Error Analysis MCQ Quiz - Objective Question with Answer for Error Analysis - Download Free PDF

The correct answer is: 2) The closeness of a measured value to the true value

Explanation:

Accuracy in measurement refers to how close a measured value is to the true or accepted reference value. It quantifies the absence of systematic errors (bias).

Key Properties of Accuracy:

• True Value Proximity: High accuracy means minimal deviation from the actual value.
• Independent of Precision: A measurement can be accurate but not precise (or vice versa).

Why Not the Other Options?

1. Variation due to environmental factors → Describes precision (repeatability) or sensitivity to disturbances, not accuracy.

2. Ability to repeat the same measurement → Defines precision, not accuracy.

3. Smallest detectable change → Refers to resolution, not accuracy.

The type of error that occurs due to unpredictable variations in measurement conditions is: 2) Random error

Explanation:
Random errors arise from uncontrollable fluctuations in measurement conditions (e.g., noise, temperature changes, observer inconsistencies).

They vary unpredictably in magnitude and direction but can be reduced by averaging multiple readings.

Unlike systematic errors, they cannot be corrected by calibration.

Other Error Types (for comparison):

• Gross error → Caused by human mistakes (e.g., misreading an instrument).
• Calibration error → A type of systematic error due to incorrect instrument calibration.
• Systematic error → Consistent, repeatable deviations (e.g., faulty equipment, bias).

Concept

The accuracy of voltmeter is given by:

% Accuracy = \({V_{mean}-V_{true}\over V_{true}}\times 100\)

\(V_{mean}={V_1+V_2.........V_N\over N}\)

Precision is based on the deviation of readings from the mean and is given by:

\(Precision={Max \space Deviation \space from \space Mean\over V_{mean}}\)

Calculation

Given readings are 204 V, 205 V, 203 V, 205 V and 203 V

\(V_{mean}={204+205+203+205+203\over 5}=204\space V\)

% Accuracy = \({204-200\over 200}\times 100\)

% Accuracy = 2.5%

Mean = 204 V 

Maximum deviation from mean = 204 - 203 = 1V

\(Precision={1\over 204}\times 100\)

Precision = 0.5%

Explanation:

A thermometer consistently reads 2°C higher than the actual temperature. This thermometer is precise but not accurate.

Understanding Precision and Accuracy:

To understand why the given thermometer is "precise but not accurate," we need to define precision and accuracy in the context of measurements.

Precision: Precision refers to the consistency and repeatability of measurements. A precise measuring instrument will give very close or the same readings when used multiple times under the same conditions. Precision is about the instrument's ability to produce the same results consistently.

Accuracy: Accuracy refers to how close a measurement is to the true or actual value. An accurate measuring instrument will give readings that are very close to the true value. Accuracy is about the correctness of the measurements.

Analyzing the Thermometer's Performance:

The thermometer in question consistently reads 2°C higher than the actual temperature. Let's break down what this means:

1. Consistency: The fact that the thermometer consistently reads 2°C higher means that it is giving repeatable and consistent results. Every time the thermometer is used, it shows a value that is always 2°C above the actual temperature. This indicates that the thermometer is precise because it shows the same deviation from the actual temperature each time.
2. Deviation from True Value: Although the thermometer is consistent, its readings are not correct. Since it consistently shows a temperature that is 2°C higher than the actual temperature, it is not providing the true temperature value. This indicates that the thermometer is not accurate.

Therefore, the thermometer is precise (because it consistently shows the same deviation) but not accurate (because the readings are not close to the true value).

The relative error er is a measure of how significant the absolute error is compared to the true value of a quantity. It provides a way to express the error as a fraction or percentage of the true value, giving a sense of the magnitude of the error relative to the size of the measurement.

CONCEPT:

• Error: The result of every measurement of experiments by any measuring instrument contains some uncertainty. This uncertainty is called error. 
• Absolute error: The magnitude of the difference between the measurement of the experimental readings and the true value of the quantity is called the absolute error.
• Final absolute error: The arithmetic mean of all the absolute errors (for different measurements, different errors will be there) is known as the final absolute error or absolute mean error.
• Relative error: the ratio of the mean absolute error (or final absolute error) to the mean value of the experimental value measured.
   

EXPLANATION:

• We measure the absolute error for every measured experimental value. 
• Then we take the mean of all these absolute errors. This final value is known as the final absolute error. (Δamean)
• Also, we calculate the mean of all the measured experimental values. (amean)
• The relative error is the ratio of the mean absolute error (or final absolute error) Δamean to the mean value amean of the experimental value measured.
   

\(Relative \;error = {Δa_{mean} \over a_{mean}}\)

Concept:

Static error (E) = Am – At

Am = Measured value

At = True value

Relative Static Error: The ratio of absolute error to the true value is called relative static error.

\(R.S.E = \frac{{\left| {{A_m} - {A_t}} \right|}}{{{A_t}}} \times 100\)

Limiting Error:

The maximum allowable error in the measurement is specified in terms of true value, is known as limiting error. It will give a range of errors. It is always with respect to true value, so it is a variable error.

Guaranteed Accuracy Error:

The allowable error in measurement is specified in terms of full-scale value is known as a guaranteed accuracy error. It is a variable error seen by the instrument since it is with respect to full-scale value.

Application:

Given-

A_m = 125 V, A_t = 125.5 V

∴ Static error (E) = 125 - 125.5

E = 0.5 V

Absolute error (ε):

The difference between the indicated or measured value and the true or actual value is called absolute error. Also known as a static error.

Absolute error (ε) = Am - At

Where

Am =  measured or indicated value

At = true or actual value

Gross error:

• Gross errors are the observational errors that happen due to the lack of observation of the observer.
• These errors vary from observer to observer.
• The gross errors may also occur due to improper selection of the instrument.

Relative error:

The relative error is the absolute error over the true or actual value.

Relative static error R.S.E = ε / At = \(\frac{{{{\bf{A}}_{\bf{m}}} - {{\bf{A}}_{\bf{t}}}}}{{{{\bf{A}}_{\bf{t}}}}}\)

% R.S.E = \(\frac{{{{\bf{A}}_{\bf{m}}} - {{\bf{A}}_{\bf{t}}}}}{{{{\bf{A}}_{\bf{t}}}}}\) × 100

Probable error is a quantity formerly used as a measure of variability which is equal to 0.6745 times the standard deviation.

Accuracy: It is the degree of closeness with which the reading approaches the true value of the quantity to be measured.

Precision: It is the measure of reproducibility i.e., given a fixed value of a quantity, precision is a measure of the degree of agreement within a group of measurements.

• The precision of an instrument does not guarantee accuracy
• An instrument with more significant figures has more precision
• Deflection factor is reciprocal of sensitivity

Resolution: The smallest change in output to the change in input is known as resolution. Resolution is the smallest measurable input change.

Sensitivity: It is defined as the ratio of the changes in the output of an instrument to a change in the value of the quantity being measured. It denotes the smallest change in the measured variable to which the instrument responds.

Deflection factor or inverse sensitivity is the reciprocal of sensitivity.

Concept:

Span: It is defined as the difference between the largest and smallest reading of the instrument.

If voltmeter scale is -V1 to V2, then the span is given by (V2 + V1)

Explanation:

The voltmeter scale is -5 V to 5 V

If voltmeter scale is -V1 to V2, then the span is given by (V2 + V1)

Now, span = 5 + 5 = 10 V

An error can be classified into three types: Gross error, Systematic error and random error.

1. Gross error: this class of error mainly cover a human mistake in reading instruments and calculating measurement results.
2. Systematic error: The systematic error can be classified into three type’s Instrumental error environmental error and observation error.

Instrumental error:

• Error due to improper zero adjustment
• Due to inherent shortcomings of the instruments
• Due to the misuse of the instruments
• Due to the loading effect

Environmental error: These errors are due to environmental factors like change in humidity, temperature and variation in pressure.

Observation error: These errors are induced only by the observer and most common error is parallax error. These parallax errors are introduced while reading a meter scale.

3. Random error: These errors are due to small factors which changes very often from instrument to the other instrument. These errors are also due to unknown cases which are also called residual error

Reproducibility: It is the degree of closeness with which given value may be repeatedly measured. It may be specified in terms of units for a given period of time.

Perfect reproducibility means that the instrument has no drift.

Given that,

Measured value = 25.34 W

Absolute error = - 0.11 W

Absolute error = Measured value – true value

⇒ -0.11 = 25.34 – true value

Gauge factor (GF) or strain factor of a strain gauge is the ratio of relative change in electrical resistance (R) to the mechanical strain (ε).

\(GF = \frac{{\Delta R/R}}{{\Delta L/L}}\)

\(GF= \frac{{\Delta R/R}}{{{\rm{\varepsilon \;}}}}\)

Where,

ε = Strain = \(\Delta L\) /L

ΔL= Absolute change in length

L = Original length

ΔR = Change in strain gauge resistance due to axial strain and lateral strain

R = Unstrained resistance of strain gauge

Calculation:

Given-

GF = 2, R = 50 Ω, ε = 0.001

Now change in resistance of an electrical strain gauge can be calculated as

ΔR = 2 x 50 x 0.001

ΔR = 0.1 Ω 

We can classify the instruments into two types

Absolute Instruments:

• These instruments give the magnitude of the quantity under measurements in terms of physical constants of the instrument
• There is no necessity of calibrating or comparing with other instruments
• Tangent Galvanometer and Rayleigh’s current balance are examples of this class

Secondary Instruments:

• These instruments are so constructed that the quantity being measured can only be measured by observing the output indicated by the instrument i.e. deflection of the instrument
• These instruments are calibrated by comparison with an absolute instrument or any other secondary instrument which has already been calibrated against an absolute instrument
• A voltmeter, a glass thermometer, and a pressure gauge are typical examples of secondary instruments

Working with absolute instruments for routine work is time-consuming. Therefore, secondary instruments are most commonly used. Absolute instruments are seldom used except in standard institutions and laboratories while secondary instruments find usage almost in every sphere of measurement.

>Integrating Instruments: These instruments record the consumption of the total quantity of electricity, energy, etc. during a particular period of time. These instruments give reading for a specific period of time but no indication of reading for a particular instant of time.

Example: Ampere-hour meter, Energy meter, kilovolt ampere-hour meter.

Indicating Instruments: These indicate the quantity being measured by means of a pointer that moves on a scale. These instruments indicate the instantaneous value of the electrical quantity being measured at the time at which it is being measured.

>Example: Ammeter, Voltmeter, Wattmeter, Speedometer

Recording Instruments: These instruments record continuously the variation of any electrical quantity with respect to time. In principle, these are indicating instruments but so arranged that a permanent continuous record of the indication is made on a chart or dial.

>Any electrical quantity like current, voltage can be recorded by a suitable recording mechanism.

>Example: A potentiometric type of recorder used for monitoring temperature records the instantaneous temperatures on a strip chart recorder.