Student guide: Statistics
The Statistics tab puts numbers on the results the Filters pick out. It is 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. The subtitle names the data and the analysis suite.
The table uses exactly the results the Charts tab is showing. Like the chart, it applies the Threshold factor and leaves out pairs below the threshold. The Data tab does not apply it. 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 and Sample 2. They describe the two results of each pair separately.
| Statistic | Formula | What it tells you |
|---|---|---|
| Count | n = Σ 1 | How many pairs are above the threshold. With fewer than 10, none of the statistics below is reliable. |
| Count of Nulls, Count of NA | None | Always 0, because results of 0 and missing results were removed when the file was read. The Data guide explains this. 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. 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 value. |
| 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 good duplicates give matching statistics. A difference in the means is a bias between Sample 1 and Sample 2, for example from unequal masses such as a half core against a quarter core. 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
For ALS in the latest year, set Analysis Suite to MINE-Core, Analyte to Cu_MEICP41s_ppm and Sample check stage to Pulverising Stage. The table reads:
| Sample 1 | Sample 2 | |
|---|---|---|
| Count | 24 | 24 |
| Minimum | 2 | 2 |
| Maximum | 9380 | 9520 |
| Median | 528.5 | 526 |
| Mean | 1336.292 | 1340.333 |
| Standard Deviation | 2371.976 | 2370.508 |
| RSD / CV (%) | 177.504 | 176.86 |
- The two columns agree closely, with medians of 528.5 and 526 ppm, means within 5 ppm and SDs within 2 ppm. The two results of each pair come from the same material, and the lab treats them alike.
- The CV of 178% is not poor precision. The 24 pairs range from 2 ppm to 9,380 ppm, so their SD is enormous compared with their mean. The CV describes that spread of grades.
- The mean of 1,336 is far above the median of 528.5, because a few very high-grade pairs pull it up. For the precision of these pairs, look at the RD v Mean Pairs chart’s statistics for these filters. They give a median RD of −1%, with 1 of 24 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. It has the same descriptive rows, then these:
| 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, each column has its own CRM’s certified value and statistics. |
| RSD / CV (%) | (s / x̄) × 100 | The lab’s repeatability on this CRM. For a CRM this is a precision measure, because every result is meant to 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 bias as a percentage of the certified value. |
| t-statistic | (x̄ − μ₀) / (s / √n) | The bias measured in standard errors. It shows 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, with 10 or more results, |t| above about 2 means the mean really is off the certified value, not just scattered around it. Below 2, the results are consistent with the certified value.
The t-statistic grows with √n, so a small bias becomes “significant” once there are enough results. The Overview’s CUSUM works on the same idea, adding up small differences over time.
A worked example: a CRM
With Analysis Suite still on MINE-Core, switch QC type to Accuracy. This table is for ALS, 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. That is 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, so no bias is shown.
- 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, for ALS, MINE-Core, Al_MEICP41s_pct and blank FB:
| Statistic | FB |
|---|---|
| Count | 12 |
| Mean | 0.411 |
| Certified Value | 0.4 |
| Mean Bias (Error) | 0.011 |
| t-statistic | 0.568 |
This blank is certified at 0.4% Al, not 0, so the table treats 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.
For a blank, though, the average is not the question. 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, which means 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 draws its warning line 5× LLD above zero, so every one of the 12 results is above it there. Read that chart with the certified value in mind. When a blank has a certified value above zero, quote the Overview’s rule.
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. | The Overview, and the Cusum chart on the Charts tab. |
| How well do duplicates agree? | The median RD and outlier counts under the chart, and the Overview’s median RD against the stage limit. | Charts tab and 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. It also gives 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 178% 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, and the Statistics tab has a column for each of the 20 CRMs, each checked against its own certified value. Find EXP-3, certified at 3.8, with a mean of 2.57, a bias of −1.23 (−32%) and a t-statistic of −13.7. Why would one set of statistics over all 20 CRMs together, with grades from 2 to 10,000 ppm, have hidden this bias?
- The threshold changes the table. Set Threshold factor to 10 on the Charts tab for the MINE-Core, Ag_MEICP41s_ppm, Crushing Stage example. 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. Which check would you quote: the Statistics bias, the chart’s warning line or the Overview’s rule?