Significant figures, precision, and accuracy

Every number in physics carries a hidden claim about how uncertain it is. When we write 4.53 m, we are implicitly saying that we’re confident of the two decimal digits and the third is unknown — the true value lies somewhere between 4.525 and 4.535. Significant figures are the convention for writing that claim, and the combination rules exist to keep us from computing our way into an accuracy that wasn’t in the input.

How many figures count?

The three basic rules:

  1. All non-zero digits are significant.
    – 4.53 — three sig figs
    – 126.7 — four sig figs

  2. Zeros between non-zero digits (interior zeros) are significant.
    – 2.005 — four
    – 10.03 — four

  3. Leading zeros are not significant — they’re just placeholders.
    – 0.00456 — three sig figs (4, 5, 6)
    – 0.0102 — three sig figs (1, 0, 2)

The tricky case — trailing zeros:

Resolving the ambiguity: scientific notation

Rewriting in scientific notation removes the ambiguity completely:

Number How many sig figs?
1.5 × 10^3 2
1.50 × 10^3 3
1.500 × 10^3 4
1.5000 × 10^3 5

Whenever there is doubt, use scientific notation. This is one of the main reasons scientific reports always use it, even when the value doesn’t demand it.

Precision vs accuracy

These two are entirely independent — a measurement can have any combination of the two.

The dartboard analogy is useful:

    accurate + precise       precise, not accurate
         🎯                      🎯
        (••)                    ( )
        (••)                    ( )
                             (••••) ← tight group, off-center

  accurate, not precise      neither
         🎯                      🎯
       (•  •)                  ( •  )
       (  • •)                 (•    )

Key point: sig figs report precision, not accuracy. A number with 6 sig figs is not necessarily more correct than one with 3 — it’s just reported more precisely. If the instrument is biased, all six digits are wrong.

Combining measurements — the quick rules

The combination rules come out of error propagation. For hand calculation, these two quick rules cover most cases:

Rule 1 — Addition and subtraction: keep the least decimal place

$$
12.51 + 3.4 = 15.9 \quad\text{(not } 15.91\text{)}
$$

Because 3.4 is only certain to one decimal place, the sum can’t be more certain than that.

Rule 2 — Multiplication and division: keep the least sig figs

$$
6.38 \times 2.1 = 13 \quad\text{(not } 13.398\text{)}
$$

2.1 has only 2 sig figs, so the product has 2. The calculator says 13.398 — those three extra digits are false precision.

Why the two rules are different

A combined example

Suppose we measure a rectangular room with a tape:

Area = $L \times W$ → Rule 2 (multiplication):

$$
A = 4.52 \times 3.10 = 14.012 \to \boxed{14.0\ \mathrm{m}^{2}}
$$

Three sig figs (both inputs have three).

Perimeter = $2L + 2W$ → Rule 1 (addition):

$$
P = 2(4.52) + 2(3.10) = 9.04 + 6.20 = 15.24 \to \boxed{15.24\ \mathrm{m}}
$$

Both terms have two decimal places, so the sum has two decimal places.

Note: the factor of 2 is an exact number — it comes from the definition of perimeter, not a measurement — and doesn’t limit sig figs. The same is true of defining constants (like $c$, which since 2019 is exactly 299,792,458 m/s) and counting constants ($N_A$ since 2019).

Rounding — which rule?

When the next digit is 5, two common conventions:

  1. Round half up (the school rule): $2.35 \to 2.4$, $2.45 \to 2.5$. Simple, but biased — always rounds up, so over long calculations the average drifts high.

  2. Round half to even (banker’s rounding, IEEE 754): $2.35 \to 2.4$ (up), $2.45 \to 2.4$ (down!) — the goal is to make the last digit even. This is the standard in scientific and financial computation because it has no long-term bias.

Python, R, MATLAB, Julia, and most numerical libraries use rule 2. If you’re surprised that round(2.5) == 2 in Python, that’s why.

Explicit uncertainty notation

Sig figs are an implicit convention. For more precision, report the uncertainty explicitly:

Any time you write a paper or record lab data, use these formats — sig figs are fine for teaching and back-of-envelope work but insufficient for a rigorous report.

Common mistakes

1. Rounding early. In a multi-step calculation, keep all the calculator digits through intermediate steps and only round the final answer. Rounding at each step compounds error.

2. Treating exact numbers as measured ones. Formula coefficients ($\tfrac{1}{2}$, $2\pi$), integer counts (5 students, 24 hours per day), and defining SI constants since 2019 (c, h, e, k, N_A) are all exact — unlimited sig figs.

3. Sig figs on measured constants. Some constants remain measured (like $G$, known to about `2.2 × 10^{-5}$ relative uncertainty), so treat them as measurements with a definite number of sig figs, not as exact.

4. Scientific notation vs sig fig display. A calculator might show 3.400000000 — that’s 3.4 with meaningless extra digits. If you want to indicate four sig figs, write 3.400, not 3.4.

5. Ambiguous trailing zeros in reports. If the Earth-Sun distance is roughly 150,000,000 km, write 1.50 × 10^{8} km to make the sig figs unambiguous — not 150,000,000 km where the reader can’t tell.

What you should be able to do

After this section you should be able to:

📚 See also: Halliday Vol 1, Ch 1 — end-of-chapter unit-conversion and sig-fig problems.
📖 Open reference: NIST — Uncertainty of Measurement Results (the formal method for reporting uncertainty, with examples).
📖 BIPM — Guide to the Expression of Uncertainty in Measurement (GUM) (the international standard).
📖 IEEE 754 — Standard for Floating-Point Arithmetic (source of the round-half-to-even rule).

سوالی دارید؟ 🤔

اگه مفهومی نامشخص بود یا سوالی داشتید، اینجا بپرسید. جوابتون در اینجا منتشر می‌شه.

💬 جواب بهتری داری؟ یا یه سؤال جدید؟

اگه به سؤالای بالا پاسخی داری که فکر می‌کنی روشن‌تر یا کامل‌تر از مال منه، یا یه سؤال جدید برای دانش‌آموزای دیگه داری — تو بخش نظرات پایین صفحه ارسال کن. هر پیامی رو می‌خونم، تأیید می‌کنم و منتشر می‌شه. این‌جوری همه از تجربه‌ی همدیگه استفاده می‌کنیم. 🌱