Errors in Measurement

Why every measurement is uncertain

NCERT opens §1.3 with a single sentence: "Every measurement involves errors." That is not a confession of carelessness — it is a statement about the physics of measurement itself. Three independent sources guarantee that no reading is ever exact.

First, the instrument has a finite resolution. A metre scale cannot resolve below a millimetre. A vernier calliper of 20 divisions on the sliding scale resolves to $0.005$ cm. A screw gauge of pitch 1 mm and 100 circular divisions resolves to $0.001$ cm. Below the least count, the instrument is silent — the reading is rounded to the nearest division and the next digit is unknowable.

Second, the observer is imperfect. Parallax shifts the apparent position of a pointer on a scale. Stopwatch reflexes vary by 0.1–0.3 s. Two students reading the same screw gauge can disagree by one division because they estimate the in-between fraction differently.

Third, the environment is unsteady. Temperature changes expand or contract the scale itself. Voltage fluctuations shift the readings of any electrically driven meter. Stray air currents disturb a balance. A pendulum loses energy to air friction and its period drifts.

Together these three sources make the reported value a range, not a point. When NCERT writes the length of a sheet as $l = 16.2 \pm 0.1$ cm in Example 1 of §1.3.3, the $\pm 0.1$ is not optional — it is the part that tells you which digits to trust. The parent Units and Measurement chapter introduces this idea; here we systematise it.

Accuracy vs precision — the dartboard

Two words students treat as synonyms, but NEET expects you to distinguish them sharply.

Accuracy is how close a measured value is to the true (or accepted) value. A high-accuracy measurement has small systematic bias — on average, it lands on the bull's-eye.

Precision is how close repeated measurements are to one another, regardless of where they cluster. A high-precision measurement has small spread — every reading lands in the same spot, even if that spot is offset from the truth.

The dartboard analogy collapses the distinction onto a 2×2 grid. NCERT does not draw the dartboard explicitly, but the underlying logic is identical to the way physicists actually use the words.

The practical takeaway: a screw gauge of LC $0.001$ cm with a $-0.004$ cm zero error is precise but inaccurate. Correcting the zero error moves the cluster to the bull's-eye without changing its tightness. Averaging more readings tightens the cluster but cannot move a systematic offset.

Types of errors

NCERT's legacy §1.6 (carried in the post-rationalised chapter only as a brief mention) classifies errors into three families. NEET 2023's random-error question lifted the textbook definition verbatim.

Systematic errors — constant bias, fixed direction

Systematic errors come from a determinable cause and push every reading in the same direction. They can in principle be removed by correction. Four sub-types appear repeatedly in NEET stems:

• Instrumental error. The instrument itself is biased — a thermometer whose mercury column is detached, a metre scale whose end is worn, a screw gauge with a non-zero zero reading. NEET 2018 tested a screw gauge with zero error $-0.004$ cm; the correction adds $0.04$ mm to every raw reading.
• Imperfect experimental technique. Measuring the temperature of a human body by placing the thermometer under the armpit gives a value slightly lower than the true body temperature, because heat loss through the skin biases the reading downward.
• Personal error. The observer introduces a systematic bias — habitual parallax, anticipatory stop-watch clicks, consistent over-reading of vernier fractions. Personal error differs between observers but is constant for a given observer.
• Environmental error. A length measurement made on a hot day reads short relative to the same scale on a cool day because the scale itself expanded. Pressure, humidity and electromagnetic interference cause similar fixed biases.

Systematic errors are not reduced by averaging — every reading carries the same bias, so the mean carries it too. They are reduced by correction (subtract the zero error) or recalibration (replace the instrument).

Random errors — fluctuating, both directions

Random errors arise from causes that cannot be controlled or predicted: micro-fluctuations in temperature, pressure or voltage, observer reflex jitter, air currents, electronic noise. By definition they scatter readings symmetrically on either side of the true value. Their hallmark, captured in NEET 2023 Q.11, is that they cannot be associated with any systematic or constant cause.

The cure for random error is repetition. Take $n$ readings and average them — the mean's spread shrinks as $1/\sqrt{n}$. NCERT Exercise 1.8(c) makes the point sharply: 100 measurements of a rod's diameter yield a more reliable estimate than 5 measurements, precisely because random errors cancel on averaging.

Gross errors — mistakes

Gross errors are blunders: a misread digit, an entry copied to the wrong row, a unit forgotten ("3" instead of "3 cm"). They do not follow any statistical pattern. The only defence is care — re-read, re-record, redo the suspect reading. NCERT does not theorise gross errors; it expects you to discard them.

Least count error — the instrument's floor

The least count (LC) of an instrument is the smallest value it can measure. A 30-cm metre scale has LC = $1$ mm; a vernier calliper of 20 sliding-scale divisions has LC = $1$ MSD$/20 = 0.005$ cm; a screw gauge of pitch 1 mm and 100 circular divisions has LC = $0.01$ mm. The least-count error is the maximum uncertainty introduced by the instrument's finite resolution — typically taken as $\pm \tfrac{1}{2}$ LC for a well-read instrument, or $\pm 1$ LC for a conservative estimate. Below the least count, the instrument simply cannot speak.

Quantifying error from a set of readings

Suppose you take $n$ readings $a_1, a_2, \dots, a_n$ of the same quantity. NCERT prescribes a five-step procedure to extract a best value and a credible uncertainty.

1. Arithmetic mean is taken as the best estimate of the true value (because random errors cancel on averaging): $$a_\text{mean} = \frac{a_1 + a_2 + \dots + a_n}{n}$$
2. Absolute error of each reading is the magnitude of its deviation from the mean: $$\Delta a_i = |a_\text{mean} - a_i|$$
3. Mean absolute error is the average of the absolute errors: $$\Delta a_\text{mean} = \frac{\Delta a_1 + \Delta a_2 + \dots + \Delta a_n}{n}$$
4. Relative error (also called fractional error) is the dimensionless ratio: $$\text{relative error} = \frac{\Delta a_\text{mean}}{a_\text{mean}}$$
5. Percentage error is the relative error expressed per hundred: $$\text{percentage error} = \frac{\Delta a_\text{mean}}{a_\text{mean}} \times 100\%$$

The final reported result is then $a = a_\text{mean} \pm \Delta a_\text{mean}$, with the absolute error rounded to one significant figure and the mean rounded to the same decimal place.

Notice three discipline points. The mean was rounded to the same precision as the original readings — not to extra spurious digits. The absolute error was rounded to one significant figure. The reported value and its uncertainty share the same last decimal place — quoting $1.328 \pm 0.02$ would be self-contradictory.

Combination of errors

Most NEET questions do not ask the error in a single directly measured quantity. They ask the error in something derived from several measured quantities — density from mass and dimensions, gravity from pendulum period and length, resistance from voltage and current. The four rules below cover every case. NCERT records them through §1.3.3 Example; we derive each.

Rule 1 — Sum and difference: absolute errors add

If $Z = A + B$ or $Z = A - B$, the maximum possible error in $Z$ is the sum of the absolute errors:

Derivation (subtraction). The largest value of $Z$ is $(A + \Delta A) - (B - \Delta B) = (A - B) + (\Delta A + \Delta B)$. The smallest is $(A - \Delta A) - (B + \Delta B) = (A - B) - (\Delta A + \Delta B)$. So $Z$ spreads by $\pm(\Delta A + \Delta B)$. Errors never subtract — they always add in the worst case.

Rule 2 — Product and quotient: relative errors add

If $Z = AB$ or $Z = A/B$, the maximum relative error in $Z$ is the sum of the relative errors:

Derivation. Write $A = A_0(1 + \delta_A)$ where $\delta_A = \Delta A/A$. Then $AB = A_0 B_0 (1 + \delta_A)(1 + \delta_B) \approx A_0 B_0 (1 + \delta_A + \delta_B)$ to first order, since $\delta_A \delta_B$ is second-order small. The relative error in the product is therefore $\delta_A + \delta_B$. The same algebra with $1/(1 + \delta_B) \approx 1 - \delta_B$ gives the quotient case — and the relative errors still add, because worst case demands the sign of $\delta_B$ flip.

Rule 3 — Power: multiply the relative error by the absolute value of the exponent

If $Z = A^p$, the relative error gets multiplied by $|p|$:

This is just Rule 2 applied $p$ times: $A^p = A \cdot A \cdot \dots \cdot A$ ($p$ factors), each contributing $\Delta A/A$. For negative or fractional powers the same formula holds with the absolute value, because the worst-case analysis is sign-insensitive.

Rule 4 — General formula (the one NEET 2023 used)

Combining Rules 2 and 3, for $Z = A^p B^q / C^r$:

Every coefficient is positive. The exponents become multipliers; sign of the exponent is dropped. This single line is the most-tested NEET formula in the chapter.

The pendulum classic — error in $g$

The simple-pendulum problem is the canonical NEET application of Rule 4. The period of a simple pendulum is

Two design lessons fall out. To shrink $\Delta g$, sharpening the timer (a stopwatch reading to 0.01 s) pays double — both the exponent and the relative scale favour it. Lengthening the pendulum increases $T$ and dilutes $\Delta T/T$. NCERT Exercise 1.8(b) makes the related point that you cannot reduce screw-gauge error arbitrarily by adding more circular-scale divisions — beyond a point, random errors dominate the least-count limit.