Descriptive Statistics for Forensic Data

Central tendency describes the location of a dataset on a measurement scale. The three classical measures are the mean, the median, and the mode, and each answers a slightly different question about the data.

The mean is calculated by summing all values and dividing by the count. If a forensic scientist measures the refractive index of ten glass fragments and records values 1.51832, 1.51840, 1.51835, 1.51828, 1.51841, 1.51833, 1.51839, 1.51836, 1.51830, and 1.51837, the mean is the sum divided by ten. The mean is the natural starting point for comparing a case sample against a reference population mean, but it has a well-known weakness: a single outlier, perhaps a contaminating fragment of quite different composition, can shift the mean substantially. The seven other fragments might all cluster near 1.51835, but one fragment at 1.52100 will pull the mean upward.

The median is the middle value when the data are sorted. For the ten values above, sorted in order, the median is the average of the fifth and sixth values. If a dataset contains that contaminating outlier at 1.52100, the median barely changes, because the middle of the sorted list is still dominated by the cluster of similar values. This resistance to outliers makes the median a useful complement to the mean. When mean and median differ substantially, the analyst should investigate why, because the difference usually signals either outliers or a skewed distribution.

The mode is the most frequently occurring value. In continuous measurement data such as refractive index measured to five decimal places, exact repeated values are uncommon, so the mode is rarely informative unless the data are rounded to fewer decimal places or are inherently discrete. Where the mode is informative is in categorical evidence: if twenty soil particles are classified into five mineralogical types and twelve of them are quartz, the mode is quartz. The mode can also reveal bimodality, which would indicate that the sample contains two distinct sub-populations, perhaps glass from two different sources mixed at the scene.

A measure of central tendency alone is insufficient for forensic comparison. Knowing that the mean refractive index of a reference glass population is 1.51835 tells you nothing about how tightly the population clusters around that value. Two populations can share the same mean but have completely different spreads: one might have all values within 0.00005 of the mean, while another has values ranging 0.0015 on either side. These populations are easily distinguishable despite sharing a mean, but only if spread is also reported.

Variance is calculated by finding the deviation of each data point from the mean, squaring each deviation, and averaging the squared deviations. Squaring serves two purposes: it makes all deviations positive regardless of direction, and it penalises large deviations more than small ones. However, squaring the deviations also squares the unit of measurement. If the original data are in micrometres (fibre diameter), the variance is in square micrometres, which is not a natural unit and is difficult to interpret directly.

Standard deviation solves this by taking the square root of the variance, returning the measure to the original units. A standard deviation of 0.3 micrometres means that a typical fibre measurement falls within about 0.3 micrometres of the mean. In a dataset that follows a normal distribution, approximately 68% of values fall within one standard deviation of the mean, and approximately 95% fall within two standard deviations. These properties underpin many of the comparison rules used in forensic laboratories.

In practice, both variance and standard deviation appear in forensic reports. Variance appears in formal statistical tests such as analysis of variance (ANOVA), which compares spread within groups to spread between groups. Standard deviation appears in descriptive summaries because it is expressed in the same units as the measurement and can be understood by non-specialist readers. A report that gives only the mean and standard deviation of a refractive index dataset, without the raw values or the count, is not complete; the underlying data should also be available for independent verification.

Glass refractive index measurement is one of the most studied applications of descriptive statistics in forensic science. The primary technique, temperature-controlled immersion microscopy using oils of known refractive index, and later automated instruments such as the Mettler Toledo refractometer, produces a series of continuous measurements for each fragment. A case sample of twenty fragments from a suspect's clothing is compared against a reference sample of thirty fragments from the crime scene window.

The analyst calculates mean and standard deviation for both the case sample and the reference sample. A simple comparison rule used historically is the two-standard-deviation overlap rule: if the mean of the case sample falls within two standard deviations of the mean of the reference, the samples are said to be consistent. This rule has known limitations, particularly when the reference population database is large and the within-source variation is small, because many different source glasses can produce overlapping ranges. More rigorous approaches, covered in related topics on visualising forensic data and common distributions in forensic science, use likelihood ratios that account for both within-source and between-source variation. Descriptive statistics remain the foundation of these more sophisticated approaches.

A forensic analyst working in the UK follows guidelines from the Forensic Science Regulator's Codes of Practice, which require that the analytical method, the number of measurements, and the summary statistics be reported in full. In the United States, SWGMAT guidelines on glass analysis similarly require reporting the refractive index range, mean, and standard deviation for each set of measurements. In Europe, ENFSI guidelines on glass examination specify the same. These requirements exist because without complete reporting, independent experts instructed by the defence cannot check the work.

Fibre evidence involves both categorical attributes (fibre type, colour, weave construction) and continuous measurements (fibre diameter, twist per centimetre). Descriptive statistics apply to the continuous measurements, while frequency counts and proportions apply to the categorical attributes.

Fibre diameter is measured in micrometres using a calibrated microscope. A batch of fibres from a crime scene garment might yield measurements with a mean of 18.2 micrometres and a standard deviation of 0.8 micrometres. A reference sample from a suspect's garment might show a mean of 18.0 micrometres and a standard deviation of 0.9 micrometres. The overlap of these ranges does not itself constitute a definitive match; it means the measurements are consistent, which is one component of a broader comparison that also includes colour, fibre type, and construction.

When the distribution of fibre diameters is plotted as a histogram, the shape of the distribution often becomes visible. A symmetric, bell-shaped distribution suggests that the variation arises from random manufacturing variation around a target diameter. A bimodal distribution, with two distinct peaks, might indicate that the fibres came from two different batches or even two different garments. The mean and standard deviation alone would not reveal this bimodality; they would describe the overall spread of a dataset that actually contains two distinct sub-populations. This is one reason why descriptive statistics should be accompanied by graphical display whenever the dataset is large enough for a histogram.

Cherry-picking in forensic statistics means selecting only the subset of descriptive statistics, or only the subset of data points, that supports a predetermined conclusion. It is a form of confirmation bias with direct consequences for the reliability of evidence presented in court. The same dataset can tell different stories depending on which summaries are reported.

Consider a forensic fibre analyst who measures fifty fibres from a crime scene and fifty from a suspect's garment. Forty-five of the crime scene fibres have diameters between 17 and 19 micrometres; five are outliers between 22 and 25 micrometres. If the analyst reports only the mean and standard deviation of the forty-five consistent fibres, excluding the five outliers without explanation, the report gives a misleadingly tight spread that makes the case sample appear more consistent with the reference than it actually is. Including the five outliers and explaining their significance, whether they suggest contamination, a secondary transfer, or genuine within-sample variation, is what complete honest reporting requires.

The remedy is straightforward: report the full set of measurements (or make them available for inspection), state the number of measurements taken, state any measurements that were excluded and the reason, and report mean, median, and standard deviation for the complete dataset unless there is a documented scientific rationale for excluding specific values. Under the Criminal Procedure Rules in England and Wales, the Bharatiya Sakshya Adhiniyam 2023 in India, the Federal Rules of Evidence in the United States, and the EU's framework for scientific evidence in criminal proceedings, an expert who omits material data from their report is exposed to serious procedural and professional consequences.

The gap between computing a statistic and communicating it effectively to a judge or jury is significant. A statement that the mean refractive index of the case glass is 1.51837 with a standard deviation of 0.00004 conveys nothing to a non-specialist unless the analyst also explains what those numbers mean: that the measurements are tightly clustered, that the standard deviation represents the typical distance of any single measurement from the central value, and how that compares to the reference population.

The preferred presentation in modern forensic evidence is the verbal equivalence scale anchored to a numerical measure. After reporting the descriptive statistics, the analyst states whether the data are consistent or inconsistent, and, where a likelihood ratio framework is used, places that conclusion on a calibrated verbal scale. The descriptive statistics support the conclusion but do not replace it: they are the factual foundation from which the inferential conclusion is drawn. Separating the factual summary from the inferential conclusion is a core requirement of ENFSI guidance on evaluative reporting.

Graphical aids, such as box plots showing the range and interquartile spread, or dot plots showing individual measurements, can help courts understand how the case sample relates to the reference population. Courts in England and Wales have accepted expert graphics as part of evidence for several decades; the US Supreme Court ruling in Daubert v Merrell Dow (1993) and its progeny have reinforced the requirement that the scientific basis of any statistical presentation be clearly explained. Descriptive statistics are the most basic such basis, and presenting them clearly is the starting point for any evaluative report.