Why careful analysis matters
Collecting data is only half of an experiment. The other half is analysing it honestly: reporting values with the right number of digits, describing the spread, and comparing results with known values.
Significant figures
Significant figures (sig figs) are the digits in a measurement that carry meaning: all the certain digits plus one estimated digit. They show the measurement’s precision.
Rules for counting significant figures:
| Rule | Example | Sig figs |
|---|---|---|
| Non-zero digits always count | 254 | 3 |
| Zeros between non-zero digits count | 1005 | 4 |
| Leading zeros never count | 0.0042 | 2 |
| Trailing zeros after a decimal point count | 2.50 | 3 |
| Trailing zeros in a whole number are ambiguous | 1500 | 2, 3 or 4 |
Scientific notation removes ambiguity: 1.50 × 10³ has 3 sig figs.
Calculating with significant figures
- Multiplying and dividing: the answer has as many sig figs as the measurement with the fewest.
- 4.56 cm × 1.4 cm = 6.384 cm² → 6.4 cm² (2 sig figs)
- Adding and subtracting: the answer has as many decimal places as the measurement with the fewest.
- 12.11 g + 0.3 g = 12.41 g → 12.4 g (1 decimal place)
Round only at the end of a calculation to avoid rounding errors.
Describing a data set
For repeated measurements, report the mean and a measure of spread.
- Mean: add all the values and divide by how many there are.
- Range: largest minus smallest.
- Standard deviation: describes how far values typically are from the mean. A small standard deviation means the data are tightly clustered (high precision).
Example: five measurements of g: 9.74, 9.81, 9.79, 9.86, 9.80 m/s².
- Mean = 49.00 ÷ 5 = 9.80 m/s²
- Range = 9.86 − 9.74 = 0.12 m/s²
- Standard deviation ≈ 0.04 m/s²
Report as 9.80 ± 0.04 m/s².
Percent error
Percent error compares your experimental value with the accepted value:
percent error = |experimental − accepted| ÷ accepted × 100%
If you measured g as 9.62 m/s² and the accepted value is 9.81 m/s²:
percent error = |9.62 − 9.81| ÷ 9.81 × 100% = 0.19 ÷ 9.81 × 100% ≈ 1.9%
A small percent error suggests good accuracy. A large one suggests a systematic error or a flaw in the method.
Related: percent difference compares two experimental values when there is no accepted value.
Graphical analysis
Plot data on a scatter graph and draw a line of best fit. It should pass as close to as many points as possible, with roughly equal numbers above and below.
- The slope often has a physical meaning. On a graph of mass vs. volume, the slope is density.
- The y-intercept may reveal a systematic error if it should be zero but isn’t.
- The R² value (from software) shows how well the line fits; values near 1 indicate a strong fit.
Sometimes data are curved. Scientists often transform them (for example, plotting y against x² or using logarithms) to get a straight line that’s easier to analyse.
Common mistakes
- Copying every calculator digit. Round to the correct sig figs.
- Rounding in the middle of a calculation.
- Forcing a line through every point instead of finding the best fit.
- Forgetting the absolute value in percent error.
Try it yourself
- How many significant figures are in 0.04050?
- Calculate 3.20 m ÷ 1.6 s with correct sig figs.
- A student measures the density of aluminum as 2.61 g/cm³. The accepted value is 2.70 g/cm³. Find the percent error.
- Four readings are 12.1, 12.4, 12.2 and 12.3 s. Find the mean and range.
(Answers: 4; 2.0 m/s; 3.3%; mean 12.25 s → 12.3 s, range 0.3 s)
Quick recap
- Significant figures show how precise a measurement is.
- Multiply/divide: fewest sig figs; add/subtract: fewest decimal places.
- Report the mean with a measure of spread, such as standard deviation.
- Percent error compares your result with the accepted value.
- Lines of best fit reveal trends, slopes and possible systematic errors.