CHEM-1003: Measurement, Accuracy, Precision, and Significant Figures

Unit 1: Measurement in Chemistry

Accuracy vs. Precision

In scientific measurement, understanding the difference between accuracy and precision is essential for evaluating data quality and reliability.

• 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: Consider a set of dart throws at a target. If all darts land close together but far from the bullseye, the throws are precise but not accurate. If darts are spread out but average near the bullseye, they are accurate but not precise.

Accuracy vs. Precision: Visual Example

The following diagram illustrates four scenarios:

• A: Low accuracy, low precision (darts scattered, far from center)

• B: Low accuracy, high precision (darts clustered together, far from center)

• C: High accuracy, low precision (darts scattered, average near center)

• D: High accuracy, high precision (darts clustered at center)

Key Point: High precision does not guarantee high accuracy, and vice versa.

Accuracy vs. Precision: Data Table Example

Suppose four students measure the density of aluminum (true value: 2.7 g/mL). Their results are:

	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

Interpretation: 'A' = Accurate, 'NA' = Not Accurate, 'P' = Precise, 'NP' = Not Precise.

Measurement Terms: Accuracy and Percent Error

Accuracy can be quantitatively expressed using percent error:

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

• Percent error indicates how close a measurement is to the accepted value.

• Smaller percent error means higher accuracy.

Example: Measuring a 100.00 g standard weight:

• Measured value: 98.89 g

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

Measurement Terms: Precision and Range

Precision is often evaluated by the range of repeated measurements:

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

• Smaller range indicates higher precision.

• Example: Four measurements of 100.01, 100.00, 99.99, 100.00 g have a range of 0.02 g, showing high precision.

Table: Precision Comparison

Trial #	Accurate (g)	Inaccurate (g)
1	100.01	98.89
2	100.00	101.00
3	99.99	100.00
4	100.00	101.09
Average	100.00	100.25
Range	±0.01	±1.51

Interpretation: The 'Accurate' set is more precise (smaller range).

Significant Figures (Sig Figs)

Significant figures reflect the precision of a measured or calculated quantity. The rules for determining the number of significant figures are:

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

• Zeros between nonzero 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 only 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)

Significant Figures in Calculations

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

• 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 value 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

Example: Round 2.3467 × 103 to 3 significant figures: 2.35 × 103

Conversion Factors

Conversion factors are used to convert between different units of measurement. They are ratios that express how many of one unit are equal to another unit.

• 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 is used to express very large or very small numbers in the form $a \times 10^n$.

• 1,000 = $1 \times 10^3$

• 0.001 = $1 \times 10^{-3}$

• 2,386 = $2.386 \times 10^3$

• 0.0123 = $1.23 \times 10^{-2}$

Summary Table: Accuracy vs. Precision

Term	Definition	How to Evaluate
Accuracy	Closeness to true value	Percent error
Precision	Closeness of repeated measurements	Range or standard deviation

Additional info: These notes cover foundational measurement concepts in General Chemistry, including how to evaluate and report data quality using accuracy, precision, significant figures, and unit conversions. Mastery of these skills is essential for all subsequent laboratory and theoretical work in chemistry.