# Statistical Diagrams in Micromine. Examples of Application

Statistical analysis, including statistical diagrams, are crucial means in geology for visualizing and interpreting data, leading to a better understanding of exploration or production project geology. Micromine provides a wide range of statistical graphs, charts, and plots to easily and quickly analyze complex geological data, identify patterns and make informed decisions for further project development. Let is consider some specific examples of the geological application of statistical diagrams, as follows:

**Box and Whisker Plot**for Fe content analysis by rock types;**Multivariate Histogram**for combined visualisation and analysis of four components (SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}and TiO_{2});**Multi-purpose Chart**for strategic planning of mining operations;**Ternary Diagram**for petrographic classification of rock samples by chemical composition;**Multiple Scattergram**for correlation analysis of Fe, Zn, Cu and Ag;**Boundary Analysis Diagram to verify the boundary of the mineralised body by Au and Ag grades**;**Top Cut**for cutting extremely high gold grades;**Swath Plot**for comparing of raw samples and the block model by gold grade;**Cross Validation**for resource model validation;**Gantt Chart**for drilling operations scheduling.

**Box and Whisker Plot (or Box Plot)** graphically describe the spread and means of one or more distributions of numerical data. These plots provide a general summary of the data distribution, highlighting essential statistical features that are important for geological analysis.

A Box and Whisker Plot It displays the distribution of a dataset based on five summary statistics: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The “box” shows the interquartile range (IQR), which represents the middle 50% of the data, and the “whiskers” extend to the smallest and largest values.

Let’s consider a box and whisker plot for the total Fe content by rock type within a deposit. This plot allows us to compare visually and statistically how rock types differ in terms of average Fe content and range of Fe content.

Box and Whisker Plot is also used to compare parameters of multiple datasets, such as geological formations, ore types, domains, etc., which can be compared not only by grades but also by density, grain size distribution and other properties. The plot can be used to analyse geochemical data to identify patterns of chemical distribution and anomalies.

**Multivariate (Comparative) Histogram** is a statistical tool used to visualise the joint distribution of multiple variables or multiple data sets simultaneously. Unlike univariate histograms, multivariate histograms allows to analyse multiple variables and relationships.

Let’s review a multivariate histogram of the four components SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, and TiO_{2}. In this case, the input data are several fields of laboratory testing results table, but the fields of several such tables can be used. The X-axis shows the value of the content of the components on a logarithmic scale, the Y-axis shows the frequency in percent. Histograms of individual values can be displayed in one window (left) or in separate windows (right). By quickly visualising several components at once, their relationships and correlations becomes easier to analyse.

Multivariate histograms are used to compare the distributions of several variables or several data sets with each other. This allows you to highlight the similarities or differences in the distribution of one parameter of different groups (raw samples, composites and model blocks by grades) or different parameters of the same group (chemical element contents, physical properties, etc.). Individual settings of bin sizes, colouring, and other parameters make the visualisation comfortable to analyse.

**Multi-purpose Chart **is a versatile graphical tool designed to display various types of data and relationships in a single chart. These charts can combine multiple data visualization techniques, such as bar charts, line graphs, scatter plots, and more, to provide a comprehensive view of the information. Multiple axes can be used to compare different data sets with different scales.

One example of multi-purpose chart use is an 8-year production plan based on a strategic mining plan (or optimization report). The X-axis shows the time periods in years, and the Y-axis shows the ore and waste rock tonnage (as bars) and net present value of the mine (as a line). This universal diagram shows the dynamics of annual mine production and the gradual increase in profits.

Multi-purpose charts are used to combined display of different values or parameters using different designs (lines, dots, columns, etc.). Such charts are used, in particular, to visualise the pit or stope optimisation results, Swath analysis, key parameters of the mine life cycle (LOM – Life of Mine), etc.

**Ternary Diagram (****Triangle Plot****)** is a graphical representation used to display the proportions of three variables that sum to a constant value, often 100%. It consists of three axes, each representing one of the components in the three-component system. The position of each point is determined by the ratio (percentage) of the three components.

The Ternary Diagram example shows the classification of rock samples by the content of the three components SiO_{2}, Al_{2}O_{3} and Fe_{2}O_{3}. As a result, the samples were classified as ferrite, bauxite, kaolinite, laterite or intermediate rocks. The petrographic variety fields are created graphically directly on the diagram, and the classes can be automatically saved to the input file and viewed in the map window. In this case, the classification results can be used, for example, for further geological mapping.

Ternary diagrams are also used in mineralogy and petrography to show phase equilibrium, phase transitions and stable states and to identify minerals and rocks types based on their phase composition or conditions. A well-known example is the QAPF Diagram – is a double ternary diagram used to classify igneous or volcanic rocks based on their mineralogical composition, which name refers to the classification minerals: Quartz, Alkali feldspar, Plagioclase, Feldspathoid.

**Multiple Scattergrams **is several graphs, each showing the relationship between two variables, and the position of each point corresponds to the values of the two variables. It shows the pairwise correlation of several variables at the same time.

The figure below shows a multiple scattergram illustrating the correlations between four chemical elements in rock samples, namely Fe, Zn, Cu and Ag. The scatter of the points characterises the correlation between two chemical elements in this population. The diagram shows a high degree of Zn-Ag correlation, medium Cu-Ag and Cu-Zn correlation, and low Fe-Ag, Fe-Cu and Fe-Zn correlation.

Multiple Scattergrams allow analysing the correlation of different parameters in a data set or different data sets for a certain parameter. For example, it can be used to compare several types of duplicates (field, coarse, analytical, etc.) during QA/QC analysis.

**Boundary (Contact) Analysis Diagram** is a tool for delineating and visualizing the boundaries between different geological units or zones. This type of diagram helps to analyse changes in geological properties at these boundaries and determine the contact nature (hard or soft). The Boundary Analysis works using spatial statistics of the investigated domains with equal binning of data from the contact (zero distance) at regular distance increments until the maximum distance is reached or no data is available.

A boundary analysis diagram shows the change in gold and silver grades at the boundary of low grade mineralisation. The input data is a assay table with the coordinates of the sampling points and a mineralisation wireframe. The X-axis shows the distance to the boundary (contact), and the Y-axis shows the corresponding grade values, columns show the number of points used for analysis. This diagram shows the soft (not strong) boundary in terms of gold and silver grades.

Boundary or contact analysis charts can be used to define boundaries between different data sets of groups according to any defined parameters. It is often used in geological and structural mapping to identify the boundaries between different lithological and stratigraphic rock types and to define the limiting tectonic structures. During the mineral resource estimation, such diagrams help to verify the correctness of the definition of mineralised (ore) bodies and structural domains.

**Top Cutting**, also known as **Capping**, is the practice of limiting the maximum grade values of samples to avoid the disproportionate affect of extremely high values on the average grade and resulting resources. Typically, extremely high grades is common for gold and other precious metals deposits.

Let’s consider the hurricane limits for gold grades using six diagrams, namely: Histogram, Cumulative Frequency, Probability Plot, gMean vs Cut, CV vs Top Cut, Relative Nugget vs Top Cut. The top cut grade level is determined by the graphical method in each of the presented diagrams, varies from 5.3 to 17.56 ppm and averages 7.73 ppm.

Cut-off grades are used in mineral resource estimation to normalise data in statistical analysis, reduce the impact of extremely high values on the quality of geostatistical analysis and variography, and avoid overestimating the average grade and resources and, as a result, potential revenues.

**Swath Plot** is a graphical display of a value or parameter distribution from a series of bands, or swaths, generated in several directions.

Let’s consider using a Swath plot to compare the raw samples and the block model by gold grade within latitude 30 m wide swaths. The input data for this plot is an assay file and a block model. The X-axis shows the exploration profile name, and the Y-axis shows the number of data points and the average Au grade. The plot shows the amount of data (samples and model blocks) as bars, and the average gold grade for the swath as a line. In this case, the number of model blocks is much higher than the number of original samples, which is common, and the changes in average grade from south to north have a similar trend.

Swath plots are commonly used in resource estimation for block model validation by comparing with raw data. But in general, such plots can be used to identify spatial trends in any geological data, as well as to compare datasets by certain parameters.

**Cross Validation (Rotation Estimation or Out-of-Sample Testing) **is a statistical method used to evaluate the precision and reliability of a predictive model. Some data points are temporarily removed from the sample, their values are estimated using the remaining data points, and the estimated values are compared with the actual values.

The example below shows four plots: True vs Estimate, Sample location plan, Standard Error vs Estimate and Standard Error distribution histogram.

The graphical part of the cross-check (diagrams) is followed by a generating report and a data file, which help to analyse and interpret the results in more detail. So, the report contains such parameters as the Mean of raw and estimated values, the Mean Error Statistic and the Standard Deviation of the Error Statistic, etc. The closer the Mean Error Statistic is to 0 and the Standard Deviation of the Error Statistic is to 1, the better the test results are. The data file contains for each point: coordinates, raw and estimated values, residual and Error statistics.

In general, cross-validation helps to evaluate the effectiveness of spatial interpolation methods and select optimal kriging parameters, such as search radius, variogram model, number of data points, etc.

**Gantt Chart **is a graphical representation of activity against time that is used to plan tasks and monitor progress. The timelines are horizontal bars and form a bar chart.

The figure below presents a drilling Gantt chart for optimal drilling rig allocation and drilling scheduling in time. The X-axis shows the time scale (days, weeks, months, etc.), with each bar representing a time period during which drilling operations are planned. The Y-axis represents the different drilling rigs, each row of the chart corresponds to the individual drilling rig and shows the drilling schedule represented by bars. The Gantt chart allows to optimise the use of drilling rigs by monitoring the drilling operations progress, identifying resource allocation conflicts and schedule delays.

The Gantt chart is also used extensively to manage geological exploration, planning and coordination of mining or any other operations. The diagram allows to visualise the sequence and timeline of any activity for effective management and control.

Micromine offers extensive statistical processing functionality, including statistical charts that allow you to visualise, analyse and interpret geological data for deeper understanding.

This review does not cover all types of diagrams, but only some examples of how they can be used to solve certain geological problems. The choice of a particular type of chart or graph should be appropriate to the specific study context and objectives.

For more details about all types of Micromine diagrams, please follow the corresponding publication “Statistical Diagrams in Micromine. Definitions and functionalities”.