Run-Sequence Plot

NIST/SEMATECH Section 1.3.3.25 Run-Sequence Plot

What It Is

A run-sequence plot, also called a run chart, displays the measured values in the order they were collected, with the horizontal axis representing the run order or time index and the vertical axis representing the response. It is the simplest and most direct graphical diagnostic for time-ordered data.

The horizontal axis shows run order index (1, 2, ..., N) and the vertical axis shows the measured response value. No smoothing or modeling is applied. The plot is intentionally raw so that any patterns -- trends, shifts, cycles, or outliers -- are visible without being obscured by statistical processing. A horizontal reference line at the overall mean is often included.

Questions This Plot Answers

Are there any shifts in location?
Are there any shifts in variation?
Are there any outliers?

Why It Matters

The run-sequence plot is the single most important first step in any time-ordered data analysis. If the process is not stable (fixed location, fixed variation), then summary statistics like the mean and standard deviation do not characterize a single, well-defined process. Process instability must be identified and addressed before meaningful statistical analysis can proceed.

When to Use a Run-Sequence Plot

Use a run-sequence plot as the first step in analyzing any time-ordered dataset to detect shifts in location, changes in spread, drifts, trends, and oscillations. It addresses the fundamental question of whether the process is stable over time, which is a prerequisite for meaningful statistical analysis. The run-sequence plot is one of the four panels in the standard 4-plot diagnostic and is often the single most informative plot for process data.

How to Interpret a Run-Sequence Plot

For a stable process, the run-sequence plot shows a horizontal band of points scattered randomly around a constant mean, with no visible patterns, trends, or shifts. An upward or downward trend indicates systematic drift in the process. A sudden jump in the level of the data indicates a shift event, such as a tool change, batch change, or environmental disturbance. Increasing or decreasing spread over time indicates non-constant variability. A recurring oscillatory pattern suggests a periodic influence such as temperature cycling, operator rotation, or seasonal effects. The run-sequence plot is intentionally simple so that any departures from randomness are immediately visible.

Examples

Stable Process

Data points scatter randomly within a horizontal band of constant width around the mean, with no visible trends, shifts, or patterns. This indicates a process with fixed location and fixed variation, suitable for standard statistical analysis.

Location Shift

An abrupt jump in the level of the data partway through the sequence, where the mean before and after the shift are visibly different. Common causes include tool changes, batch changes, or environmental disturbances.

Trend

A gradual upward or downward drift in the data over the observation period, indicating that the process mean is not constant. This can result from tool wear, temperature drift, or material degradation.

Assumptions and Limitations

The run-sequence plot requires that the data be recorded in the order of collection. It makes no distributional assumptions and is appropriate for any measurement data. The plot is purely visual and does not provide formal tests; it should be complemented with quantitative methods such as the runs test, CUSUM chart, or control chart when formal change detection is needed.

See It In Action

This technique is demonstrated in the following case studies:

Filter Transmittance Case Study

Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.3.25

Python Example

import numpy as np
import matplotlib.pyplot as plt

# Generate normal data with constant mean and variance
rng = np.random.default_rng(42)
n = 100
data = rng.normal(loc=50, scale=5, size=n)

fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(range(1, n + 1), data, "o-", markersize=3, linewidth=0.8)
ax.axhline(y=np.mean(data), color="r", linestyle="--", label="Mean")
ax.set_xlabel("Run Order")
ax.set_ylabel("Response")
ax.set_title("Run Sequence Plot")
ax.legend()
plt.tight_layout()
plt.show()