A histogram that’s skewed to one side, a control chart that’s outside its limits, a supplier reporting a defect rate that you can’t verify—these are situations that every quality manager or production manager encounters on a regular basis. The descriptive statistics are the tool that allows us to turn these observations into well-reasoned decisions rather than mere impressions. They are not used to do math for the sake of it: they are used to determine whether a process is under control, whether a part is truly out of specification, or whether a batch of 10,000 parts needs to be sorted after inspecting 60 samples.
This guide covers the practical applications of descriptive statistics in industrial production, based on the methodology developed by Maurice Pillet, co-founder of Ellistat, professor emeritus, and a leading authority in France on quality and Six Sigma.
Descriptive, inferential, and multivariate statistics: Where does your day-to-day analysis fit in?
When collecting production data, you typically end up with a table containing inputs (the Xs, or process parameters) and outputs (the Ys, or product characteristics). There are three main types of statistical analysis you can use to make sense of this data, and it’s helpful to know which one you use on a daily basis.
- Descriptive Statistics focus on a single variable at a time, either X or Y, whether it is quantitative (a measurement) or qualitative (a defect, a category). This is the most common and most immediate level of analysis.
- Inferential Statistics seek a non-random relationship between variations in a Y and one or more Xs, generally to explain or model the behavior of Y.
- Multivariate Statistics process the entire table without distinguishing between input and output, in order to classify individuals or identify relationships between variables.
Descriptive statistics are therefore the essential starting point: before looking for a (inferential) cause or an overall (multivariate) structure, you must first know how to correctly interpret a single column of data. This is where most day-to-day quality decisions are made.
Before you analyze, really take a close look at your data
Visual Identification Using Conditional Formatting
The first step—which is often overlooked—is simply to look at the data table with conditional formatting applied. The principle is simple: the values are colored according to their position in the distribution, ranging from blue to red, with green in between.
A trained eye immediately spots a problem: a column dominated by blue, with little green, indicates a distribution that likely deviates from a normal distribution. A column dominated by green, with a high concentration in the center and the extremes in blue and red, on the other hand, resembles a well-centered normal distribution. This quick glance, before any calculations are made, already sets the direction for the analysis to come.
Automatic Outlier Detection
A point located outside the “whiskers” of a box plot is not necessarily an outlier in the statistical sense. The distinction matters: a true outlier must be identified by a dedicated statistical test, not just by a rough visual assessment. It is this test—not visual impression—that should trigger an investigation or the exclusion of the data from your capability calculations.
Graphical Analysis: Histograms, Box-and-Whisker Plots, and Control Charts
Histogram and box-and-whisker plot for assessing the shape of a distribution
The histogram remains the starting point for determining whether a distribution resembles a normal distribution, particularly when a probability density curve is overlaid on it. The box-and-whisker plot complements this analysis, especially when used for comparison: it allows you to place several characteristics side by side (multiple diameters, multiple batches, multiple machines) and identify at a glance which one behaves differently from the others.
Control Charts: Continuously Monitoring Process Stability
Control charts provide a view over time, whereas histograms provide a static view. They must be able to adapt to real-world conditions rather than the other way around:
- monitoring based on the mean, the median, or a weighted moving average (EWMA), either individually or in combination; ;
- centering can be performed relative to the target or relative to the mean of the data; ;
- Support for variable sample sizes: If the number of replicates changes from 3 to 2, then to 1 depending on the samples, the card must automatically adjust its limits without requiring manual recalculation.
It is this flexibility that distinguishes a control chart that truly reflects your process from a theoretical chart that no longer matches the reality on the shop floor.
Test for normality before calculating your capability indices
Capability indices (Cp, Cpk, Pp, Ppk) and most calculations of the proportion of nonconforming items are based on an assumption about the distribution, most often the normal distribution. Before calculating them, it is therefore necessary to verify that this assumption holds—otherwise, the resulting indices are meaningless.
Three normality tests cover virtually all situations encountered in production:
- the chi-square test, appropriate when the data is stratified and individual values are not available
- the Shapiro-Wilk test, which is appropriate for small samples of up to about 30 items
- the Anderson-Darling test, more robust beyond this sample size
A good analysis tool doesn’t just calculate all three; it identifies and highlights the one that is truly relevant based on the size and structure of your data, to avoid misinterpretation. And if the assumption of normality is rejected, you shouldn’t stop there: certain characteristics—particularly those bounded at zero (flatness, circularity, out-of-roundness)—naturally follow a different distribution, such as the Rayleigh distribution, which then models their actual behavior much better.
MAD: A robust metric that few quality tools offer
The standard deviation is highly sensitive to outliers: a single incorrectly measured or genuinely defective item can artificially inflate it, even multiplying it by a significant factor in an otherwise perfectly stable sample. The MAD (median absolute deviation) does not have this drawback: it remains virtually unchanged in the presence of an outlier, precisely because it is based on medians rather than means.
In practice, the MAD is the nonparametric equivalent of the standard deviation. It is particularly useful for tests comparing variability between batches, machines, or suppliers when your data contains—or is likely to contain—outliers that would skew a comparison based on the traditional standard deviation.
Attribute Charts: Managing Quality When Counting Defects, Not Measurements
Not all quality data are continuous measurements. Count, defect rates, and nonconformities per batch: these attribute data are managed using a dedicated set of charts (P, NP, C, U), and choosing the right chart has a direct impact on the reliability of the monitoring process.
The P-chart, for example, assumes that the only source of variability is sampling. However, as soon as lot sizes vary from one period to the next, this assumption no longer holds, and the P-chart triggers out-of-control alerts that do not reflect the reality of the process. The U-chart, which relates the number of defects to a unit of opportunity, is therefore much better suited to most real-world industrial contexts—in fact, it is the most commonly used chart in practice whenever lot sizes are not strictly constant.
Using discrete distribution laws to validate a sorting or acceptance decision
There are two questions that come up very often in the field, and the laws of discrete distributions provide direct answers to them, without the need to consult a statistical table.
First scenario: Decide whether to sort inventory. You have 10,000 parts in stock. You begin an inspection and take a sample of 60 parts, 2 of which are nonconforming. Do you need to sort through the entire batch? The hypergeometric distribution (adapted when considering the number of units in a finished lot) allows us to provide a confidence interval: In this example, we can estimate that with a 95% confidence level, the inventory would contain no more than approximately 1,010 nonconforming units out of 10,000. This information is far more useful for decision-making than a simple rule of three based on the observed rate.
Case 2: Assessing the risk of accepting a shipment from a supplier. A supplier reports a defect rate of 3 %. You select 50 units for inspection. The binomial distribution This calculation shows that the probability of finding no defective parts in this sample—even if the true defect rate of 3 % is accurate—is approximately 21 %. In other words, finding nothing during the inspection does not in any way guarantee that the lot is in compliance. It is precisely this type of calculation that allows for the proper design of a sampling plan rather than setting it at random.
Predicting a Periodic Phenomenon Using Fourier Decomposition
Some quality issues exhibit a periodic pattern that a simple graph cannot explain, even when it is clearly visible. This is the case, for example, with a watch movement whose amplitude is measured every six seconds: the graph shows regular oscillations, but the cause is not obvious.
Visit Fourier decomposition allows us to reconstruct the signal based on its dominant frequencies and identify which one contributes most to the observed deviations. In this example, the analysis reveals a periodicity corresponding to a rotation of about one and a half minutes—which directly points to the mechanical component in question, a wheel rotating at that frequency and generating the measured disturbances. This is a valuable tool whenever a cyclic mechanical cause is suspected behind an apparently random drift.
Setting Control Limits Properly: Between Traditional and Expanded Limits
This is a point that is often not properly understood, even though it directly affects the effectiveness of a control chart. Conventional control limits, calculated at plus or minus 3 standard deviations based on the law of averages, should never be tightened further: doing so risks disrupting a process that is actually functioning correctly, simply because it is reacting to normal statistical noise.
Conversely, one can allow for some process drift by setting a minimum capability target (a threshold Cpk, for example) and an acceptable beta risk—the percentage of cases in which one tolerates not immediately detecting an exceedance of that target. This makes it possible to define expanded limits, broader than the traditional boundaries.
Best practice is to set control limits between these two boundaries: never narrower than the standard limits, never wider than the expanded limits. This type of calculation also makes it possible to determine the sample size actually needed to guarantee a given capability objective—and it is not uncommon to find that a sample of just one part is sufficient, when the process’s short-term standard deviation is low enough, thereby avoiding unnecessary sampling.
Go Further: Confidence and Prediction Intervals
Here is a common mistake: Based on a preproduction sample (15 units, for example), we calculate a mean and a standard deviation, then estimate the variation in the actual batch by simply applying these values to a normal distribution, as if the sample’s mean and standard deviation were exactly those of the entire population.
This assumption is too strong: the mean and standard deviation calculated from 15 pieces can themselves vary from one sample to another. The method defined by ISO 16269 takes this into account, and the’range of variation The result obtained in this way is consistently wider than that obtained by a simple calculation based on approximately X standard deviations. Ignoring this nuance leads to an underestimation of the true expected dispersion in production, with the risk of discovering the problem after the series has been launched rather than beforehand.
There are two related concepts: the dispersion interval indicates the range containing a given proportion of individuals, whereas the’prediction interval estimates the expected range in a future population of a specified size (for example, a future batch of 500 items). A nonparametric calculation also exists for cases where the normality assumption does not hold.
Why centralize these analyses in a quality management tool rather than in Excel spreadsheets?
Each of these analyses can be accessed individually; many quality managers already perform them, often using Excel formulas they’ve built from scratch or paper statistical tables. The challenge isn’t the method itself, but making it available quickly, reliably, and reproducibly on a daily basis, using data that is constantly changing.
That is the module's role Data Analysis from the Ellistat suite: it centralizes all of these descriptive statistics: graphical analysis, control charts, normality tests, MAD, attribute charts, distribution laws, Fourier decomposition, sampling plan calculators compliant with ISO 2859 and ISO 3951 standards, control limits, and dispersion intervals… all within a single interface designed to remain easy to use, with no more than two levels of menus. The goal is not to add statistical complexity, but rather to reduce it, by making these methods directly usable by quality and production teams without requiring prior statistical expertise.
Conclusion
Descriptive statistics are not an academic exercise reserved for quality experts: they answer very practical questions that production and quality managers ask themselves every day: Is this value really an outlier? Is this process stable? Is this supplier’s batch reliable? Should we sort this inventory? When used properly, industrial descriptive statistics transform raw data into quick, well-reasoned decisions, rather than unverifiable hunches. The challenge is never the statistical theory itself, but rather its simple and reliable application, day after day, to the actual data from your production floor.
