Measurement errors and statistical analysis

Gross, systematic and random errors, limiting error at part scale, combination of errors, and the statistics (mean, deviation, standard deviation, probable error) used to treat repeated readings.

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Why it matters

No instrument reads the true value. Before you trust a reading — or a power, resistance or efficiency calculated from several readings — you must know how large the error can be and whether it is a fixed bias or random scatter. Specifying meters, writing test reports and every lab numerical in this subject starts here.

Key ideas

Error and correction. The static error is δA = Am − At (measured minus true). The static correction is its negative, δC = At − Am; add it to the reading to get the true value. Always state which sign convention you use.

Accuracy and precision are different things.

  • Accuracy is closeness of the reading to the true value.
  • Precision is closeness of repeated readings to each other (repeatability, and the number of significant figures an instrument can resolve).
  • A precise instrument with a zero offset is consistently wrong: precise but not accurate. Calibration removes the bias; it cannot remove scatter.

Classes of error

  • Gross errors — human blunders: misreading a scale, wrong range, wrong connection, arithmetic slips, a voltmeter loading a high-resistance circuit. Reduced by care and by taking at least three independent readings.
  • Systematic errors — same size and sign every time under the same conditions:
    • Instrumental: friction in bearings, weakened spring, ageing magnet, zero not set, wrong calibration.
    • Environmental: temperature, humidity, stray magnetic and electric fields, pressure.
    • Observational: parallax, an observer's habitual bias. Systematic errors are found by calibration against a better standard and removed by corrections, shielding, temperature compensation and good technique. Averaging does not remove them.
  • Random (residual) errors — small, unpredictable variations that remain after gross and systematic errors are removed (noise, small fluctuations in conditions). They are treated statistically; averaging n readings reduces their effect on the mean by a factor √n.

Limiting (guarantee) error. Manufacturers guarantee that the error does not exceed a stated percentage of full-scale deflection. The absolute limit is fixed, so the relative error grows as the reading falls. That is why you choose a range on which the reading is above roughly two-thirds of full scale.

Combination of errors. When a result is calculated from several measured quantities:

  • Sum or difference: the absolute limiting errors add.
  • Product or quotient: the relative (percentage) limiting errors add.
  • Power: the relative error is multiplied by the magnitude of the exponent.
  • The worst-case (limiting) value assumes all errors take their extreme values with the worst signs. If the errors are independent and random, the root-sum-square combination gives a more realistic (smaller) estimate.

Statistical treatment of random errors. With n readings x₁ … xₙ:

  • Arithmetic mean x̄ is the best estimate of the true value.
  • Deviation of each reading dᵢ = xᵢ − x̄; the algebraic sum of deviations is zero.
  • Average deviation D = Σ|dᵢ| / n indicates precision.
  • Standard deviation σ: use n in the denominator for a large population (n ≥ 20 or so) and n − 1 for a small sample. Variance V = σ².
  • For a Gaussian (normal) distribution about 68.3 % of readings lie within ±σ, 95.4 % within ±2σ and 99.7 % within ±3σ. The probable error r = 0.6745σ is the band containing half the readings.
  • The standard deviation of the mean is σ/√n: the mean is more reliable than any single reading.

Formulas

δA = Am − At ; εr = δA / At Am = measured value; At = true value (same unit); εr = relative error (dimensionless, ×100 for %).

δA = ±(εFS × Full-scale value) ; % error at reading = δA / Am × 100 εFS = guaranteed accuracy as a fraction of full scale. Valid anywhere on the scale.

X = x₁ ± x₂ → δX = ±(δx₁ + δx₂) X = x₁·x₂ or x₁/x₂ → δX/X = ±(δx₁/x₁ + δx₂/x₂) X = x₁ⁿ·x₂ᵐ → δX/X = ±(|n|·δx₁/x₁ + |m|·δx₂/x₂) Worst-case limiting error.

δX/X = √((δx₁/x₁)² + (δx₂/x₂)²) Root-sum-square estimate for independent random errors of a product or quotient.

x̄ = Σxᵢ / n ; D = Σ|xᵢ − x̄| / n σ = √(Σ(xᵢ − x̄)² / (n − 1)) (small sample) ; σ = √(Σ(xᵢ − x̄)² / n) (large population) V = σ² ; r = 0.6745·σ ; σm = σ / √n ; rm = r / √n All in the unit of the measured quantity (V in unit²). σm = standard deviation of the mean; rm = probable error of the mean.

Worked examples

Example 1 — limiting error at part scale. A 0–150 V voltmeter has a guaranteed accuracy of ±1 % of full scale. It reads 75 V. Find the limiting error in volts and as a percentage of the reading.

  1. δA = ±0.01 × 150 = ±1.5 V.
  2. % error = 1.5 / 75 × 100 = ±2 %.

Answer: ±1.5 V, i.e. ±2 % of the reading — twice the nameplate figure, because the reading is at half scale.

Example 2 — combination of errors (GATE level). Power is found from P = V·I. The 0–150 V voltmeter of Example 1 reads 75 V and a 0–10 A ammeter, also ±1 % of full scale, reads 5 A. Find P, its worst-case limiting error and the root-sum-square estimate.

  1. P = 75 × 5 = 375 W.
  2. Voltmeter: δV/V = 1.5/75 = 2 %. Ammeter: δI = 0.01 × 10 = 0.1 A, δI/I = 0.1/5 = 2 %.
  3. Worst case: δP/P = 2 % + 2 % = 4 % → δP = 0.04 × 375 = 15 W.
  4. RSS: δP/P = √(2² + 2²) = 2.83 % → δP = 10.6 W.

Answer: P = 375 W ± 15 W (±4 %) worst case; about ±10.6 W (±2.83 %) as an RSS estimate.

Example 3 — statistics of a set of readings. Ten readings of a frequency (Hz): 101.2, 101.4, 101.7, 101.3, 101.3, 101.2, 101.0, 101.3, 101.5, 101.1. Find the mean, average deviation, sample standard deviation, probable error of one reading and probable error of the mean.

  1. x̄ = 1013.0 / 10 = 101.3 Hz.
  2. Deviations: −0.1, 0.1, 0.4, 0, 0, −0.1, −0.3, 0, 0.2, −0.2. Σ|d| = 1.4 → D = 0.14 Hz.
  3. Σd² = 0.36 → σ = √(0.36/9) = 0.20 Hz.
  4. r = 0.6745 × 0.20 = 0.135 Hz.
  5. rm = 0.135 / √10 = 0.043 Hz.

Answer: x̄ = 101.3 Hz, D = 0.14 Hz, σ = 0.20 Hz, r = 0.135 Hz, rm = 0.043 Hz.

Common mistakes

  • Treating the nameplate "±1 %" as ±1 % of the reading. It is of full scale; at low readings the relative error is much larger.
  • Adding absolute errors for a product, or relative errors for a sum. Products and quotients add percentages; sums and differences add absolute values.
  • Subtracting errors for a quotient (R = V/I): limiting errors always add, whatever the operation.
  • Forgetting to multiply by the exponent, e.g. P = I²R doubles the relative error of I.
  • Believing averaging removes systematic error. It only reduces random error.
  • Mixing up n and n − 1 in σ; check which the question expects (small sample → n − 1).
  • Confusing probable error (0.6745σ) with standard deviation.

For GATE IN

Expect NAT questions on limiting error at a part-scale reading, the limiting error of a quantity computed from two or three meter readings (power, resistance by V/I, energy), and mean, standard deviation and probable error from a short data set. MCQs test the classification of errors and accuracy versus precision. Practise doing error combinations quickly in percentages and watch whether the question wants worst case or RSS.

Quick check

  1. A 0–10 A ammeter of ±2 % FS accuracy reads 2.5 A. What is the limiting error as a percentage of reading?
  2. R = V/I with V ±1 % and I ±1.5 %. What is the worst-case limiting error in R?
  3. P = I²R with I ±1 % and R ±0.5 %. Limiting error in P?
  4. Averaging 100 readings reduces the standard deviation of the mean by what factor?
  5. Which class of error is reduced by calibration? Answers: 1. ±0.2 A = ±8 %; 2. ±2.5 %; 3. ±2.5 %; 4. 10; 5. systematic error.

Interview questions

All Electrical and Electronic Measurements interview questions

Try answering each one aloud before you open it.

  1. 1.What is a measurement error in the context of electrical and electronic measurements?Concept

    A measurement error is the difference between the measured value and the true value of the quantity being measured. It can arise from various sources such as instrument inaccuracies, environmental conditions, or human error.

  2. 2.Explain the difference between systematic errors and random errors.Concept

    Systematic errors are consistent, repeatable errors associated with faulty equipment or flawed experiment design, leading to a bias in measurements. Random errors, on the other hand, are unpredictable and arise from unknown or unpredictable variations in the measurement process, affecting the precision of measurements.

  3. 3.How can statistical analysis be used to minimise the impact of random errors in measurements?Concept

    Take many readings and use their arithmetic mean as the best estimate of the true value, because random errors are equally likely to be positive or negative and tend to cancel. The spread is quantified by the standard deviation σ, and the standard deviation of the mean is σ/√n, so 100 readings make the mean ten times more reliable than a single reading. The probable error 0.6745σ gives the band that contains half the readings. Statistics only treats random error; a systematic bias stays in the mean and must be removed by calibration.

  4. 4.Why is calibration important in reducing measurement errors?Application

    Calibration is important because it ensures that instruments provide accurate measurements by comparing them with a standard reference. Regular calibration helps to identify and correct systematic errors, thereby improving the accuracy and reliability of measurements.

  5. 5.Explain how environmental factors can introduce errors in electrical measurements.Application

    Environmental factors such as temperature, humidity, and electromagnetic interference can introduce errors in electrical measurements. For example, temperature changes can affect the resistance of materials, leading to inaccurate readings, while electromagnetic interference can cause noise in electronic signals.

  6. 6.Why is it important to consider the uncertainty of a measurement?Application

    Considering the uncertainty of a measurement is important because it provides a range within which the true value is expected to lie. This helps in assessing the reliability and accuracy of the measurement, allowing for better decision-making and risk assessment in engineering applications.

  7. 7.A voltmeter is known to read 0.5 V high (static error +0.5 V). If it reads 12.5 V, what is the true voltage?Numerical

    Static error is defined as measured minus true value, δA = Am − At, so a +0.5 V error means the meter reads high. The static correction is −0.5 V, and the true voltage is 12.5 − 0.5 = 12.0 V. Interviewers check that you state the sign convention, because a correction has the opposite sign to the error.

  8. 8.An ammeter shows readings of 5.1 A, 5.0 A, 5.2 A and 5.1 A. Calculate the average current and the standard deviation.Numerical

    The mean is (5.1 + 5.0 + 5.2 + 5.1)/4 = 5.1 A. The deviations are 0, −0.1, 0.1 and 0 A, so the sum of squares is 0.02 A². Treating the four readings as a small sample, σ = √(0.02/3) = 0.082 A; using n in the denominator gives 0.071 A. For a small number of readings the n − 1 form is the correct one.

  9. 9.How is error propagation applied when a result is calculated from several measurements?Application

    For a sum or difference the absolute limiting errors add; for a product or quotient the relative (percentage) errors add; for a power the relative error is multiplied by the exponent, so P = I²R has twice the percentage error of I plus that of R. These worst-case rules give the guaranteed limit. If the errors are independent and random, the root-sum-square of the relative errors gives a more realistic estimate. Errors never subtract, even when the quantities are divided.

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