Error Analysis and Significant Figures
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Direct answer
Every measurement carries an uncertainty at least as large as the instrument's least count, and these uncertainties propagate through every calculation built on them: in sums and differences absolute errors add; in products, quotients and powers relative errors add, weighted by the exponents. Results must finally be reported to a number of significant figures matched to the least precise input, because digits quoted beyond the uncertainty are meaningless.
What you must remember
- Absolute error is the magnitude of deviation from the true value; relative error = absolute error divided by the value; percentage error = 100 times the relative error.
- Sum or difference (Z = A plus or minus B): delta Z = delta A + delta B — absolute errors always add.
- Product or quotient (Z = A B or A/B): delta Z/Z = delta A/A + delta B/B — relative errors add.
- Power rule: for Z = A^p B^q/(C^r), delta Z/Z = p (delta A/A) + q (delta B/B) + r (delta C/C); a squared quantity contributes double its percentage error — so in g = 4 pi^2 l/T^2, T's error counts twice.
- Significant figures: all non-zero digits count; captive zeros count (1005 has four); leading zeros do not (0.023 has two); trailing zeros count only with a decimal point (2.300 has four, 200 alone has one unless written 2.00 × 10^2).
- In multiplication and division the result keeps the fewest significant figures among the inputs; in addition and subtraction it keeps the fewest decimal places.
- Least count: vernier callipers = 1 MSD − 1 VSD (typically 0.01 cm); screw gauge = pitch divided by circular-scale divisions (typically 0.01 mm); zero error must be corrected with its sign before any reading is used.
Common confusion
The classic slip is adding percentage errors in a sum — sums propagate absolute errors, products propagate relative ones. The second confusion is trailing zeros: 200 is ambiguous between one and three significant figures until a decimal point or scientific notation settles it. Distinguish accuracy (closeness to the true value, spoiled by systematic errors such as zero error) from precision (closeness of repeated readings, limited by least count and random scatter).
Exam-focused takeaway
JEE Main asks propagation numericals — percentage error in density, resistance or g — plus least-count arithmetic with zero-error correction and significant-figure counting: quick, certain marks. JEE Advanced adds the judgement layer: choosing the instrument whose least count suits the tolerance, minimising the dominant error, rounding numerical-value answers correctly, and dimensional checks that catch algebraic slips. Two rules — absolutes add for sums, relatives add for products — make this chapter free marks.
Frequently asked questions
What is the difference between accuracy and precision?
Accuracy is closeness to the true value (spoiled by systematic errors like a zero error); precision is the closeness of repeated readings to one another (limited by least count and random errors).
How do errors combine in addition versus multiplication?
Absolute errors add in sums and differences; relative errors add in products, quotients and powers, each quantity weighted by its exponent.
Which zeros are significant?
Captive zeros and trailing zeros after a decimal point count; leading zeros never do; trailing zeros without a decimal point are ambiguous until scientific notation fixes them.
What is an instrument's least count?
The smallest measurable change: for vernier callipers, one main-scale division minus one vernier division; for a screw gauge, pitch over the number of circular-scale divisions.
How is a trailing 5 rounded?
By the NCERT even-digit rule: dropping an exact 5 leaves the preceding digit even (2.35 and 2.45 both round to 2.4).