Student guide: Charts
The Charts tab draws the results the Filters pick out. The menu on the left (Charting) chooses the chart and its settings; the chart is on the right, with How this is calculated above it and its statistics below.
Which charts are offered depends on the QC type in the Filters: eleven for duplicates (Precision), six for CRMs (Accuracy) and one for blanks (Contamination). This page takes them in that order. Every chart has a menu at its top right for full screen, printing, and downloading the picture or the data behind it.
Controls on every chart
- Point size (1 to 6, default 2): the size of the points.
- How this is calculated opens a window with the method behind the chart and, under This chart, the formula with this chart’s own numbers: the RD formula and threshold in use, or the centre line and limits. Read it whenever a chart surprises you.
- Under the chart, a statistics panel gives the numbers the chart was drawn from. On the Precision charts it turns orange when there are more outliers than a normal distribution would give.
Precision: duplicates
Every Precision chart starts from the same thing: for each duplicate pair, the original result a (Sample 1) and the duplicate b (Sample 2), and a relative difference (RD) between them.
RD to use
| Choice | Formula | Sign | Notes |
|---|---|---|---|
| MPRD% (default) | (a − b) / ((a + b) / 2) × 100 | Kept | The difference as a percentage of the pair’s mean. Neither result is treated as the reference. The Overview uses its absolute value. |
| ARD% | |a − b| / a × 100 | Absolute | The difference as a percentage of the original. |
| CV% | |a − b| / (√2 × (a + b) / 2) × 100 | Absolute | The pair’s standard deviation as a percentage of its mean. |
| HARD% | |a − b| / (a + b) × 100 | Absolute | Exactly half the absolute MPRD%: a HARD% limit of 10% is an MPRD% limit of ±20%. |
| RDHYSLOP | (a − b) / (√2 × (a + b) / 2) × 100 | Kept | CV% with its sign (Hyslop and White, 2009). |
Example. For a = 2.50 and b = 2.40: ARD% 4.00, CV% 2.89, HARD% 2.04, MPRD% +4.08, RDHYSLOP +2.89. The same pair, five different numbers: always say which you used. The RD statistics page works each one through.
Why the sign matters: MPRD% is positive when the original is higher. If a whole chart of pairs sits on one side of zero, the duplicates are systematically lower (or higher) than the originals: a sampling bias in how the duplicate was taken, not random scatter.
Threshold factor
Threshold factor (× LLD) (0 to 50, default 0) multiplies the lower detection limit to give a threshold. Pairs whose mean is below it are left out of the statistics and the control lines. On the Scatter and RD v Mean Pairs charts they are drawn grey; the other charts don’t draw them. After the chart is drawn the label shows the threshold in units: Threshold factor (Threshold = 10).
Why: near the detection limit, results are rounded to the last digit, so a pair of 0.1 and 0.2 is a 67% relative difference that says nothing about the laboratory. The Overview uses a factor of 10; on the charts it starts at 0 so that you see everything, and choose.
Apply upper detection limit leaves out pairs where either result is above the method’s upper limit (where the data gives one): such a result reads at the limit and is not a measurement. The caption under the chart says what was applied.
Control lines, bands and outliers
Most RD charts draw control lines at ±2 SD (the warning limit) and ±3 SD (the action limit) of the RDs shown, worked out separately for each check stage. Control line method chooses how:
| Method | Centre and spread | Use it when |
|---|---|---|
| SQC (default) | The mean RD and its standard deviation. | The RDs are roughly normal. |
| MAD | The median, and a robust SD of 1.4826 × the median absolute deviation. | A few wild pairs would otherwise inflate the SD and hide everything else. |
Show colour bands shades the chart green within ±2 SD, amber between 2 and 3 SD, red beyond. Points beyond 3 SD are outliers and coloured as such.
The statistics panel under these charts reads, for Cu_MEICP41s_ppm at the Pulverising Stage (ALS, ACA-Core, the latest year):
Data count: 25 · LLD: 1 · Threshold: 0 · Mean: −1.7 · Median: −1 · Skew: −2 · Kurtosis: 9.9 · Beyond 2 SD: 2 (8%) · Outliers (beyond 3 SD): 1 (4%)
- The mean RD (−1.7%) and median (−1%) are near zero: no bias between original and duplicate.
- Skew −2 and kurtosis 9.9 say the RDs are not normal: one tail is long and the peak is sharp. A normal distribution has skew 0 and kurtosis 0.
- With 25 pairs, a normal distribution would put about 1 pair beyond 2 SD and none beyond 3 SD. There are 2 and 1, so the panel turns orange and suggests a threshold. Look at which pairs they are before blaming the laboratory: a low-grade pair is the usual cause.
Why ±2 and ±3 SD: for a normal distribution 4.6% of results fall beyond 2 SD and 0.3% beyond 3 SD. More than that means the pairs are not just scattering randomly. These are limits on the data’s own spread; the stage limits on the Overview (±10%, ±20%, ±30%) are fixed limits on precision itself. A laboratory can pass one and fail the other.
Facet by check stage draws one panel per check stage, two across (one below the other for RD v Sequence), each with its own statistics. It is offered on every Precision chart except Thompson Howarth, RD Rolling Median and RD Box and Whiskers.
The charts
Scatter: Sample 1 along the bottom, Sample 2 up the side, one point per pair, with the X = Y line where perfect duplicates would sit. Two best-fit lines are drawn: ordinary least squares, and a robust Theil-Sen line (the median of all pairwise slopes) that one wild pair can’t swing. Outliers are found one of two ways:
- Control line (%) (5 to 40, default 20) draws manual lines at y = x ± that percentage, and a pair whose difference is more than that percentage of the original is an outlier.
- Use SQC to detect outliers replaces them with the ±2 and ±3 SD lines of the RD in use (as a wedge that widens with grade), with bands and the SQC/MAD choice.
Why start here: the scatter shows grade and agreement together. Pairs far from X = Y at high grade are the ones that matter to a resource; a cloud of scatter near the origin is the detection limit.
RD v Mean Pairs: the RD of each pair against the pair’s mean. The X = Y line is gone; instead the pairs are a band around zero, with the mean (purple dashed) and median (green) RD lines and the control lines. The threshold is drawn as a vertical line when the factor is above 0.
Why: this is the standard precision chart. It shows at once whether precision depends on grade (the band widens to the left, towards the detection limit) and whether there is a bias (the band sits above or below zero).
RD v Sequence: the RD of each pair in order, with X axis choosing the order: Check ID (sample order), Lab Job No, Despatch No or Date. With a Date axis a smoothed trend line is added; Trend line smoothing (%) sets how smooth. Why: precision that changes through time, a step when a crusher was replaced, or a run of one sign that means a bias, only show in sequence.
RD v Density/Histogram: the RDs as a histogram, with the density curve. Charts to display chooses the histogram alone or with the density; Number of bins by Sturge’s Rule sets the bins from the count, or untick it to choose 10 to 60. Why: the shape of the distribution. A normal bell centred on zero is what good duplicates give; a shoulder or a second peak means two populations, usually two grades or two stages mixed.
RD v Percentile: the RDs ranked, with lines at the 50th, 95.4th and 99.7th percentiles, the percentiles that match ±1, ±2 and ±3 SD for a normal distribution. Why: to read off what RD a given share of pairs stays within: “90% of pairs agree within 8%” is a statement a report can use.
RD Rolling Median: the Overview’s own precision test, drawn. Each light point is one pair’s |MPRD%|, each dark point a despatch’s median, and the line is the median of all the pairs in the latest Rolling window of despatches (5 to 30, default 20). The stage limit and the action limit (twice it) come from the Rules. For Cu at the Pulverising Stage:
Despatches: 7 · Pairs: 25 · Latest rolling median: 2.2% · Latest despatch median: 0.1% · Stage limit: 10.0% (standard stage) · Despatches over the stage limit: 0
Precision here is comfortably inside its 10% limit. The RD Rolling Median page has a worked example of the rolling median itself.
Why a rolling median: it follows the typical pair, so one bad pair doesn’t move it, and it answers the question a manager asks: is precision getting better or worse?
RD Box and Whiskers: one box per analyte, every analyte at once, in one panel per check stage. The whiskers reach 2 × the interquartile range. The Analyte filter is hidden because every analyte is drawn. Why: a survey of the whole suite: which elements are noisy at which stage.
Thompson Howarth: a different idea. Instead of RDs, it sorts the pairs by their mean concentration into groups of Partition window pairs (default 11), takes each group’s median |a − b| and mean concentration, and fits a line through those points. The intercept is the precision at zero concentration (the practical detection limit follows from it) and the slope is the precision at high grade. What to chart draws the pairs, the fitted line, or both; the robust fit (Theil-Sen) is recommended. The chart ignores the check-stage filter and draws every stage as its own panel, each headed with its pair count and intercept: for Cu_MEICP41s_ppm in the latest year, Pulverising Stage 25 pairs Intercept: 0.044, while the Field Prep Stage’s 7 pairs are too few groups for a line. A curve of Precision% against concentration is drawn only when a stage has at least 5 × the window pairs (55 with the default window), so widen the date range to see one. It is the slowest chart in the app; give it half a minute. The Thompson-Howarth page gives the arithmetic.
Why: precision is not one number; it depends on grade. Thompson-Howarth gives precision as a function of grade, which is what a resource estimate needs, and it works below the threshold where RDs don’t.
Quantile-Quantile - Normal Distn: the RDs’ quantiles against a normal distribution’s, with a 95% band. Points on the line mean the RDs are normal; tails peeling away mean outliers. Quantile-Quantile: Sample 1’s sorted values against Sample 2’s. Empirical Cumulative Density Function: the cumulative share of pairs up to each RD, against the normal curve with the same mean and SD. These three are ways of asking whether the SQC control lines, which assume a normal distribution, are the right ones; if they aren’t, use MAD.
Accuracy: CRMs
Every Accuracy chart plots a CRM’s results against its certified value. The CRM is chosen in the Filters (CRM type and CRM); one at a time, except for the Z-Score and Box and whiskers charts.
Use CRM provided statistics
Unticked, the control lines are built from the computed mean and standard deviation of the results on the chart. Ticked, they use the CRM’s certified value and certified SD from the file. Why it matters: computed lines describe the laboratory’s own scatter, so a laboratory that is consistently 10% low still looks “in control”. Certified lines ask whether the results are right. The Overview’s jumps tick the box for that reason. If the CRM has no certified SD, the app says so and you must untick it.
The worked example: Mo on EXP-3
From the Overview, click Investigate on the first Current Risk (Mo_MEICP41s_ppm: the latest EXP-3 result is 9.0 SD below certified). The Filters are set to Accuracy, ALS, All Suites, Mo_MEICP41s_ppm and EXP-3, a note above the chart says where you came from, and the chart is a Shewhart with the certified statistics.
Shewhart: each result in order, against the centre line and limits. This chart reads:
Centre line 3.8 and σ = 0.2. Warning limits (±2σ): 3.4 and 4.2. Action limits (±3σ): 3.2 and 4.4.
and the statistics panel:
Expected value: 3.8 · Certified SD: 0.2 · Data count: 79 · % outside 2SD: 93.7 · % outside 3SD: 93.7
74 of the 79 results are below 3.2 ppm, the lower action limit. This is not scatter; the laboratory reports this CRM about a third low, every time. The green Certified value line, the Mean and the Median are drawn whichever statistics you use, so you can see the gap.
Run Nelson 8 rules adds a table of eight patterns that mean a process is out of control, each PASS or FAIL, and clicking a row marks the points that break it. For these results rules 1 (a point beyond 3σ), 2 (nine in a row on one side), 5 (2 of 3 beyond 2σ on one side), 6 (4 of 5 beyond 1σ) and 8 (eight in a row outside 1σ) all fail. Why the rules: a bias too small to cross the 3σ line still shows as a run on one side (rule 2) or a cluster beyond 2σ (rule 5). The Nelson 8 rules page explains each.
Cusum: a running sum of each result’s distance from the target, in two halves: C⁺ grows while results sit above the target, C⁻ while they sit below, and each resets to zero when the results come back. Shift Detection (k, × SD) (default 0.5) is the allowance subtracted at every step, so that random scatter doesn’t accumulate; Decision Interval (H, × SD) (default 3) is the control line. With the certified statistics, k = 0.5 × 0.2 = 0.1 and H = 3 × 0.2 = 0.6 ppm. For Mo on EXP-3 the C⁻ line dives through −0.6 almost at once and keeps falling: a cumulative sum of 79 results each about 1.2 ppm low. This is the chart on the Overview’s featured card, and the Overview’s bias test is a CUSUM of the same kind (on z-scores pooled across the analyte’s company CRMs, with H = 5). The CUSUM page has a worked table.
Why a CUSUM when a Shewhart would do here: it wouldn’t always. A bias of 0.7 SD never crosses a Shewhart limit; a CUSUM finds it in about ten results.
Exponentially Weighted Moving Average: a smoothed version of the results, each new one weighted by Lambda (0.05 to 1, default 0.2) and the running average by 1 − λ, against limits of Number of Sigmas (L) (default 2.5) that widen over the first few results and settle. The data points turn red when the EWMA is outside its limits. Why: like the CUSUM it finds small, sustained shifts; unlike it, it reads in the units of the result. The EWMA page gives the formula.
All Time Series: the Shewhart, CUSUM and EWMA stacked, hovering linked, each with its statistics in its subtitle. Use it to show the same bias three ways.
Z-Score: every result as (result − certified value) / SD, so that several CRMs of different grades sit on one chart, with lines at 0, ±2 and ±3. The SD is the certified one when the box is ticked and the CRM has one; otherwise the SD of that CRM’s own results, and the legend says which (ACA-04 (computed SD)). The subtitle counts the results beyond ±2σ and ±3σ and names any CRM left out for having too few results. Why: to compare CRMs. If every CRM of an element is low, the laboratory’s calibration is low; if one is, that CRM’s certificate may be wrong.
Box and whiskers: one box per CRM, whiskers at 2 × IQR, every result jittered beside it. No certified line is drawn; it is a picture of consistency, not bias. A tall box or long whiskers mean the CRM is reporting inconsistently.
Contamination: blanks
The one chart, Bar, draws each blank result as a bar in order of Check ID, with a warning line at Warning factor (× LLD) (1 to 20, default 5) times the detection limit, and the certified value if the blank has one. Bars above the warning line are orange. The subtitle counts them: for the demo’s blank FB on Al_MEICP41s_pct, warning line 0.05 (5 × LLD 0.01) · Above the warning line: 12 of 12 (100%).
That blank is certified at 0.4% Al, so every result is far above a line drawn 0.05% above zero. The Overview’s blank check measures the 5× LLD from the certified value instead, and counts the results above 0.45%. When a blank has a certified value, read the Overview’s count, not the chart’s.
Why 5× LLD: a blank reads a little above zero by chance; five detection limits is beyond that noise, and means material from another sample, or from the crusher or pulveriser, got in.
Reading a chart from the Overview
When you arrive from an Overview button, a note above the chart says From the Overview:
Exercises
- Five numbers for one pair. Find a pair in the Data tab, and work out its ARD%, CV%, HARD%, MPRD% and RDHYSLOP by hand. Which would you report for a resource estimate, and why?
- Threshold. On the Scatter chart for Cu_MEICP41s_ppm at the Pulverising Stage, raise Threshold factor from 0 to 10. How do Data count, Beyond 2 SD and the outlier count change? Is the orange warning still there? What grade did the threshold remove?
- SQC or MAD? On RD v Mean Pairs for the same filters, switch Control line method to MAD. Which way do the control lines move, and why? Check the skew and kurtosis in the panel before you answer.
- Find the bias three ways. From the Overview, Investigate Mo_MEICP41s_ppm on EXP-3, then look at the Shewhart, Cusum and EWMA in turn. On each, say what shows the bias. Then untick Use CRM provided statistics on the Shewhart: what happens to the limits, and why is the chart now misleading?
- One CRM or all? On the Z-Score chart for Mo_MEICP41s_ppm, choose every CRM in the Filters. Are all the CRMs low, or only EXP-3? What does each answer tell you about where the problem is?