Significant Figures, Precision, and Accuracy in Measurements

Measurement in Chemistry

Uncertainty in Measurement

In chemistry, measurements are made using various devices such as graduated cylinders, burettes, balances, and scales. Every measurement inherently contains some degree of uncertainty due to limitations in the measuring instrument and human estimation.

• Certain digits: Digits in a measurement that are known with certainty based on the instrument's calibration.

• Uncertain digit: The rightmost digit in a recorded measurement, which is estimated. Only one uncertain digit is typically recorded.

Example: If a graduated cylinder reads 25.3 mL, the '3' is the uncertain digit.

Significant Figures

Definition and Importance

The collection of all certain digits plus the one uncertain digit in a measurement is called significant figures. Significant figures communicate the precision of a measurement and help ensure that calculations do not imply greater accuracy than the data supports.

• Uncertainty is implied in the rightmost digit, typically understood as ±1 in the last digit unless otherwise specified.

Precision and Accuracy

Definitions and Differences

Although often used interchangeably in everyday language, precision and accuracy have distinct meanings in scientific measurement:

• Accuracy: The closeness of a measured value to the true or accepted value.

• Precision: The degree of agreement among several measurements of the same quantity (i.e., reproducibility).

Example: If you weigh a sample three times and get values very close to each other, your measurements are precise. If those values are also close to the actual mass, they are accurate.

Visual Representation

The following dartboard analogy illustrates the difference between accuracy and precision:

• (a) Low accuracy, low precision: Darts are scattered and far from the bullseye.

• (b) Low accuracy, high precision: Darts are grouped together but far from the bullseye.

• (c) High accuracy, high precision: Darts are grouped together and close to the bullseye.

Rules for Counting Significant Figures

1. Nonzero Integers

• All nonzero digits always count as significant figures.

• Example: 123 has three significant figures.

2. Zeros

• Leading zeros: Zeros that precede all nonzero digits. Never count as significant figures. Examples: 0.0075 (2 sig figs), 0.123 (3 sig figs), 0.456 × 102 (3 sig figs)

• Captive zeros: Zeros between nonzero digits. Always count as significant figures. Examples: 10.567 (5 sig figs), 1.045 (4 sig figs), 10.005 (5 sig figs)

• Trailing zeros: Zeros at the end of a number. Count as significant figures only if the number contains a decimal point. Examples: 500 (1 sig fig), 500. (3 sig figs), 5.00 × 102 (3 sig figs)

3. Exact Numbers

• Numbers obtained by counting or by definition (not by measurement) are considered to have an infinite number of significant figures and do not limit the number of significant figures in calculations.

• Examples: 1 in. = 2.54 cm (exact), 5280 ft. = 1 mi. (exact), 4 experiments (exact), 453.59 g = 1 lb. (exact)

Significant Figures in Calculations

Multiplication and Division

The result should have the same number of significant figures as the number with the least significant figures among the values used in the calculation.

• Example: $4.56 \times 1.4 = 6.4$ (rounded to 2 significant figures)

Addition and Subtraction

The result should have the same number of decimal places as the number with the least decimal places among the values used in the calculation.

• Example: $12.11 + 18.0 + 1.013 = 31.1$ (rounded to 1 decimal place)

Rules for Rounding

General Guidelines

• Carry extra digits through the final result, then round as needed.

Specific Rounding Rules

• If the digit to be removed is less than 5, the preceding digit stays the same. Example: 15.22 rounded to three significant figures is 15.2.

• If the digit to be removed is 5 or greater, the preceding digit is increased by 1. Example: 121.45 rounded to four significant figures is 121.5.

Summary Table: Significant Figure Rules

Type of Digit	Counts as Significant?	Example	Number of Significant Figures
Nonzero integers	Always	123	3
Leading zeros	Never	0.0075	2
Captive zeros	Always	1.045	4
Trailing zeros (with decimal)	Yes	5.00	3
Trailing zeros (no decimal)	No	500	1
Exact numbers	Infinite	1 in. = 2.54 cm	Infinite