Purpose:
Check underlying statistical assumptions The 4-plot is a collection of 4 specific EDA graphical techniques whose purpose is to test the assumptions which underlie most measurement processes. A 4-plot consists of a
  1. run sequence plot;
  2. lag plot;
  3. histogram;
  4. normal probability plot.
If the 4 underlying assumptions of a typical measurement process hold, then the above 4 plots will have a characteristic appearance (see the normal random numbers example below) ; if any of the underlying assumptions fail to hold, then it will be revealed by anomalous appearance in one or more of the plots.
(Go to examples.) Sample Plot:
This 4-plot reveals a process which has fixed location, fixed variation, is non-random (oscillatory), has a non-normal, U-shaped distribution, and has 3 outliers. Importance:
Testing the 4 underlying assumptions helps ensure the validity of the final scientific and engineering conclusions

There are 4 assumptions which typically underlie all measurement processes, namely, that the data from the process at hand "behave like":

  1. random drawings;
  2. from a fixed distribution;
  3. with that distribution having a fixed location; and
  4. with that distribution having a fixed variation.
Predictability is an all-important goal in science and engineering. If the above 4 assumptions hold, then we have achieved probabilistic predictability--the ability to make probability statements not only about the process in the past, but also about the process in the future. In short, such processes are said to be "statistically in control". If the 4 assumptions do not hold, then we have a process which is drifting (with respect to location, variation, or distribution), is unpredictable, and is out of control. A simple characterization of such processes by a location estimate, a variation estimate, or a distribution "estimate" inevitably leads to optimistic and grossly invalid engineering conclusions.

Inasmuch as the validity of the final scientific and engineering conclusions is inextricably linked to the validity of these same 4 underlying assumptions, it naturally follows that there is a real necessity that each and every one of the above 4 assumptions be routinely tested. The 4-plot (run sequence plot, lag plot, histogram, and normal probability plot) is seen as a simple, efficient, and powerful way of carrying out this routine checking. Interpretation:
Flat, equi-banded, random, bell-shaped, and linear Of the 4 underlying assumptions:

  1. If the fixed location assumption holds, then the run sequence plot will be flat and non-drifting.
  2. If the fixed variation assumption holds, then the vertical spread in the run sequence plot will be the approximately the same over the entire horizontal axis.
  3. If the randomness assumption holds, then the lag plot will be structureless and random.
  4. If the fixed distribution assumption holds (in particular, if the fixed normal distribution holds), then the histogram will be bell-shaped and the normal probability plot will be linear.
If all 4 of the assumptions hold, then the process is "statistically in control". In practice, many processes fall short of achieving this ideal. Questions 4-plots can provide answers to many questions:
  1. Is the process in-control, stable, and predictable?
  2. Is the process drifting with respect to location?
  3. Is the process drifting with respect to variation?
  4. Is the data random?
  5. Is an observation related to an adjacent observation?
  6. If a time series, is is white noise?
  7. If not white noise, is it sinusoidal, autoregressive, etc.?
  8. If non-random, what is a better model?
  9. Does the process follow a normal distribution?
  10. If non-normal, what distribution does the process follow?
  11. Is the model     Y = constant + error     valid and sufficient?
  12. If the default model is insufficient, what is a better model?
  13. Is the formula SD(Ybar) = SD(y) / sqrt(n) valid?
  14. Is the sample mean a good estimator of the process location?
  15. If not, what would be a better estimator?
  16. Are there any outliers?
Input:
1 Variable: Y A single variable Y--that is, a single column of numbers. These numbers need not have been collected equi-spaced in time. Definition:
1. Run sequence plot;
2. Lag plot;
3. Histogram;
4. Normal probability plot The 4-plot consists of the following:
  1. Run sequence plot (to test fixed location & variation)
  2. Lag Plot (to test randomness)
  3. Histogram (to test (normal) distribution)
  4. Normal) probability plot (to test (normal distribution)
Examples:
1. Normal random numbers (the ideal)
2. Uniform random numbers
3. Random walk
4. Flicker noise
5. Josephson junction cryothermometry
6. Beam deflections
7. Filter transmittance
8. Standard resistor
9. Heat flow meter 1
10. Heat flow meter 2
11. Heat flow meter 3
Stat Category:
Time Series & Univariate The 4-plot should be used for the routine analysis of both univariate data sets and time series. Its scope is much broader, however, since the 4-plot can be applied to the residuals from any regression or ANOVA to ascertain as to whether the broader underlying assumptions hold for these more general categories. Usage:
Moderate The 4-plot receives moderate usage; due to the importance of assumption testing on the validity of the final scientific and engineering conclusions, the 4-plot should receive much heavier usage. The 4-plot is arguably the most important EDA tool for checking underlying assumptions in the univariate case. Other Issues References Filliben ... Dataplot Command 4-PLOT Related Techniques Run Sequence Plot
Lag Plot
Histogram
Normal Probability Plot

Autocorrelation Plot
Spectral Plot
PPCC Plot
Do Worked Example 1. With our data
2. With you own data