Before you start
Estimated time 25–35 minutes.
Prerequisites Basic unit conversion and arithmetic with square roots. Use a calculator or spreadsheet. The companion calculator accepts one group of values at a time.
Learning objectives
- Convert measurements to common units and calculate the mean and sample standard deviation for a group.
- Explain how a zero or near-zero mean affects a percentage measure of relative spread.
- Use the reference-check example to distinguish the spread of readings from the difference between their mean and the reference.
Choose the right comparison
Analyze one group at a time. The CSV spells micrograms as ug and uses AU for arbitrary signal units. The mixed-unit case repeats the first dataset with different units; treat it as a conversion exercise, never as additional observations.
| Group | Values as recorded | Measurement context |
|---|---|---|
| same_preparation | 9.8 mg/L; 10.1 mg/L; 10.1 mg/L | Three readings of one prepared sample |
| separate_preparations | 9.50 mg/L; 10.30 mg/L; 10.20 mg/L | One reading from each of three separately prepared samples |
| mixed_units | 9.8 mg/L; 10100 μg/L; 0.0101 g/L | Same three readings as same_preparation, expressed in different units |
| zero_blank | -0.02 AU; 0.00 AU; 0.02 AU | Signed baseline-corrected blank signal |
| near_zero_blank | -0.02 AU; 0.01 AU; 0.04 AU | Signed baseline-corrected blank signal |
| reference_check | 9.79 mg/L; 9.80 mg/L; 9.81 mg/L | Three readings against a fictional reference value of 10.00 mg/L |
Practice questions
- Work group by group. For each, compute the mean and sample standard deviation, then decide whether the CV is meaningful.
- Before calculating the mixed-unit case, convert the readings to mg/L. How do the results compare with the first group?
- Compare the spread in the two concentration groups. Explain why the second group covers a different part of the measurement process.
- Compare the reference check’s mean with the supplied reference value. Does a small SD rule out a difference from that value?
- For the same-preparation readings, compute the standard uncertainty of the mean due to repeat readings alone, assuming independence and stable measurement conditions. Identify two components this number leaves out.
Calculation rules
Mean = sum of readings / n. Sample SD = square root of the sum of squared deviations from the mean divided by (n − 1). The companion calculator displays a CV percentage using 100 × sample SD / absolute mean, often called percent relative standard deviation (RSD). Compare relative spread on a scale with a meaningful zero and a mean sufficiently far from zero.
Worked answer key
| Group | n | Mean | Sample SD | CV | Mean/SD unit |
|---|---|---|---|---|---|
| same_preparation | 3 | 10.0000 | 0.1732 | 1.732% | mg/L |
| separate_preparations | 3 | 10.0000 | 0.4359 | 4.359% | mg/L |
| mixed_units | 3 | 10.0000 | 0.1732 | 1.732% | mg/L |
| zero_blank | 3 | 0.0000 | 0.0200 | Undefined | AU |
| near_zero_blank | 3 | 0.0100 | 0.0300 | 300.000% (not useful here) | AU |
| reference_check | 3 | 9.8000 | 0.0100 | 0.102% | mg/L |
Same preparation: The sum is 30.0 mg/L, giving a mean of 10.0 mg/L. Deviations are −0.2, +0.1 and +0.1 mg/L. Their squared sum is 0.06 (mg/L)²; dividing by 2 and taking the square root gives 0.1732 mg/L. CV is about 1.732%.
Mixed units: 10,100 μg/L = 10.1 mg/L and 0.0101 g/L = 10.1 mg/L. The converted readings and results equal the first group.
Separate preparations: The observed SD is larger here. Separate preparations can include variation from preparation as well as reading. Three observations do not establish the source or size of each variation component.
Blank signals: The exact zero mean makes CV undefined. The near-zero case gives 300% arithmetically, but the baseline-corrected signal and small mean make it unhelpful for judging relative precision. Report the mean and SD with the baseline context.
Reference check: The mean is 0.20 mg/L below the stated reference value even though the sample SD is 0.01 mg/L. The data alone cannot establish the cause or statistical significance of the difference; a fuller assessment would need the reference uncertainty and measurement design.
Uncertainty from repeat readings: Under the stated assumptions, s/√n ≈ 0.100 mg/L. That is a component for the mean, not a complete uncertainty estimate. Contributions such as calibration, sample preparation and any drift still need consideration for the stated measurement.
Instructor notes
- Have learners show their unit conversions for the mixed-unit group and their squared deviations for the first group.
- Compare the blank groups with the reference check. Ask learners to write which statistics they would report for each and why.