Replicate measurement datasets and worked statistics

Practice mean, sample standard deviation and coefficient of variation on six synthetic datasets, including mixed units, blank signals and a reference check.

Download the materials

Printable worksheet
Standalone resource with instructions and a worked answer key.
Six replicate measurement sets (CSV)
Eighteen readings with group, unit, measurement context and origin.
License and attribution
CC BY 4.0 terms and a suggested credit.

Use this resource

Open the measurements CSV and analyze each group separately.
Convert mixed units before calculating and record why CV may not be meaningful for blank signals.
Compare your calculations and interpretation with the worked answer key.

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.

Practice cases in the CSV
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

Expected outputs (rounded for checking, not report precision)
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.

Provenance and reuse

All values and measurement contexts were constructed for instruction. They do not describe a real instrument, method or sample.

Applies to the original teaching text, worksheets and synthetic data in this resource. Linked third-party standards and publications keep their original terms. Standard symbols and other facts remain subject to any existing public-domain status.

Reuse the original materials under Creative Commons Attribution 4.0 International (CC BY 4.0). Attribution — Replicate measurement datasets and worked statistics — Lab Data Field Guide, version 1.0

Sources

Open replicate statistics calculator: mean, sample sd and %rsd or read the related guides.