Student guide: Statistics
The Statistics tab puts numbers on the results the Filters pick out: one table, with a row for each statistic, its formula, and its value. The title above it says what the numbers describe, for example Laboratory ALS - Cu_MEICP41s_ppm - Pulverising Stage - 10/11/2024 to 10/11/2025, and the subtitle names the data and the analysis suite.
The table uses exactly the results the Charts tab is showing, with one difference: the chart’s Threshold factor is applied here (pairs below the threshold are left out), as it is on the chart, but not on the Data tab. So if the Data tab lists 30 pairs and the table counts 25, the threshold is why.
Copy, CSV and Excel above the table save it.
1. Precision: the duplicates table
For Precision the table has two value columns, Sample 1 (the original results) and Sample 2 (the duplicates), each described on its own.
| Statistic | Formula | What it tells you |
|---|---|---|
| Count | n = Σ 1 | How many pairs are above the threshold. Fewer than 10 and nothing below is reliable. |
| Count of Nulls, Count of NA | — | Always 0: results of 0 and missing results were removed when the file was read (see the Data guide). They are shown for completeness. |
| Minimum, Maximum, Range | min(x), max(x), max − min | The spread of concentrations. A wide range usually means the pairs span low grades and high grades, whose precision differs. |
| Sum | Σ x | Used in the mean. |
| Median | x̃ | The middle result. Compare it with the mean: a mean far above the median means a few high-grade pairs dominate. |
| Mean | x̄ = Σ x / n | The average concentration. |
| SE. Mean | s / √n | How well the mean is known. |
| CI Mean 0.95 | x̄ ± t · SE | The half-width of the 95% confidence interval of the mean: the mean is known to within ± this. |
| Variance, Standard Deviation | s² = Σ(x − x̄)² / (n − 1), s = √s² | The spread of the results around their mean. |
| RSD / CV (%) | (s / x̄) × 100 | The standard deviation as a percentage of the mean. |
Read Sample 1 and Sample 2 together. They are the same material twice, so their statistics should agree. A difference in the means is a bias between original and duplicate (for example a sampling bias in how the duplicate was taken); a difference in the SDs means one of the two is noisier.
What this table is not. It describes the concentrations, not the agreement between the pairs. The relative differences, their control lines and the outlier counts are under the chart on the Charts tab, and the stage-limit check is on the Overview. The CV here is the spread of grades across the pairs, not the precision of the assay.
A worked example
Set Analyte to Cu_MEICP41s_ppm and Sample check stage to Pulverising Stage (ALS, ACA-Core, the latest year). The table reads:
| Sample 1 | Sample 2 | |
|---|---|---|
| Count | 25 | 25 |
| Minimum | 0.5 | 1 |
| Maximum | 9380 | 9520 |
| Median | 495 | 496 |
| Mean | 1282.86 | 1286.76 |
| Standard Deviation | 2337.352 | 2336.005 |
| RSD / CV (%) | 182.2 | 181.5 |
- The two columns agree closely: medians of 495 and 496 ppm, means within 4 ppm, SDs within 2 ppm. Original and duplicate are drawn from the same material and the lab treats them alike.
- The CV of 182% is not poor precision. The 25 pairs range from 0.5 ppm to 9,380 ppm, so their SD is enormous compared with their mean. The CV describes that spread of grades.
- The mean (1,283) is far above the median (495): a few very high-grade pairs pull it up. For the precision of these pairs, look at the Scatter chart’s statistics for these filters: a median RD of −1%, with 1 of 25 pairs beyond 3 SD.
2. Accuracy and Contamination: the CRM table
For Accuracy and Contamination the table has one value column per CRM or blank chosen (headed with its ID), the same descriptive rows, then:
| Statistic | Formula | What it tells you |
|---|---|---|
| Certified Value | μ₀ | The CRM’s certified value from the file (STANDARDVALUE). With several CRMs chosen on the Z-Score or Box and whiskers chart, it is their average, which means little; choose one CRM to read it. |
| RSD / CV (%) | (s / x̄) × 100 | The lab’s repeatability on this CRM. For a CRM this is a precision measure, because every result should be the same. |
| Mean Bias (Error) | x̄ − μ₀ | How far the lab’s average sits from the certified value, in the result’s units. |
| Relative Bias / % Bias (%) | ((x̄ − μ₀) / μ₀) × 100 | The same as a percentage of the certified value. |
| t-statistic | (x̄ − μ₀) / (s / √n) | The bias measured in standard errors: how confidently the mean differs from the certified value. |
Reading the t-statistic. It compares the bias with how precisely the mean is known. As a rule of thumb, |t| above about 2 (with 10 or more results) means the mean really is off the certified value, not just scattered around it; |t| below 2 means the results are consistent with the certified value. It grows with √n, so a small bias becomes “significant” once there are enough results: that is the same idea as the Overview’s CUSUM, which adds small differences up over time.
A worked example: a CRM
With the demo’s default Accuracy filters (ALS, ACA-Core, Ag_MEICP41s_ppm, CRM ACA-04, the latest year):
| Statistic | ACA-04 |
|---|---|
| Count | 5 |
| Minimum / Maximum | 47.2 / 49.7 |
| Mean | 48.06 |
| Standard Deviation | 0.976 |
| Certified Value | 48.4 |
| RSD / CV (%) | 2.031 |
| Mean Bias (Error) | −0.34 |
| Relative Bias / % Bias (%) | −0.702 |
| t-statistic | −0.779 |
- Five results of silver on this CRM averaged 48.06 ppm against a certified 48.4 ppm: a bias of −0.34 ppm, or −0.7%.
- The t-statistic of −0.78 says that bias is less than one standard error: five results scattered by about 1 ppm can’t tell a mean of 48.06 from 48.4. No bias is demonstrated.
- The CV of 2% is the lab’s repeatability on this material at this grade.
Compare the Shewhart chart for the same filters: all five results are within the warning limits, and the certified value sits between the mean and the median.
A worked example: a blank
Switch QC type to Contamination (ALS, ACA-Core, Al_MEICP41s_pct, blank FB):
| Statistic | FB |
|---|---|
| Count | 12 |
| Mean | 0.411 |
| Certified Value | 0.4 |
| Mean Bias (Error) | 0.011 |
| t-statistic | 0.577 |
This blank is certified at 0.4% Al, not 0, so the table judges it like a CRM: on average the lab reports it at 0.411%, a bias of 0.011% that the t-statistic says is not significant. The average is not the question for a blank, though. Contamination is a single result that is too high, so the Overview asks whether any result is more than 5× the detection limit (0.05%) above the certified value, that is above 0.45%. The results run from 0.29% to 0.54%, and the Overview counts 12 exceedances in the last 20 despatches for this blank. The Contamination chart’s warning line is drawn 5× LLD above zero, so there every one of the 12 results is above it; read it with the certified value in mind. When a blank has a certified value above zero, the Overview’s rule is the one to quote.
3. Which numbers to report
| Question | Use | Where |
|---|---|---|
| How repeatable is the lab on a known material? | The CRM’s RSD / CV. | Statistics tab, Accuracy. |
| Is the lab biased on a CRM? | Mean Bias, % Bias and the t-statistic, with n. | Statistics tab, Accuracy. |
| Is the bias sustained over time? | The CUSUM and the Overview’s bias in SD. | Overview; Charts tab, Cusum. |
| How well do duplicates agree? | The median RD and the outlier counts under the chart; the Overview’s median RD against the stage limit. | Charts tab; Overview. |
| What grades do the pairs cover? | Minimum, median, maximum of Sample 1. | Statistics tab, Precision. |
A good QC report states the number of results, the period, the suite and the stage or CRM, the bias with its t-statistic or the median RD with its limit, and the threshold used. The title above the table gives the first four.
Exercises
- Grade, not precision. With the Cu_MEICP41s_ppm Pulverising Stage example above, explain why a CV of 182% does not mean the lab’s precision is poor. What statistic on the Charts tab answers the precision question?
- A bias that isn’t. For ACA-04 above, the mean is 0.34 ppm below the certified value, yet the t-statistic is −0.78. What would have to change for this bias to become significant: a larger bias, a smaller SD, or more results? Why does the Overview use a CUSUM for the same purpose?
- Several CRMs at once. From the Overview, Investigate the Mo_MEICP41s_ppm finding, then change the chart to Z-Score: the CRM filter now holds every CRM. Look at the Statistics tab: the Certified Value has become 433.454 and the mean 430.531, with a t-statistic of −0.05. What are those numbers the average of, and why do they say nothing about the bias the Overview found on EXP-3 (certified 3.8)?
- The threshold changes the table. Set Threshold factor to 10 on the Charts tab for the Ag_MEICP41s_ppm Crushing Stage defaults. How does the Count change, and does the CV go up or down? Explain which pairs were removed.
- Blank or CRM? The demo’s blank FB has a certified value of 0.4% Al. Using the numbers above, say whether the laboratory is contaminating it, and which check (the Statistics bias, the chart’s warning line or the Overview’s rule) you would quote.