Precision and accuracy

Accuracy vs precision. Accuracy is how close a measured value is to the true (accepted) value. Precision is how close repeated measurements are to each other, regardless of whether they are close to the true value. Four combinations are possible for a set of repeated readings: precise and accurate (tightly clustered near the true value — the goal of good experimental work); precise but not accurate (tightly clustered but shifted away from the true value, almost always the signature of a systematic error, such as an uncalibrated instrument or a consistent zero-error); accurate but not precise (scattered widely, but the scatter happens to average out near the true value); and neither (scattered and off-target, usually from a mix of random error and a faulty method). It is worth being explicit that these two ideas are independent — a set of readings can be excellent by one standard and poor by the other, so "precise" is never a synonym for "correct."

Significant figures are the digits in a measured number that carry real information about how finely the measuring instrument could resolve the quantity. Counting them uses a small set of rules. All non-zero digits are always significant. Zeros sandwiched between non-zero digits are significant — $205$ has 3 significant figures. Leading zeros, the ones that only fix the position of the decimal point, are never significant — $0.0025$ has 2 significant figures, not 4. Trailing zeros written after a decimal point are significant, because they represent a real claim about precision — $2.500$ has 4 significant figures, telling the reader the measurement was good to the thousandths place. Trailing zeros in a number with no decimal point are ambiguous — $100$ could mean 1, 2, or 3 significant figures depending on how it was measured — and the ambiguity is resolved only by writing the number in scientific notation, where $1.00\times10^{2}$ is unambiguously 3 significant figures.

Combining measurements changes how many significant figures the answer is allowed to keep, and the rule depends on the operation. In addition or subtraction, the result is rounded to the same number of decimal places as the term with the fewest decimal places — decimal places, not digit count, because the least-precise term fixes the smallest place value the sum can be trusted to. In multiplication or division, the result is rounded to the same number of significant figures as the term with the fewest significant figures, because a product or quotient can only be as relatively precise as its least precise factor. Exact numbers — counting numbers like "12 eggs" or defined conversion factors like $1\ \text{m} = 100\ \text{cm}$ — are treated as having infinite significant figures and never limit a calculation's precision.

where: "fewest decimal places" and "fewest significant figures" are counted across all the terms or factors entering that single operation, and only exact numbers are exempt from limiting the result.

Rounding-off convention. When a digit must be dropped, if it is less than 5 the preceding digit is left unchanged; if it is greater than 5 the preceding digit is increased by 1; and if it is exactly 5 (with nothing non-zero after it), the preceding digit is rounded to the nearest even number — this "round-half-to-even" rule prevents a small upward bias from accumulating when many numbers are rounded in a long calculation.

There is no single algebraic formula behind precision and accuracy — the topic is a reporting convention, not a physical law — so the reasoning that matters is why the significant-figures rules protect a calculation from claiming false precision. Precision analysis of scientific measurement requires understanding why a calculated result can never be more trustworthy than the least trustworthy number that went into it, which is why the addition rule looks at decimal places while the multiplication rule looks at significant-figure counts, and why the two must never be swapped.

Step 1. Think of each measured digit as carrying a small, unavoidable uncertainty in its last recorded place — a balance reading $2.53\ \text{g}$ really means "somewhere between about $2.525$ and $2.535\ \text{g}$." Any digit beyond that last recorded place is unknown, not zero, so it cannot be written down as if it were known.

Step 2. In addition or subtraction, the absolute uncertainty of each term (fixed by its last decimal place) adds up, so the sum's uncertainty is at least as large as the least precise term's uncertainty — hence the sum can only be trusted to that same decimal place, no further. This is why decimal places, not the total count of digits, govern rounding in $+/-$.

Step 3. In multiplication or division, uncertainties combine relatively (as fractions of the value), so the answer's relative uncertainty is dominated by whichever factor has the largest relative uncertainty — which is the factor with the fewest significant figures. This is why sig-fig count, not decimal places, governs rounding in $\times/\div$.

Step 4. Exact numbers (a count of "12 eggs," or a defined relation like $1\ \text{m} = 100\ \text{cm}$) carry zero uncertainty by definition, so they never limit either rule — they are effectively infinitely precise and simply pass through the calculation.

Step 5. Rounding only the final answer, and never an intermediate result, matters because rounding at each step throws away real information and lets small errors compound across a multi-step calculation — the correct practice is to carry extra digits through intermediate steps and apply the significant-figure rule once, at the end.

A student weighs a sample of copper sulphate crystals three times on a digital balance and obtains $2.53\ \text{g}$, $2.54\ \text{g}$, and $2.52\ \text{g}$. The true mass, found later using a calibrated reference balance, is $2.80\ \text{g}$. The student is then asked to carry out two further calculations using correctly rounded significant figures: (a) $12.11 + 18.0 + 1.014$, and (b) $6.626\times10^{-34}\ \text{J s}$ divided by $2.5\times10^{-9}\ \text{s}$.

Step 1 — Check precision of the mass readings. The three readings ($2.53$, $2.54$, $2.52\ \text{g}$) are all within $0.01\ \text{g}$ of each other, so they are tightly clustered. This means the measurements are precise.

Step 2 — Check accuracy of the mass readings. The average of the three readings is

Comparing this to the true value of $2.80\ \text{g}$, the difference is $0.27\ \text{g}$, which is large — far outside the $0.01\ \text{g}$ spread of the readings themselves.

Step 3 — Conclude on precision and accuracy. The measurements are precise (they agree with each other) but not accurate (they do not agree with the true value). This pattern points to a systematic error — for example, an uncalibrated or "zero-error" balance that consistently under-reads by roughly the same amount each time.

Step 4 — Apply the addition rule to part (a). The three numbers $12.11$, $18.0$, and $1.014$ have $2$, $1$, and $3$ decimal places respectively. The result must be rounded to the fewest decimal places among the terms, which is $1$ (fixed by $18.0$).

Rounded to 1 decimal place: $31.1$.

Step 5 — Apply the division rule to part (b). $6.626\times10^{-34}$ has 4 significant figures; $2.5\times10^{-9}$ has 2 significant figures. The multiplication/division rule rounds to the fewest significant figures among the operands, which is $2$.

Rounded to 2 significant figures: $2.7\times10^{-25}$.

Competitive stretch

Suppose a fourth balance reading of $2.803\ \text{g}$ is added to the original three, and it happens to be very close to the true value $2.80\ \text{g}$. The new average shifts toward $\frac{2.53+2.54+2.52+2.803}{4} \approx 2.62\ \text{g}$, which is closer to the truth than before but still off by about $0.18\ \text{g}$ — a single accurate reading pulls the average toward accuracy but cannot, by itself, fix a systematic error affecting the other three readings; only recalibrating the balance removes the bias.

• Treating "precise" and "accurate" as synonyms — a tightly grouped set of readings can still be systematically wrong, as in Worked example Step 3.
• Forgetting that leading zeros in decimals like $0.00450$ do not count, while the trailing zero after the $5$ does count (this number has 3 significant figures, not 5 or 1).
• Applying the significant-figures counting rule to addition/subtraction — that operation is governed by decimal places, not significant-figure count, exactly as in Step 4 above.
• Rounding intermediate steps too early in a multi-step calculation, which compounds small errors — round only the final answer, per Step 5 of the reasoning above.

Ask the tutor: unsure whether a calculation should be rounded by decimal places or significant figures, or want more practice spotting significant figures in tricky numbers — ask the AI tutor to walk through extra examples with you.