Student guide: Charts
The Charts tab draws the results the Filters pick out. The Charting menu on the left chooses the chart and its settings. The chart is on the right, with How this is calculated above it and its statistics below.
The charts on offer depend on the QC type in the Filters. There are eleven for duplicates (Precision), six for CRMs (Accuracy) and one for blanks (Contamination), and 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 sets the size of the points, from 1 to 6. The default is 2.
- How this is calculated opens a window with the method behind the chart. Under This chart it gives the formula with this chart’s own numbers, such as the RD formula and threshold in use, or the centre line and limits. Read it whenever a chart surprises you.
- The statistics panel under the chart 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. Each duplicate pair has two results, a (Sample 1) and b (Sample 2), and a relative difference (RD) between them. Neither result is the original. A difference between them can come from the geology as well as from sampling or the laboratory.
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 Sample 1. It is the one form that treats the two results differently. |
| 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% equals 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, the five forms give ARD% 4.00, CV% 2.89, HARD% 2.04, MPRD% +4.08 and RDHYSLOP +2.89. The same pair gives five different numbers, so always say which you used. The RD statistics page works each one through.
Why the sign matters: MPRD% is positive when Sample 1 is higher. If a whole chart of pairs sits on one side of zero, Sample 2 is systematically lower or higher than Sample 1. That is a bias in how the pair was taken or prepared, not random scatter. Unequal masses, such as a half core against a quarter core, are a common cause.
Threshold factor
Threshold factor (× LLD) multiplies the lower detection limit to give a threshold. It runs from 0 to 50, and the default is 0.
Pairs whose mean is below the threshold are left out of the statistics and the control lines. The Scatter and RD v Mean Pairs charts draw them grey, and the other charts don’t draw them. Once the chart is drawn, the label shows the threshold in units, for example Threshold factor (Threshold = 10).
Why: near the detection limit, results are rounded to the last digit. A pair of 0.1 and 0.2 is then 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 you see everything and choose for yourself.
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. The app works them 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, and red beyond. Points beyond 3 SD are outliers, and are coloured as such.
Set Analysis Suite to MINE-Core and look at Cu_MEICP41s_ppm at the Pulverising Stage, for ALS in the latest year. The statistics panel under these charts reads:
Data count: 24 · LLD: 1 · Threshold: 0 · Mean: 0.98 · Median: −1 · Skew: 3.4 · Kurtosis: 11.6 · Beyond 2 SD: 1 (4.2%) · Outliers (beyond 3 SD): 1 (4.2%)
- The mean RD of 0.98% and the median of −1% are near zero, so there is no bias between Sample 1 and Sample 2.
- Skew 3.4 and kurtosis 11.6 say the RDs are not normal. One tail is long and the peak is sharp, where a normal distribution has skew 0 and kurtosis 0.
- With 24 pairs, a normal distribution would put about 1 pair beyond 2 SD and none beyond 3 SD. There is 1 beyond 3 SD, so the panel turns orange and suggests a threshold. Check which pairs they are before blaming the laboratory, because 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 Overview’s stage limits of ±10%, ±20% and ±30% are fixed limits on precision itself, and a laboratory can pass one and fail the other.
Facet by check stage draws one panel per check stage, each with its own statistics. The panels sit two across, or one below the other for RD v Sequence. The option is on every Precision chart except Thompson Howarth, RD Rolling Median and RD Box and Whiskers.
The charts
Scatter plots Sample 1 along the bottom and Sample 2 up the side, one point per pair. The X = Y line shows where perfect duplicates would sit. The chart draws two best-fit lines. One is ordinary least squares, and the other is a robust Theil-Sen line, the median of all pairwise slopes, which one wild pair can’t swing.
Outliers are found in one of two ways:
- Control line (%) draws manual lines at y = x ± that percentage, from 5 to 40 with a default of 20. A pair whose difference is more than that percentage of Sample 1 is an outlier.
- Use SQC to detect outliers replaces them with the ±2 and ±3 SD lines of the RD in use. These form a wedge that widens with grade, with colour bands and the SQC or 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 plots the RD of each pair against the pair’s mean. There is no X = Y line. The pairs form a band around zero, with the control lines, a purple dashed line for the mean RD and a green line for the median. When the factor is above 0, a vertical line marks the threshold.
Why: this is the standard precision chart. It shows at once whether precision depends on grade and whether there is a bias. A band that widens to the left, towards the detection limit, means precision depends on grade. A band above or below zero means a bias.
RD v Sequence plots the RD of each pair in order. X axis chooses the order: Check ID for sample order, Lab Job No, Despatch No or Date. A Date axis adds a smoothed trend line, and Trend line smoothing (%) sets how smooth it is. Why: some patterns only show in sequence, such as precision that changes over time, a step when a crusher was replaced, or a run of one sign that means a bias.
RD v Density/Histogram draws the RDs as a histogram, with the density curve. Charts to display shows the histogram alone or with the density. Number of bins by Sturges’ rule sets the bins from the count. Untick it to choose 10 to 60 bins.
Why: the chart shows the shape of the distribution. Good duplicates give a normal bell centred on zero. A shoulder or a second peak means two populations mixed, usually two grades or two stages.
RD v Percentile ranks the RDs, with lines at the 50th percentile, which is the median, and at the 95.4th and 99.7th. These are the shares a normal distribution keeps within ±2 and ±3 SD. Why: you can read off the RD that a given share of pairs stays within. A statement such as “90% of pairs agree within 8%” is one a report can use.
RD Rolling Median draws the Overview’s own precision test. Each light point is one pair’s |MPRD%|, and each dark point is a despatch’s median. The line is the median of all the pairs in the latest Rolling window of despatches, from 5 to 30 with a default of 20.
The chart draws every despatch in the date range and shades the last window of them. The latest rolling median and the Overview’s verdict are based on those despatches. The stage limit and the action limit, twice the stage limit, come from the Rules. For Cu at the Pulverising Stage:
Despatches: 7 · Pairs: 24 · Rolling window: 20 despatches · Latest rolling median: 1.9% · Latest despatch median: 0.1% · Median of all pairs: 1.9% · Stage limit: 10.0% (standard stage) · Despatches with the rolling median over the stage limit: 0 · Despatches with the rolling median over the action 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. It answers the question a manager asks, whether precision is getting better or worse.
RD Box and Whiskers draws one box per analyte, every analyte at once, with one panel per check stage. The whiskers reach 2 × the interquartile range. The Analyte filter is hidden, because every analyte is drawn. Why: the chart surveys the whole suite and shows which elements are noisy at which stage.
Thompson Howarth works differently and doesn’t use RDs. It sorts the pairs by their mean concentration into groups of Partition window pairs, 11 by default. It 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, which gives the practical detection limit, and the slope is the precision at high grade.
What to chart draws the Precision v Concentration curve, the Thompson Howarth fit, or Both, which is the default. The robust Theil-Sen fit is recommended.
The chart ignores the check-stage filter and draws each stage as its own panel, headed with its pair count and intercept. For Cu_MEICP41s_ppm in the latest year, one panel reads Pulverising Stage · 24 pairs · Intercept: -0.545. That intercept is below zero because, with so few low-grade pairs, the line says little about precision near zero. The Field Prep Stage’s 7 pairs are too few groups for a line.
A curve of Precision% against concentration needs at least 5 × the window pairs in a stage, which is 55 with the default window. Widen the date range to see one. This is the slowest chart in the app, so give it half a minute. The Thompson-Howarth page gives the arithmetic.
Why: precision is not one number, because it depends on grade. Thompson-Howarth gives precision as a function of grade, which is what a resource estimate needs. It also works below the threshold, where RDs don’t.
Quantile-Quantile - Normal Distn plots 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 plots Sample 1’s sorted values against Sample 2’s. Empirical Cumulative Density Function plots the cumulative share of pairs up to each RD, against the normal curve with the same mean and SD.
These three charts test whether the SQC control lines suit your data, because those lines assume a normal distribution. If they don’t, use MAD.
Accuracy: CRMs
Every Accuracy chart plots a CRM’s results against its certified value. You choose the CRM in the Filters, with CRM type and CRM. The charts show one CRM at a time, except the Z-Score and Box and whiskers charts.
Use CRM provided statistics
Unticked, the control lines use 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. If the CRM has no certified SD, the app says so and you must untick it.
Why it matters: computed lines describe the laboratory’s own scatter. A laboratory that is consistently 10% low still looks “in control”. Certified lines ask whether the results are right, which is why the buttons on the Overview tick the box.
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 app sets the Filters to Accuracy, ALS, All Suites, Mo_MEICP41s_ppm and EXP-3. It opens a Shewhart chart with the certified statistics, and a note above it says where you came from.
Shewhart plots 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.
The statistics panel reads:
Expected value: 3.8 · Certified SD: 0.2 · Data count: 79 · % outside 2SD: 93.7 · % outside 3SD: 93.7
Of the 79 results, 74 are outside the action limits. 73 are below the lower limit of 3.2 ppm, and one is at 7 ppm. This is not scatter, because the laboratory reports this CRM about a third low every time. The chart always draws the green Certified value line, the Mean and the Median, 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 marked PASS or FAIL. Click a row to mark the points that break it. For these results, five rules fail: 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).
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 keeps a running sum of each result’s distance from the target, in two halves. C⁺ grows while results sit above the target, and C⁻ while they sit below. Each resets to zero when the results come back.
Shift Detection (k, × SD) is the allowance subtracted at every step, so that random scatter doesn’t build up. Its default is 0.5. Decision Interval (H, × SD) is the control line, with a default of 3. 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. It is a cumulative sum of 79 results, each about 1.2 ppm low. This is the chart on the Overview’s featured card. The Overview’s bias test is a CUSUM of the same kind, run on z-scores pooled across the analyte’s company CRMs, with H = 5.
Why a CUSUM when a Shewhart would do here: a Shewhart chart doesn’t always do. A bias of 0.7 SD never crosses a Shewhart limit, but a CUSUM finds it in about ten results. The CUSUM page has a worked table.
Exponentially Weighted Moving Average draws a smoothed version of the results. Each new result is weighted by Lambda, from 0.05 to 1 with a default of 0.2, and the running average by 1 − λ. Number of Sigmas (L), default 2.5, sets the limits, which widen over the first few results and then settle. The data points turn red when the EWMA is outside its limits.
Why: like the CUSUM, the EWMA finds small, sustained shifts. Unlike the CUSUM, it reads in the units of the result. The EWMA page gives the formula.
All Time Series stacks the Shewhart, CUSUM and EWMA, each with its statistics in its subtitle. Pointing at one chart highlights the same result on the others. Use it to show the same bias three ways.
Z-Score plots every result as (result − certified value) / SD, with lines at 0, ±2 and ±3. Several CRMs of different grades can then sit on one chart. The SD is the certified one when the box is ticked and the CRM has one. Otherwise it is the SD of that CRM’s own results, and the legend says so, for example ACA-04 (computed SD).
The subtitle counts the results beyond ±2σ and ±3σ, and names any CRM left out for having too few results. The ring beside the chart counts the results as passed within ±2, warning from ±2 to ±3, and failed beyond ±3.
Group by CRM is ticked to start. It puts each CRM’s results together along the X axis, one colour each. Untick it to see every CRM in one sequence, coloured by band, which shows a change that hit all the CRMs at once.
Why: the chart compares CRMs. If every CRM of an element is low, the laboratory’s calibration is low. If only one is, suspect that CRM’s certificate.
Box and whiskers draws one box per CRM, with whiskers at 2 × IQR and every result jittered beside it. There is no certified line, because the chart shows 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. A warning line sits at Warning factor (× LLD) times the detection limit, with the factor from 1 to 20 and a default of 5. The chart also draws the certified value if the blank has one.
Bars above the warning line are orange, and the subtitle counts them. For the demo’s blank FB on Al_MEICP41s_pct it reads 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, and five detection limits is beyond that noise. A result that high means material got in from another sample, or from the crusher or pulveriser.
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?