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Multi-vari chart and run chart

Find which family holds the variation — within the part, part to part or over time — and whether the data sequence hides clusters, mixtures, trends or oscillation.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

CheckRevised 11/03/2026
Analysis
Data
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Role of each column

One row per measurement. In the multi-vari chart, the 1st text column becomes Factor 1 (the innermost) and the last numeric column the response; change the roles above if needed. In the run chart, values are read in row order.

Multi-vari options
Run chart options

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Multi-vari chart

Families of variation

Share of the total variation (sum of squares) explained by each factor, from the outermost to the innermost; the rest is repetition within the same combination.

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Means by level and by combination

The numbers behind the plotted points: N, mean, standard deviation, minimum and maximum.

Run chart

Points in collection order with the median. Shaded band = longest run about the median; thick line = longest run up or down.

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P-values of the four tests

Bars below the α line indicate a non-random pattern. Log scale.

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Randomness tests

Runs about the median (clustering and mixtures) and up or down (trends and oscillation), with normal-approximation P-values.

Next step

Take this page’s results to another calculator with the fields already filled in.

The calculation report gathers this page's inputs, results, equations, assumptions and references, with fields for project, engineer and revision. The full report also includes the charts and interpretations, in color. In the print dialog, choose Save as PDF.

Method, assumptions and references

Equations used

Multi-vari: mean of each level combination; Factor 1 connected within each Factor 2 level, Factor 2 connected across groups, Factors 3 and 4 in panels
Families of variation (nested, sequential): SS(Fₖ) = Σ nᵢ(ȳᵢ − ȳ)²; SS(Fⱼ within Fⱼ₊₁…Fₖ) = Σ n(ȳ cell − ȳ parent group)²; SS(repetition) = Σ (y − ȳ combination)²; % = SS ÷ total SS
Runs about the median: m = points > median, n = points ≤ median, N = m + n; E(R) = 1 + 2mn/N; Var(R) = 2mn(2mn − N)/[N²(N − 1)]; Z = (R − E)/√Var; p clustering = Φ(Z), p mixtures = 1 − Φ(Z)
Runs up or down (equal values count as a fall): E(V) = (2N − 1)/3; Var(V) = (16N − 29)/90; Z = (V − E)/√Var; p trends = Φ(Z), p oscillation = 1 − Φ(Z)

Assumptions and limits

  • The breakdown into families depends on the factor order (outermost to innermost) and is descriptive; with crossed factors, each family includes the interactions with the outer factors. To test effects, use ANOVA.
  • Run chart: with subgroups, the tests use each subgroup’s mean (or median); the center line is the median of those points. P-values use the normal approximation, adequate with 10 or more points on each side of the median.
  • Data must be in production or collection order; sorting by value destroys the test.

Standards and technical documents

  • Seder, L. A. Diagnosis with diagrams. Industrial Quality Control 6(4–5), 1950
  • Bhote, K. R.; Bhote, A. K. World Class Quality, 2nd ed. AMACOM, 2000 (multi-vari)
  • Swed, F. S.; Eisenhart, C. Tables for testing randomness of grouping in a sequence of alternatives. Annals of Mathematical Statistics 14:66–87, 1943
  • Wallis, W. A.; Moore, G. H. A significance test for time series analysis. Journal of the American Statistical Association 36:401–409, 1941
  • NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.13 Runs Test for Detecting Non-randomness
  • Montgomery, D. C. Introduction to Statistical Quality Control, 8th ed. Wiley, 2019

U.S. regulations

  • ISO 9001:2015, 9.1.3: analysis and evaluation
  • ISO 7870-1:2019 Control charts — General guidelines

Frequently asked questions

What is a multi-vari chart?

A chart that shows at once how a measurement changes within the part, from part to part and over time. Each point is the mean of a factor combination, and lines join the means of the same group. It helps decide which family of variation to investigate before running experiments.

How do I set up the data collection?

Measure a few consecutive parts (3 to 5) at several positions (ends and center, for example) and repeat at several times (hourly, by shift) until you capture at least 80% of the normal process variation. Record one row per measurement with the position, part, time and value.

How do I read the families of variation?

The page splits the total variation among the factors, from the outermost (hour, shift) to the innermost (position). In the shaft example, the hour-to-hour drift holds most of it: the problem is temporal, typical of tool wear or warm-up, rather than clamping or material.

Does the multi-vari chart replace ANOVA?

No: it is a visual, descriptive tool with no P-value. Use it to choose where to look and then confirm with ANOVA or an experiment. The next-step button takes the same data to ANOVA.

What does the run chart test?

Whether the order of the points looks random. It counts runs of points on the same side of the median (too few = clustering, too many = mixtures) and runs that only rise or only fall (too few = trend, too many = oscillation), with a P-value for each pattern.

How is it different from a control chart?

The run chart uses no control limits; it detects order patterns even when all points are inside the limits. The control chart shows whether the process is stable relative to its natural variation. The two complement each other.

Why use subgroups?

When several parts are measured at the same time (for example, 5 per hour), each subgroup becomes one point with its mean or median, and the tests look at the sequence of times. In the crankshaft example, 125 measurements form 25 subgroups of 5.