^{-t/\tau}

Time to about 95 percent of a final value:

t_{95}\approx3\tau

Approximate 10 to 90 percent rise time:

t_r\approx2.2\tau

Single-pole low-pass magnitude:

\displaystyle |H(f)|=\frac{1}{\sqrt{1+(f/f_c)^2}}

Use

A filter can improve noise but damage evidence. It can delay alarms, reduce peak values, hide transients, or distort phase. Dynamic response must match the decision: trip, trend, integrated dose, vibration spectrum, thermal soak, or control feedback.

Uncertainty Components

Type A standard uncertainty from repeated readings:

\displaystyle u_A=\frac{s}{\sqrt{n}}

Rectangular Type B uncertainty from a bound $\pm a$ :

\displaystyle u_B=\frac{a}{\sqrt{3}}

Triangular Type B uncertainty from a bound $\pm a$ :

\displaystyle u_B=\frac{a}{\sqrt{6}}

Combined standard uncertainty for independent contributors:

u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}

Expanded uncertainty:

U=ku_c

Use

All components must be expressed as standard uncertainties before combining. Keep units consistent. Do not combine a data-sheet maximum error, a standard deviation, and an expanded uncertainty as if they were the same kind of quantity.

Uncertainty Propagation

For:

z=f(x_1,x_2,\ldots,x_n)

linearized propagation for independent inputs is:

\displaystyle u_z=\sqrt{\left(\frac{\partial f}{\partial x_1}u_1\right)^2+\left(\frac{\partial f}{\partial x_2}u_2\right)^2+\cdots+\left(\frac{\partial f}{\partial x_n}u_n\right)^2}

For multiplication or division:

\displaystyle \left(\frac{u_z}{z}\right)^2\approx\left(\frac{u_a}{a}\right)^2+\left(\frac{u_b}{b}\right)^2+\cdots

Use

Linearized propagation is a local approximation. Use care for nonlinear transformations, thresholds, ratios near zero, correlated inputs, clipping, and one-sided physical limits. Monte Carlo simulation can be more appropriate when the model is nonlinear or distributions are not simple.

Guard Bands and Decision Limits

Conservative lower-limit acceptance:

x_m-U\geq L_L

Conservative upper-limit acceptance:

x_m+U\leq L_U

Guarded margin above a lower limit:

M_L=x_m-U-L_L

Guarded margin below an upper limit:

M_U=L_U-(x_m+U)

Use

Guard bands convert measurement uncertainty into a decision rule. They do not prove the true value. They state whether the available evidence is strong enough to accept under the chosen risk posture.

Drift and Calibration Interval Screening

Linear drift estimate:

\displaystyle D=\frac{x_2-x_1}{\Delta t}

Projected drift over a future interval:

\Delta x_D=D\Delta t_f

Guarded future value for a capacity-like measurement:

x_{g,f}=x_m-U-|\Delta x_D|

Use

Drift should be based on instrument history, environment, aging, radiation, contamination, mechanical shock, thermal cycling, or reference checks. A calendar interval alone is weak if the installation is harsh or the decision is critical.

Worked Example: Guarded Pressure Measurement Release

A pressure transducer is calibrated and used to verify a hydraulic preload. The acceptance rule is:

p\geq6.45\ \text{bar}

The linear calibration model is:

y=b+mp

where:

b=0.011\ \text{V}

and:

m=0.502\ \text{V/bar}

The measured output during release is:

y=3.278\ \text{V}

The acquisition system is 16-bit over a 0 to 5 V span. Standard uncertainty contributors expressed in pressure units are:

Contributor	Standard uncertainty
reference standard	$0.012\ \text{bar}$
calibration fit residual	$0.018\ \text{bar}$
short-term repeatability	$0.010\ \text{bar}$
temperature effect after compensation	$0.020\ \text{bar}$

Step 1: Convert Voltage to Pressure

Use:

\displaystyle \hat{p}=\frac{y-b}{m}

Substitute:

\displaystyle \hat{p}=\frac{3.278-0.011}{0.502}=6.508\ \text{bar}

Step 2: Check ADC Quantization

ADC step in voltage:

\displaystyle q_V=\frac{5}{2^{16}}=7.63\times10^{-5}\ \text{V}

Convert this step to pressure:

\displaystyle q_p=\frac{q_V}{m}=\frac{7.63\times10^{-5}}{0.502}=1.52\times10^{-4}\ \text{bar}

Quantization standard uncertainty:

\displaystyle u_q=\frac{q_p}{\sqrt{12}}=4.39\times10^{-5}\ \text{bar}

Engineering comment: ADC quantization is negligible compared with reference, calibration, repeatability, and temperature uncertainty. More bits would not materially improve this decision.

Step 3: Combine Standard Uncertainty

Combined standard uncertainty:

u_c=\sqrt{0.012^2+0.018^2+0.010^2+0.020^2+(4.39\times10^{-5})^2}

u_c=0.031\ \text{bar}

With coverage factor:

k=2

expanded uncertainty is:

U=2u_c=0.062\ \text{bar}

Step 4: Apply the Guard Band

For a lower-limit acceptance:

p_g=\hat{p}-U

p_g=6.508-0.062=6.446\ \text{bar}

Compare with the lower limit:

6.446<6.45

The measured value is slightly above the limit, but the guarded value is below the limit.

Decision

The measurement does not support guarded acceptance. The engineer should not round the result to “6.51 bar” and release it. The appropriate response is to hold the release decision, improve the dominant uncertainty contributors if practical, repeat the measurement under controlled temperature, or use a measurement chain with lower calibration and installation uncertainty.

Engineering Comment

This example shows why measurement decisions are not only about displayed digits. The ADC has excellent resolution, but the release decision is controlled by calibration, repeatability, and temperature effects. The evidence is close enough to the limit that uncertainty changes the decision.

Common Mistakes

Common mistakes include treating resolution as accuracy, using calibration outside its range, ignoring installation error, combining uncertainty components with inconsistent definitions, sampling too slowly, filtering away the event of interest, and accepting a marginal value without guard bands.

Other errors include improving ADC resolution while the reference standard dominates, averaging biased data, ignoring sensor drift, failing to document software scaling, and using a bench calibration to justify an installed measurement in a different thermal, vibration, optical, pressure, or electromagnetic environment.

REF

Disciplines