Fundamental Concepts in Measurement: Accuracy, Precision, and Significant Figures

Unit 1: Measurement in Chemistry

Introduction

Accurate and precise measurement is foundational to all chemical experiments. Understanding the difference between accuracy and precision, as well as the correct use of significant figures and conversion factors, ensures reliable and reproducible results in the laboratory.

Accuracy vs. Precision

Accuracy and precision are two key concepts in measurement:

• Accuracy: Describes how close a measured value is to the true or accepted value.

• Precision: Describes how close repeated measurements are to each other, regardless of their closeness to the true value.

Example: In a target diagram, high precision but low accuracy is shown when repeated shots cluster together but are far from the bullseye.

Table: Accuracy and Precision in Density Measurements

Consider four students measuring the density of aluminum (true value: 2.7 g/mL):

	A	B	C	D
Trial 1	2.924	2.316	2.649	2.701
Trial 2	2.923	2.527	2.731	2.699
Trial 3	2.925	2.941	2.695	2.702
Trial 4	2.926	2.136	2.742	2.698
Accuracy	NA	NA	A	A
Precision	P	NP	NP	P

Key: A = Accurate, NA = Not Accurate, P = Precise, NP = Not Precise

Measurement Terms

• Percent Error (Accuracy): Quantifies how close a measurement is to the true value.

The formula for percent error is:

$\text{Percent Error} = \frac{\text{Experimental Value} - \text{True Value}}{\text{True Value}} \times 100\%$

Example: Measuring a 100.00 g standard weight on two balances:

• Accurate balance: 100.00 g

• Inaccurate balance: 98.89 g

• Percent deviation: $\frac{98.89 - 100.00}{100.00} \times 100\% = -1.11\%$

• Precision: Assessed by the range of repeated measurements.

Range is calculated as:

$\text{Range} = \text{Highest Value} - \text{Lowest Value}$

For example, if four measurements are 100.01, 100.00, 99.99, and 100.00 g, the range is $100.01 - 99.99 = 0.02$ g, or $\pm 0.01$ g.

Significant Figures

Significant figures (sig figs) indicate the precision of a measured or calculated quantity.

• All non-zero digits are significant. (e.g., 1234 has 4 sig figs)

• Zeros between non-zero digits are significant. (e.g., 1.003 has 4 sig figs)

• Leading zeros are not significant. (e.g., 0.0025 has 2 sig figs)

• Trailing zeros are significant if there is a decimal point. (e.g., 2.300 has 4 sig figs; 2300 has 2 sig figs unless written as 2.300 × 103)

Rules for Calculations

• Multiplication/Division: The result should have as many significant figures as the value with the least number of sig figs.

• Addition/Subtraction: The result should have as many decimal places as the value with the least number of decimal places.

Example (Addition):

• 3.461728 + 14.91 + 0.980001 + 5.2631 = 24.614829

• The number with the least decimal places is 14.91 (2 decimal places), so the answer is rounded to 24.61.

Example (Multiplication):

• 64 × 12.458 = 796.352

• 64 has 2 sig figs, so the answer is rounded to 2 sig figs: 800

Rounding Off Rules

• Identify the digit to round to (based on required significant figures).

• If the next digit is 0–4, round down (leave the digit the same).

• If the next digit is 5–9, round up (increase the digit by 1).

Example: Round 3.34237 × 104 to 2 significant figures: 3.3 × 104

Conversion Factors

Conversion factors are used to convert between units. They are ratios equal to one, allowing units to be changed without affecting the value.

• General formula:

$\text{Given Quantity} \times \frac{\text{Conversion Factor}}{1} = \text{New Quantity}$

Example: Convert 54 cm to meters:

• $54 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 0.54 \text{ m}$

Example: Convert 0.53 kg to mg:

• $0.53 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1000 \text{ mg}}{1 \text{ g}} = 530,000 \text{ mg}$

Exponential (Scientific) Notation

Scientific notation expresses numbers as a product of a coefficient and a power of ten.

• 1000 = $1 \times 10^3$

• 0.001 = $1 \times 10^{-3}$

• 2386 = $2.386 \times 10^3$

• 0.0123 = $1.23 \times 10^{-2}$

Summary Table: Accuracy vs. Precision

Term	Definition	Example
Accuracy	Closeness to true value	Measuring 100.00 g for a 100.00 g standard
Precision	Closeness of repeated measurements	All measurements are 99.99 g, 100.00 g, 100.01 g

Key Takeaways:

• Always report measurements and calculations with the correct number of significant figures.

• Use percent error to assess accuracy and range to assess precision.

• Apply conversion factors and scientific notation for unit conversions and expressing large/small numbers.