GATE Biomedical Engineering Measurements and Control Systems Syllabus

Define expanded uncertainty $U$ and the coverage factor $k$.

Expanded uncertainty defines an interval expected to contain a large fraction of the distribution: $$U = k\,u_{c}$$ The coverage factor $k$ (commonly $k=2$ for ~95% confidence) scales the combined standard uncertainty.

For a rectangular (uniform) distribution of half-width $a$, what is the standard uncertainty?

$$u = \frac{a}{\sqrt{3}}$$ This applies to Type B sources such as digital resolution or specified tolerance limits with no preferred value inside the range.

For a triangular distribution of half-width $a$, what is the standard uncertainty?

$$u = \frac{a}{\sqrt{6}}$$

What standard uncertainty corresponds to a quoted value of $a$ at a stated 95% confidence assuming a normal distribution?

Divide by the coverage factor $k \approx 2$: $$u = \frac{a}{2}$$ (more precisely $a/1.96$).

State the general law of propagation of uncertainty for $y = f(x_1, x_2, \dots, x_n)$ with independent variables.

$$u_{y}^{2} = \sum_{i=1}^{n}\left(\frac{\partial f}{\partial x_{i}}\right)^{2} u_{x_{i}}^{2}$$ The partial derivatives $\partial f/\partial x_i$ are the sensitivity coefficients.

For a sum or difference $y = x_1 \pm x_2$, how do the absolute uncertainties propagate (independent errors)?

Absolute uncertainties add in quadrature: $$u_{y} = \sqrt{u_{x_{1}}^{2} + u_{x_{2}}^{2}}$$ regardless of whether the operation is addition or subtraction.

For a product or quotient $y = x_1 \cdot x_2$ or $y = x_1/x_2$, how do uncertainties propagate (independent errors)?

Relative (fractional) uncertainties add in quadrature: $$\frac{u_{y}}{y} = \sqrt{\left(\frac{u_{x_{1}}}{x_{1}}\right)^{2} + \left(\frac{u_{x_{2}}}{x_{2}}\right)^{2}}$$

For a power law $y = x^{n}$, how does the relative uncertainty propagate?

$$\frac{u_{y}}{y} = |n|\,\frac{u_{x}}{x}$$ The relative uncertainty is multiplied by the magnitude of the exponent.

For $y = k x$ where $k$ is an exact constant, what is the uncertainty $u_y$?

$$u_{y} = |k|\,u_{x}$$ The absolute uncertainty scales by the constant; the relative uncertainty is unchanged.

How do you find the worst-case (maximum) absolute uncertainty for a sum $y = x_1 + x_2$?

In the worst case errors add directly (linear/arithmetic addition): $$u_{y,\max} = u_{x_{1}} + u_{x_{2}}$$ This is more conservative than quadrature addition.

For a general function $y=f(x_i)$, how is the worst-case absolute uncertainty estimated?

Sum the magnitudes of all sensitivity-weighted terms: $$u_{y,\max} = \sum_{i}\left|\frac{\partial f}{\partial x_{i}}\right| u_{x_{i}}$$

A resistance is found from $R = V/I$. If $V$ has 2% and $I$ has 3% relative uncertainty (independent), what is the relative uncertainty in $R$?

$$\frac{u_{R}}{R} = \sqrt{(0.02)^{2} + (0.03)^{2}} = \sqrt{0.0004 + 0.0009} = \sqrt{0.0013} \approx 0.036 = 3.6\%$$

Power dissipated is $P = I^{2}R$. Express the relative uncertainty in $P$ (independent errors).

$$\frac{u_{P}}{P} = \sqrt{\left(2\frac{u_{I}}{I}\right)^{2} + \left(\frac{u_{R}}{R}\right)^{2}}$$ The current contributes with a factor of 2 because of the squared dependence.

What is the sensitivity coefficient in uncertainty propagation?

The sensitivity coefficient is the partial derivative $c_i = \partial f/\partial x_i$, expressing how strongly the output $y$ responds to a small change in input $x_i$. It weights each input uncertainty in the propagation law.

Define the precision index $h$ (precision constant) of a Gaussian error distribution.

The precision index appears in the normal error law and is related to the standard deviation by $$h = \frac{1}{\sigma\sqrt{2}}$$ A larger $h$ means a sharper, narrower distribution and therefore higher precision.

Write the Gaussian (normal) probability density for measurement errors in terms of the precision index $h$.

$$y = \frac{h}{\sqrt{\pi}}\,e^{-h^{2}x^{2}}$$ where $x$ is the deviation (error) and larger $h$ gives a taller, narrower curve.

How does the precision index $h$ relate qualitatively to the spread of measurements?

Since $h = 1/(\sigma\sqrt{2})$, $h$ is inversely related to $\sigma$. High $h$ corresponds to small standard deviation (tight, precise data); low $h$ corresponds to large scatter.

Define the probable error of a single observation in terms of standard deviation.

$$r = 0.6745\,\sigma$$ It is the error value such that half the observations have error smaller and half larger (the 50% confidence limit) for a normal distribution.

Define the probable error of the mean.

$$r_{m} = 0.6745\,\frac{\sigma}{\sqrt{n}} = \frac{r}{\sqrt{n}}$$ It expresses the 50% confidence interval for the estimated mean.

Relate probable error to the precision index $h$.

$$r = \frac{0.4769}{h}$$ Equivalently, since $r = 0.6745\,\sigma$ and $h = 1/(\sigma\sqrt{2})$, a higher precision index gives a smaller probable error.

What fraction of normally distributed data falls within $\pm 1\sigma$, $\pm 2\sigma$, and $\pm 3\sigma$ of the mean?

Approximately 68.27% within $\pm 1\sigma$, 95.45% within $\pm 2\sigma$, and 99.73% within $\pm 3\sigma$ (the empirical 68–95–99.7 rule).

What is a gross error in measurement and how is it handled?

A gross error is a large mistake caused by the observer (misreading, wrong recording, incorrect instrument setup). It is not statistical; it is handled by careful procedure and by discarding identified erroneous readings (outliers).

List the three principal classifications of measurement errors.

1) Gross errors (human mistakes), 2) Systematic errors (instrumental, environmental, observational/loading), and 3) Random (residual) errors.

What are the main sub-types of systematic error?

Instrumental errors (inherent to the instrument, e.g. friction, calibration, wear), environmental errors (temperature, humidity, pressure, fields), and observational errors (e.g. parallax, loading effect).

What is a loading error in measurement?

A loading error is a systematic error in which the measuring instrument draws energy/current from the measured system, altering the very quantity being measured (e.g. a low-impedance voltmeter loading a circuit and reading low).

Define resolution of an instrument and relate it to quantization uncertainty.

Resolution is the smallest change in input that produces a detectable change in output. For a digital instrument with resolution $\Delta$, the quantization standard uncertainty (rectangular distribution) is $$u = \frac{\Delta}{2\sqrt{3}}$$

How do you compute the percentage error contribution of each term to the total uncertainty budget?

Each term's contribution is its variance share: $$\text{share}_{i} = \frac{c_{i}^{2}u_{x_{i}}^{2}}{u_{c}^{2}} \times 100\%$$ where $c_i$ is the sensitivity coefficient. Terms add to 100% of the combined variance.

Two lengths $L_1 = 50.0 \pm 0.2$ mm and $L_2 = 30.0 \pm 0.1$ mm are added. What is the result with its uncertainty (quadrature)?

Sum $= 80.0$ mm. Uncertainty: $$u = \sqrt{0.2^{2} + 0.1^{2}} = \sqrt{0.05} \approx 0.22\text{ mm}$$ Result: $80.0 \pm 0.2$ mm.

Why can systematic errors NOT be reduced by taking the average of many readings, unlike random errors?

Systematic errors bias every reading by the same amount in the same direction, so they persist in the mean. Random errors have zero mean and partially cancel on averaging, so the mean's random uncertainty shrinks as $\sigma/\sqrt{n}$.

For $y = \frac{x_1 x_2}{x_3}$ with independent inputs, write the relative combined uncertainty.

$$\frac{u_{y}}{y} = \sqrt{\left(\frac{u_{x_{1}}}{x_{1}}\right)^{2} + \left(\frac{u_{x_{2}}}{x_{2}}\right)^{2} + \left(\frac{u_{x_{3}}}{x_{3}}\right)^{2}}$$ All multiplicative/divisive factors contribute their relative uncertainties in quadrature.