Chapter 6: Significant Figures and Measurement Errors

6.1 Significant Figures

No physical measurement can ever be made with perfect, infinite precision — every measuring instrument has a limit to how finely it can distinguish values. Significant figures (or significant digits) are the digits in a measured or calculated value that carry real meaning about its precision, that is, all the digits that are reliably known plus one final digit that is estimated or uncertain. For example, a length measured as 24.5 cm has three significant figures, indicating that the measurement is reliable to the nearest 0.1 cm, with the last digit being somewhat uncertain.

Rules for counting significant figures

• All non-zero digits are significant (for example, 245 has three significant figures).
• Zeros between two non-zero digits are significant (for example, 1005 has four significant figures).
• Leading zeros (zeros to the left of the first non-zero digit) are never significant; they only locate the decimal point (for example, 0.0025 has two significant figures).
• Trailing zeros after a decimal point are significant (for example, 2.500 has four significant figures).
• Trailing zeros in a number without a decimal point are ambiguous and are best clarified by expressing the number in scientific notation (for example, 100 could have one, two, or three significant figures; writing it as 1.00 × 10² makes it unambiguously three).

Rules for significant figures in calculations

When measured quantities are combined through arithmetic, the result cannot be reported with more precision than the least precise measurement that went into it. In addition and subtraction, the result should be rounded off to the same number of decimal places as the quantity with the fewest decimal places among those being added or subtracted — for example, adding 12.63 cm, 3.2 cm, and 5.417 cm gives a raw sum of 21.247 cm, but since 3.2 cm has only one decimal place, the answer must be rounded to 21.2 cm. In multiplication and division, the result should be rounded off to the same number of significant figures as the quantity with the fewest significant figures among those involved in the calculation — for example, multiplying 4.51 (three significant figures) by 3.2 (two significant figures) gives a raw product of 14.432, which must be rounded to two significant figures, 14. This rule prevents a calculation from creating a false impression of greater precision than the original measurements actually support, and it is a favourite basis for numerical questions in competitive exams.

6.2 Errors in Measurement

The difference between the measured value of a quantity and its true value is called an error. Errors are classified mainly as systematic errors and random errors. Systematic errors are consistent, repeatable errors caused by a flaw in the instrument, the experimental method, or the observer (such as zero error in an instrument, poor calibration, or personal bias); because they are consistent in direction and magnitude, they can often be identified and corrected. Random errors, by contrast, arise from unpredictable fluctuations in experimental conditions (such as small variations in temperature, vibration, or the observer's judgement) and cause repeated measurements to scatter irregularly around the true value; they can be reduced by taking a large number of readings and averaging them, but cannot be entirely eliminated.

Common quantitative measures of error

• Absolute error: the magnitude of the difference between the measured value and the true (or mean) value of a quantity, expressed in the same unit as the quantity itself.
• Mean absolute error: the average of the absolute errors of a set of repeated measurements.
• Relative error: the ratio of the mean absolute error to the true (or mean) value of the quantity; it is a dimensionless, unit-independent measure of precision.
• Percentage error: the relative error expressed as a percentage, obtained by multiplying the relative error by 100.

6.3 Accuracy vs Precision

Accuracy refers to how close a measured value is to the true or accepted value of the quantity, whereas precision refers to how close repeated measurements of the same quantity are to one another, regardless of whether they are close to the true value. A set of measurements can therefore be precise (tightly clustered together) without being accurate (if they are all clustered around the wrong value due to a systematic error), and conversely a single accurate measurement made without repetition tells us little about precision. Good experimental practice aims to achieve both high accuracy and high precision.

6.4 Common Instruments Used for Precise Measurement

Beyond the ordinary metre scale, which can typically measure length only to the nearest millimetre, several precision instruments are used in the laboratory to measure very small lengths reliably. The vernier callipers uses a main scale together with a sliding vernier scale to measure lengths to a precision of about 0.01 cm (0.1 mm); it is commonly used to measure the internal and external diameters of objects and the depth of small holes. The least count of a vernier callipers — the smallest length it can measure directly — is found using the relation: least count = value of one main scale division divided by the total number of divisions on the vernier scale. The screw gauge (micrometer screw gauge) achieves still finer precision, typically 0.001 cm, by using the linear motion produced by rotating a finely threaded screw, and its least count is found using: least count = pitch of the screw divided by the number of divisions on the circular (head) scale, where the pitch is the linear distance moved by the screw for one complete rotation. Both instruments are frequently referenced in SSC/RRB questions asking for their least count formula or their typical use.