Time for 20 oscillations of a pendulum is measured as $t_1 = 39.6$ s; $t_2 = 39.9$ s; $t_3 = 39.5$ s. What is the precision in the measurements? What is the accuracy of the measurement?

Time for 20 oscillations of a pendulum is measured as t₁ = 39.6 s; t₂ = 39.9 s; t₃ = 39.5 s. What is the preci

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t_1 = 39.6

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Concept understanding — Least Count Precision

The Problem: How Fine Can You Measure?

Imagine you have a standard 15 cm ruler. You look at the markings — there's a line for every centimetre, and between those, smaller lines for every millimetre. Now try to measure the thickness of a single sheet of paper. You place it against the ruler. Does it line up exactly with a millimetre mark? Almost certainly not. It falls somewhere between two millimetre lines.

What do you report? You can't say "2.3 mm" because your ruler doesn't show tenths of a millimetre. The best you can honestly say is "about 2 mm" or "between 2 and 3 mm". That limitation — the smallest change in the quantity that your instrument can reliably detect and display — is its least count.

Note

Least count is not about how good you are at estimating. It is a property of the instrument itself. Even a perfect observer cannot extract more detail than the instrument's scale allows.

The Precise Definition

Least count is the smallest value of a physical quantity that can be measured accurately using a given instrument. It is the resolution of the measuring device.

For a simple scale (like a ruler, voltmeter, or thermometer) with equally spaced markings, the least count is:

\text{Least Count} = \frac{\text{Value of one main scale division}}{\text{Number of divisions on the vernier (or sub-scale)}}

For instruments without a vernier (like a plain ruler), the least count is simply the value of the smallest division on the scale.

Examples you already know:

Instrument	Smallest Division	Least Count
Metre ruler	1 mm	1 mm
Stopwatch (digital)	0.01 s	0.01 s
Laboratory thermometer	1 °C	1 °C
Screw gauge	0.01 mm	0.01 mm

Why This Matters in Exams

When you record a measurement, you must write it to the correct number of decimal places — the least count determines that. If a ruler has a least count of 1 mm, you cannot report a length as 12.35 cm. The correct recording is 12.3 cm or 12.4 cm (the last digit is uncertain, but it must be at the place of the least count).

Watch out

A common mistake: writing 2.50 cm when your ruler only shows millimetres. That implies you measured to 0.01 cm (0.1 mm), which you didn't. Write 2.5 cm — the last digit is at the tenths place, matching your least count of 0.1 cm.

The Core Intuition …

Why this formula?

Least Count & Precision: Why the Formula Holds

Let's build this from first principles — understanding why the least count formula works, not just memorizing it.

1. What is Least Count?

Least Count (LC) is the smallest measurement an instrument can reliably indicate.

Think of a ruler with 1 cm marks but no mm marks — you can't measure 0.5 cm precisely. The least count is 1 cm.

2. The Core Formula

For a scale-type instrument (ruler, vernier caliper, micrometer):

\text{Least Count} = \frac{\text{Value of 1 main scale division}}{\text{Number of divisions on the vernier/circular scale}}

Why this formula?

Reasoning step-by-step:

The main scale has fixed divisions (e.g., 1 mm each).
The vernier scale has $N$ divisions that exactly span $(N-1)$ main scale divisions.
So, 1 vernier division = $\frac{N-1}{N}$ main scale divisions.

The difference between 1 main scale division and 1 vernier division is:

\text{LC} = 1 \text{ MSD} - 1 \text{ VSD} = 1 - \frac{N-1}{N} = \frac{1}{N} \text{ MSD}

That's exactly the formula above.

3. Example: Vernier Caliper

Main scale: 1 mm per division
Vernier scale: 10 divisions covering 9 mm

Then:

\text{LC} = \frac{1 \text{ mm}}{10} = 0.1 \text{ mm}

Why 0.1 mm? Because the 10th vernier mark aligns with the 9th main scale mark — the smallest shift you can detect is 0.1 mm.

4. For Circular Scales (Micrometer Screw Gauge)

Same logic, different geometry:

\text{LC} = \frac{\text{Pitch}}{\text{Number of circular scale divisions}}

Pitch = distance moved by spindle in one full rotation.

Why? One full rotation moves the spindle by the pitch. If the circular scale has $N$ divisions, each division corresponds to $\frac{\text{pitch}}{N}$ linear movement.

5. Precision vs. Least Count

Precision is half the least count (or sometimes $\pm$ LC/2).

\text{Precision} = \pm \frac{\text{LC}}{2}

Why half? …

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The precision of these readings is set by the stopwatch's least count, $\pm 0.1\ \text{s}$ . Since no independently known true value of the period is given, the best available estimate of accuracy is the mean deviation of the three readings from their own mean, which works out to $\approx \pm 0.2\ \text{s}$ .

Precision vs. accuracy

Precision tells you how finely an instrument can resolve a measurement — it is fixed by the instrument's least count, independent of whether the reading is close to the truth. Accuracy tells you how close a measured value is to the true (accepted) value — and that requires knowing the true value to compare against. When no independently known true value is available, the mean of several readings is taken as the best available estimate, and the spread of the individual readings about that mean gives a practical estimate of the accuracy.

The data

t_1 = 39.6\ \text{s}, \quad t_2 = 39.9\ \text{s}, \quad t_3 = 39.5\ \text{s}

Each reading is recorded to one decimal place, so the stopwatch's least count is $0.1\ \text{s}$ .

Precision

Since every reading is limited by the same least count, the precision of the measurement is $\pm 0.1\ \text{s}$ — that is the finest interval the instrument can distinguish, regardless of how the three readings happen to scatter.

The mean of the three readings is:

\bar{t} = \frac{39.6 + 39.9 + 39.5}{3} = \frac{119.0}{3} \approx 39.7\ \text{s (rounded to the instrument's resolution)}

This is our best estimate of the time for 20 oscillations.

Accuracy

No independently known true/accepted value of the period is given in this problem, so we cannot compare against one directly. In this situation, the mean of the readings, $\bar t \approx 39.7\ \text{s}$ , is taken as the best available estimate of the true value, and the accuracy is estimated from how far the individual readings deviate from it: …

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Concept: Precision vs. Accuracy in Measurements

Precision is set by the resolution (least count) of the measuring instrument — it tells you how finely repeated readings can be told apart, regardless of how they scatter. Accuracy tells you how close the measurement is to the true value; without an independently known true value, the best available estimate of accuracy is how far the individual readings deviate from their own mean.

Method: Least Count + Mean Deviation Method

Steps:

Step 1: Identify the instrument's resolution (this IS the precision)

Each reading (39.6 s, 39.9 s, 39.5 s) is recorded to one decimal place, so the stopwatch's least count is $0.1\ \text{s}$ .

\text{Precision} = \pm 0.1\ \text{s}

Step 2: Find the mean (average) time

\bar{t} = \frac{t_1+t_2+t_3}{3} = \frac{39.6+39.9+39.5}{3} = \frac{119.0}{3} = 39.67\ \text{s} \approx 39.7\ \text{s (to the instrument's resolution)}

Step 3: Find the deviation of each reading from the mean

Measurement	Value (s)	Deviation = $\lvert t_i - \bar{t} \rvert$
$t_1$	39.6	$
$t_2$	39.9	$
$t_3$	39.5	$

Step 4: Average the deviations — this gives the accuracy …

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Common Mistakes & How to Avoid Them

This question tests your understanding of precision vs. accuracy — two terms students often confuse. Let's break down the common errors.

✗ Mistake 1: Confusing Precision with Accuracy

What students do wrong:

They say "precision = closeness to true value" or "accuracy = how consistent the readings are." This is backwards.

Why it's wrong:

Precision = how close the measurements are to each other (repeatability), and in practice is set by the resolution (least count) of the measuring instrument.
Accuracy = how close the measurement is to the true value (correctness).

How to avoid:

Remember the mnemonic:

Precision → Proximity of readings to Peers (each other) → fixed by the instrument's least count.
Accuracy → Agreement with the Actual (true) value.

✗ Mistake 2: Inventing a percentage formula for precision instead of using the instrument's least count

What students do wrong:

They try to compute precision as $\dfrac{\text{Range}}{2\times\text{Mean}}\times100\%$ or some other relative/percentage figure.

Why it's wrong:

Precision in a repeated-reading experiment like this one is simply the resolution (least count) of the measuring instrument — here, the stopwatch reads to $0.1\ \text{s}$ , so the precision is $\pm0.1\ \text{s}$ . It does not depend on how the three readings happen to scatter, and it is not normally expressed as a percentage in this kind of problem.

How to avoid:

Check how many decimal places every reading is recorded to — that IS the instrument's least count, and hence the precision.

✗ Mistake 3: Claiming Accuracy Cannot Be Found at All Without a True Value

What students do wrong:

They calculate a raw range or standard deviation and call it "accuracy" without a clear reference point, or they say accuracy cannot be found at all because no true value is stated.

Why it's wrong:

Accuracy strictly requires a reference (true/accepted value). This problem does not state an independently known true period. The standard way to proceed — and what full marks expect here — is to take the mean of the readings as the best available estimate of the true value, and report the mean deviation of the individual readings from that mean as the accuracy. Leaving accuracy completely unanswered loses marks that are available for this step.

How to avoid:

If a true value is given: $\text{Accuracy} = \left(1 - \dfrac{|\bar{t} - t_{\text{true}}|}{t_{\text{true}}}\right) \times 100\%$
If not given (as here): use the mean as the reference and report the mean deviation of the readings from it.

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Q.Time for 20 oscillations of a pendulum is measured as t1=39.6t_1 = 39.6t1​=39.6 s; t2=39.9t_2 = 39.9t2​=39.9 s; t3=39.5t_3 = 39.5t3​=39.5 s. What is the precision in the measurements? What is the accuracy of the measurement?

Precision vs. accuracy

The data

Precision

Accuracy

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Q.Time for 20 oscillations of a pendulum is measured as $t_1 = 39.6$ s; $t_2 = 39.9$ s; $t_3 = 39.5$ s. What is the precision in the measurements? What is the accuracy of the measurement?

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