Reporting Mean and Standard Deviation With Correct Precision

When reporting experimental results, the mean and standard deviation (SD) are the most common summary statistics. However, the precision with which these values are reported is often mishandled, leading to misleading or ambiguous data. This article provides a definitive reference on how to report the mean and standard deviation with the correct number of significant figures, based on established standards and best practices. Whether you are a student, engineer, or researcher, understanding these rules is essential for clear and honest data presentation.

Rule Statement

The fundamental rule is that the standard deviation determines the precision of the mean. The mean should be reported to the same decimal place as the first significant digit of the standard deviation. In practice, this often means reporting the mean to one more decimal place than the raw data, but the exact rule depends on the magnitude of the SD.

More formally, the GUM (Guide to the Expression of Uncertainty in Measurement) recommends that the uncertainty (here, the standard deviation) be reported with at most two significant digits, and the result (mean) be reported with the corresponding number of decimal places. For example, if the SD is 0.0345, the first significant digit is in the hundredths place, so the mean should be reported to the hundredths place (e.g., 12.34). If the SD is 0.003, the mean should be reported to the thousandths place (e.g., 12.345).

When the SD is very large relative to the mean, the mean may be reported with fewer decimal places. The key is to avoid implying more precision than the data actually possess.

Worked Examples

Example 1: Simple Dataset

Consider the measurements: 10.2, 10.4, 10.3, 10.1, 10.5. The mean is 10.30 and the standard deviation is 0.1581. The first significant digit of the SD is in the tenths place (0.1), so the mean should be reported to the tenths place: 10.3 ± 0.2 (rounding the SD to one significant digit). Alternatively, if you keep two significant digits for the SD, you would report 10.30 ± 0.16.

Example 2: Small SD

Measurements: 1.234, 1.235, 1.233, 1.236. Mean = 1.2345, SD = 0.00129. The first significant digit of the SD is in the thousandths place, so the mean should be reported to the thousandths place: 1.234 ± 0.001 (or 1.2345 ± 0.0013 if using two SD digits).

Example 3: Large SD

Measurements: 100, 120, 80, 90, 110. Mean = 100, SD = 15.8. The first significant digit of the SD is in the tens place, so the mean should be reported to the tens place: 100 ± 20 (or 100 ± 16).

Counter-Examples

Common errors include reporting the mean with too many decimal places, or reporting the SD with excessive precision. For instance:

• Over-reporting the mean: Reporting 10.3000 ± 0.2 implies that the mean is known to four decimal places, but the SD only justifies one decimal place.
• Over-reporting the SD: Reporting 0.158113883 ± 0.2 is absurd; the SD should be rounded to one or two significant digits.
• Inconsistent decimal places: Reporting 10.3 ± 0.16 is inconsistent because the mean has one decimal place while the SD has two. The mean should be reported to the same decimal place as the SD’s last significant digit.

Convention Comparison Table

Standards Citation

For authoritative guidance, consult the following standards:

• GUM (JCGM 100:2008), Clause 7.2.6: “The numerical value of the uncertainty should be given to at most two significant digits.”
• ASTM E29-13, Section 6.1: “The number of significant figures retained in a reported value shall be consistent with the precision of the measurement.”
• ISO 80000-1:2009, Annex C: “The uncertainty of a measurement result should be expressed with the same number of decimal places as the result.”
• NIST Technical Note 1297, Section 7.5: “The uncertainty should be reported with one or two significant digits, and the result should be rounded to the same decimal place.”

Common Mistakes

1. Ignoring the SD when rounding the mean. Always let the SD dictate the decimal place.
2. Reporting the SD with more than two significant digits. This overstates the precision of the SD itself.
3. Using the raw data precision instead of the SD. For example, if raw data are to 0.1, but the SD is 0.5, the mean should be reported to the tenths place, not the hundredths.
4. Inconsistent rounding between mean and SD. Both should be rounded to the same decimal place (or the SD to one/two sig figs and the mean to match).
5. Forgetting to round the SD before reporting. Always round the SD first, then round the mean to match.

Software Behavior Note

Statistical software often outputs many decimal places by default. For example, Excel’s AVERAGE and STDEV.S functions return values with up to 15 significant digits. Python’s numpy.mean and numpy.std similarly produce full precision. It is the user’s responsibility to apply the rounding rules. Some tools like R’s summary() or t.test() may use a default of 4 significant digits, but this is not always appropriate. Always apply the SD rule manually or via a custom function. For a reliable rounding tool, use our significant figures calculator to ensure correct precision.

Discipline Note

Different fields have specific conventions. In physics, the GUM is the standard, and the SD is often reported with one significant digit unless the first digit is 1 or 2, in which case two digits are used. In chemistry, the IUPAC guidelines recommend reporting the mean with the same number of decimal places as the SD, and the SD with one significant digit unless it is 1 or 2. In engineering, ASTM E29 is commonly applied, and the rule is to round the mean to the same decimal place as the SD’s first significant digit. In biology, many journals follow the “one more decimal place than the raw data” rule, but this can be misleading if the SD is large. Always check the target journal’s or discipline’s specific guidelines.

Quick Reference Table

Related Rules

For more on rounding and significant figures, see our guides on Significant Figures, Rounding Methods, and Measurement Uncertainty.