Rounding, Estimation & Error Intervals

SPEC N13, N14, N15 · NUMBER

Decimal places · Significant figures · Truncation

1 s.f. estimation · Error intervals from rounding and truncation

Key Definitions

DECIMAL PLACES

The number of digits after the decimal point.

3.147 has 3 d.p.

SIGNIFICANT FIGURES

The digits contributing to precision, counted from the first non-zero digit.

0.00284 has 3 s.f. (2, 8, 4).

TRUNCATION

Cutting off digits after a target position without rounding.

3.789 truncated to 1 d.p. = 3.7 (not 3.8).

ERROR INTERVAL

The range of possible values of a rounded/truncated quantity, written as a ≤ x < b.

High-priority topic

This lesson covers three separate spec points (N13–N15) that appear on every paper. Error intervals in particular have a distinctive inequality format that examiners check strictly.

Rounding to Decimal Places

Look at the NEXT digit → 5 or more: round UP · 4 or less: round DOWN

Example — round 3.148 to 2 d.p.

Target: the 2nd decimal place (the "4").

Next digit: 8. 8 ≥ 5 → round up.

Answer: 3.15

Example — round 0.6273 to 3 d.p.

Target: the 3rd decimal place (the "7").

Next digit: 3. 3 ≤ 4 → round down (keep the same).

Answer: 0.627

⚠️ EXAMINER ALERT — carrying the 9. Rounding 4.2996 to 3 d.p.: the next digit is 6 → round up. But 9 rounded up becomes 10, carrying over: 4.299 + 0.001 = 4.300. Keep the trailing zero — it signals the precision.

Rounding to Significant Figures

Count from the first non-zero digit. Leading zeros are place-holders — they do NOT count as significant.

Example — 402 800 to 2 s.f.

First sig fig: 4. Second sig fig: 0. Next digit: 2 → round down.

Answer: 400 000 (the trailing zeros are place-holders).

Example — 0.04567 to 1 s.f.

First sig fig: 4. Next digit: 5 → round up.

Answer: 0.05

d.p. vs s.f. — the Classic Trap

⚠️ EXAMINER ALERT — counting d.p. instead of s.f.

"Round 0.00284 to 2 s.f." — students write 0.00.

This has zero sig figs (just place-holder zeros).

What to write instead: count from the first non-zero digit. 0.00284 to 2 s.f. = 0.0028.

NumberTo 2 d.p.To 2 s.f.

0.002840.00 ✗0.0028 ✓

0.37860.380.38

45714571 (no decimals)4600

Quick test

Does the question say "decimal places" or "significant figures"? The mark scheme is strict — write the wrong type and lose the mark even if the digits are correct.

Truncation — Different from Rounding

Truncation means cutting off digits after a position without rounding. Whatever comes after the cut is simply discarded.

Truncation never rounds up

3.789 truncated → 3.7 (even though 89 is closer to 8)

5.999 truncated to 1 d.p. → 5.9 (even though 999 is nearly 1 whole)

Why it matters

Calculators often truncate long decimals to fit the display. If you take that as the "true" value, you lose accuracy.

⚠️ EXAMINER ALERT — Nov24 P1 Q16. Truncation is "not regularly tested" and "students struggle with both the required reason and the correct error interval." Learn the distinction from rounding — it is worth revision time.

Estimation — Round to 1 s.f.

Round EVERY value to 1 s.f., then compute

Example

Estimate (19.3 × 4.87) ÷ 0.203

19.3 ≈ 20 | 4.87 ≈ 5 | 0.203 ≈ 0.2

Estimate: (20 × 5) ÷ 0.2 = 100 ÷ 0.2 = 500

Exact value: 19.3 × 4.87 ÷ 0.203 ≈ 463. The estimate 500 is in the right ballpark.

Example

Estimate 197 ÷ 4.9

197 ≈ 200 | 4.9 ≈ 5

Estimate: 200 ÷ 5 = 40

⚠️ EXAMINER ALERT — rounding to more than 1 s.f. Students round 19.3 to 19 and 4.87 to 4.9, then compute "properly." This is NOT an estimation — it is a full calculation with slight rounding. Mark schemes want every value rounded to 1 s.f. before any arithmetic.

Practice Question

Estimate the value of (59.2 × 3.14) ÷ 0.078.

Show your working. 2 marks

M:

Round every value to 1 s.f.

C:

59.2 ≈ 60 | 3.14 ≈ 3 | 0.078 ≈ 0.08

(60 × 3) ÷ 0.08 = 180 ÷ 0.08 = 2250

E:

Estimate ≈ 2250

M1: all values rounded to 1 s.f. A1: correct final answer.

Error Intervals — Rounded Values

A value stated as a rounded number could have come from a range of true values. That range is the error interval.

a ≤ x < b

For a value rounded to the nearest unit u:

Rule for rounding

Half the rounding unit goes each side: LB = stated − u/2, UB = stated + u/2.

The lower bound is included (≤). The upper bound is excluded (<).

MCE Worked Example

A mass m is given as 6.4 kg, correct to 1 d.p.

Write down the error interval for m. 2 marks

M Rounding unit u = 0.1 kg.

Half-unit = 0.05.

LB = 6.4 − 0.05, UB = 6.4 + 0.05.

C Lower bound: 6.35

Upper bound: 6.45

Lower bound included (≤).

Upper bound excluded (<).

E6.35 ≤ m < 6.45

Both inequality signs correct. Both bounds to the right precision (2 d.p.).

Mark scheme

B1 lower bound (6.35 with ≤). B1 upper bound (6.45 with strict <). Both parts needed for 2 marks. Using ≤ at both ends is B1 only.

Strict Inequality at the Top

⚠️ EXAMINER ALERT — using ≤ at both ends.

Students write: 6.35 ≤ m ≤ 6.45

This is wrong. At exactly 6.45, the value would round up to 6.5, not stay at 6.4.

What to write instead: the top must be strictly less than.

6.35 ≤ m < 6.45

Why 6.35 is included

At m = 6.35, the "round half up" convention rounds to 6.4. So 6.35 itself rounds to 6.4 → it is a valid value for m.

Why 6.45 is excluded

At m = 6.45, the convention rounds to 6.5, not 6.4. So 6.45 is NOT a valid value for m → excluded.

Memory device

"LB is in, UB is out." Lower bound included, upper bound excluded. One ≤, one <.

Practice Question

A length L is given as 4.7 m, correct to 1 d.p.

Write down the error interval for L. 2 marks

M:

Rounding unit = 0.1 m. Half-unit = 0.05. LB = 4.7 − 0.05. UB = 4.7 + 0.05.

C:

Lower bound: 4.65 (included). Upper bound: 4.75 (excluded).

E:

4.65 ≤ L < 4.75 ✓

B1 lower; B1 upper with strict <.

Error Intervals — Truncated Values

Truncation discards the digits after the cut — it does NOT split the rounding unit into halves. The range is different from a rounded value.

Truncated to x: x ≤ true value < x + u

where u is the truncation unit.

Why truncation is different

Rounded 3.14 came from any true value in the range 3.135 up to (but not including) 3.145 — values in that range are close enough to 3.14 that rounding lands on 3.14.

Truncated 3.14 came from any true value ≥ 3.14. Anything below 3.14 truncates to 3.13 or lower. Anything from 3.14 up to (but not including) 3.15 truncates to 3.14.

Exam-Style: Nov24 P1 Q16

A value x is truncated to 2 d.p. as 5.28.

Write down the error interval for x. 2 marks

M:

Truncation means anything ≥ 5.28 but < 5.29 would be cut down to 5.28.

C:

Lower bound: 5.28 (included — the value itself truncates to 5.28).

Upper bound: 5.29 (excluded — 5.29 truncates to 5.29, not 5.28).

E:

5.28 ≤ x < 5.29 ✓

Different from rounded case — lower bound is 5.28 itself (not 5.275), upper bound is 5.29 (not 5.285).

Practice Question

Round 0.04972 to 2 significant figures. 1 mark

M:

First non-zero digit is 4 (the "4" in 0.04...). Count 2 sig figs: 4 and 9. Look at next digit (7) → round up.

C:

0.049|72 — the 7 is ≥ 5, so round the 9 up.

9 + 1 = 10, so we carry: 0.049 → 0.050.

E:

0.050 ✓

Keep the trailing zero — it shows the precision (2 s.f.).

Common Errors — Don't Lose These Marks

Students often writeCorrect versionError type

0.00284 to 2 s.f. = 0.000.0028Counted d.p., not s.f.

6.35 ≤ m ≤ 6.456.35 ≤ m < 6.45Upper bound must be strict (<)

Estimated 197 ÷ 4.9 without rounding200 ÷ 5 = 40Must round to 1 s.f. first

Rounded 3.148 to 2 d.p. = 3.143.15Next digit (8) → round up

Truncated 3.14 → interval 3.135 to 3.1453.14 ≤ x < 3.15Truncation not rounding

Summary

Key Points — Lesson 1.10

Rounding rule: next digit ≥ 5 → round up; ≤ 4 → round down.

Sig figs: count from first non-zero digit. 0.00284 to 2 s.f. = 0.0028.

Truncation: cut off digits, never round up. 3.789 truncated to 1 d.p. = 3.7.

Estimation: round EVERY value to 1 s.f., then compute.

Error interval (rounded): LB = stated − u/2, UB = stated + u/2.

Error interval (truncated): LB = stated, UB = stated + u.

Inequality format: a ≤ x < b. Lower in, upper out.

Two most dangerous traps

1. Using ≤ at both ends of an error interval.

2. Counting decimal places when the question asks for significant figures.

Rounded vs truncated interval

Both have width u (the precision unit).

Rounded: symmetric around stated value.

Truncated: all above stated value.

Exam Tip

"Premature approximation" — rounding too early in multi-step questions — costs 1 mark. Keep full precision throughout working, then round only at the final step (unless the question specifically asks for an estimate).