Uncertainty in Measurement

NCERT Reference: Chapter 1, Pages 8–10

Quick Notes:

All measurements have uncertainty – due to instrument limitations and human estimation.
Large and small numbers are written using scientific notation: N × 10ⁿ, where 1 ≤ N < 10.
Significant figures are all meaningful digits: known + one estimated.
Rules for significant figures:
- All non-zero digits → significant.
- Zeros between non-zero digits → significant.
- Leading zeros → not significant.
- Trailing zeros → significant if there’s a decimal.
In multiplication/division the final answer = least number of significant figures.
In addition/subtraction the final answer = least number of decimal places.
Dimensional analysis is used to convert units using relationships like: 1 inch = 2.54 cm, 1 foot = 12 inches, etc.
Conversion factors are treated as fractions equal to 1 and used to cancel units.

Full Notes:

1.4.1 Scientific Notation

In chemistry, we often work with extremely large or small numbers.
For example:

One mole of particles = 602000000000000000000000
Radius of a hydrogen atom = 0.000000000052 m

Writing these repeatedly is impractical. So, scientists use scientific notation, which expresses numbers as: N × 10ⁿ, where N is a decimal number between 1 and 10, and n is an integer.

Examples:

602000000000000000000000 = 6.022 × 10²³
0.000000000052 = 5.2 × 10⁻¹¹

The exponent n is:

Positive for large numbers (move decimal left)
Negative for small numbers (move decimal right)

Why use it?
Scientific notation shortens calculations, improves clarity, and reduces error in lab work and data analysis. It's particularly useful in expressing atomic and molecular quantities.

1.4.2 Significant Figures

Every measurement contains some degree of uncertainty. This is why scientists do not just record any number of digits — they carefully report significant figures, which include:

All digits known with certainty, plus
One digit that is estimated

This estimated digit is where uncertainty lies — and it limits the reliability of any calculated result.

Rules for Counting Significant Figures (NCERT Rules)

All non-zero digits are significant
285 cm has 3 significant figures
Zeros between non-zero digits are significant
5.007 g has 4 significant figures
Leading zeros (before the first non-zero digit) are not significant
0.00016 m has 2 significant figures
Trailing zeros:
- Are significant if there is a decimal point
  3.00 has 3 significant figures
- Are not significant without a decimal 4000 has 1 significant figure
Exact numbers (from counting objects or defined relationships) have infinite significant figures
1 litre = 1000 mL (exactly)

Tip: Use scientific notation to make significant figures explicit.
E.g. 4000 =
4 × 10³ to 1 sig. fig.
4.0 × 10³ to 2 sig. figs.
4.00 × 10³ t- 3 sig. figs.

Rounding Off Rules

If the digit dropped is < 5, the preceding digit remains unchanged
If the digit dropped is ≥ 5, the preceding digit is increased by 1

Matt’s exam tip

Round only at the end of calculations, never during intermediate steps.

Rules for Calculations

Multiplication/Division: Round to least number of significant figures
2.5 × 1.25 = 3.125 → 3.1 (2 sig. figs.)
Addition/Subtraction: Round to least number of decimal places
12.11 + 18.0 = 30.11 → 30.1

Accuracy vs Precision (Now Included)

Apart from the number of digits, NCERT introduces two very important terms:

Precision refers to how close repeated measurements are to each other.
Accuracy refers to how close a measurement is to the true or accepted value.

To illustrate this, NCERT gives the following example:

If the true value of a substance’s mass is 2.00 g:
- Student A measures: 1.95 g and 1.93 g
  These are precise (close to each other) but not accurate.
- Student B measures: 1.94 g and 2.05 g
  These are neither precise nor accurate .
- Student C measures: 2.01 g and 1.99 g
  These are both precise and accurate.

Student	Measurement 1	Measurement 2	Precision	Accuracy
A	1.95 g	1.93 g	High	Low
B	1.94 g	2.05 g	Low	Low
C	2.01 g	1.99 g	High	High

Why This Matters
Understanding the difference between precision and accuracy is critical in real lab work. A set of results can seem reliable because they are close together, but still be wrong. Likewise, results might be close to the correct value but vary too much to be trusted. This concept is tested often in conceptual MCQs, especially in NEET and Olympiad settings.

1.4.3 Dimensional Analysis

This is a method used to convert one unit to another by applying unit relationships as conversion factors.

These relationships are treated as fractions equal to 1, so they can be multiplied without changing the value.

Example Convert 2.5 m to cm

2.5 m × (100 cm / 1 m) = 250 cm
The unit ‘m’ cancels out, leaving the result in cm.

Example Combined Conversion: Convert 2.0 inches to feet

Step 1: Use known conversion factors
1 inch = 2.54 cm
100 cm = 1 m
1 m = 3.281 feet

Step 2: Multiply stepwise
2.0 inch × (2.54 cm / 1 inch) × (1 m / 100 cm) × (3.281 ft / 1 m) = 0.167 ft

NCERT Emphasis: Use dimensional consistency – all units must cancel correctly. This is key to ensuring correct results in chemistry formulas and reactions.

Worked Example

Question: Express 0.00016 in scientific notation and identify significant figures.

Solution: 0.00016 = 1.6 × 10⁻⁴
It has 2 significant figures

Matt’s exam tip

Students often lose marks by: Miscounting zeros, rounding too early, and applying sig fig rules from addition to multiplication (and vice versa). Be sure to finish the calculation fully, then round. Memorise the rules separately for each type of operation — this is a guaranteed CBSE/NEET test area.

Summary

Every measurement has uncertainty and should be recorded with correct significant figures.
Scientific notation expresses very large or small numbers compactly.
Rounding rules depend on operation type and should be applied only at the end.
Accuracy is closeness to true value; precision is closeness of repeated results.
Dimensional analysis converts units using cancellation with exact conversion factors.