When a process variable has only random variation each output is independent of the previous ones. This is what happens in a lottery.  In some processes this independence does not happen. If we control our daily weight, for instance, our weight today is correlated to the weight of the previous days: it has autocorrelation

A similar effect happens when you control a heavy aircraft or a ship: the weight prevents you from making a sharp turning to change the course. This opposing force to change is what is called Inertia

The inertia definition applies to moving objects and it is proportional to the object mass. But inertia also applies to fluids: a tank accumulating a fluid will also have this inertia effect.

 If we try to control a process with only random variation by reacting to every output we can see in Process Control that the process will get worse: variation will increase. 

We will now experience how to control a process with inertia with a simulator of the tilt control in a plane:

Download Inertia simulator.xlsm

Close other Excels and allow macros to run it.

Press Start to start simulating and place the cursor on top of the vertical arrows to adjust the tilt by shifting up/ down (do not click).

You should try to keep tilt as close to zero as possible.

The graphs below will show you the adjustments you made and the actual tilt evolution along the 50 runs

The Average and StdDev on top will tell you the extent of your success.

Response delay

The first thing you will notice is that there is a delay between your actions and the tilt response 

This is the result of inertia: the response is slow. 

Stabilising effect of inertia

 In this graph we can see the random source of variation and the resulting tilt without adjustments. We notice that inertia has produced a stabilising effect reducing drastically variability. This can be confirmed by the histograms:

We notice that inertia has caused a reduction of standard deviation from 18.4 to 2.6. We also notice that Tilt (with autocorrelation) passes the normality test (p = 0.21).

Now let's look at the stability of tilt: 

Variation has been drastically reduced but the source of variation was in control and now we have situations of out of control as shown by the Individuals Control Chart.

These tilt out-of-control situations have very limited range compared with the source range of variation.


Another effect of inertia is autocorrelation:

We notice that the random source of variation had no autocorrelation but inertia has caused significant tilt autocorrelation.

Automatic Control of Process with Inertia

If we automatically balance by making Adjustment = - Tilt: 

the result will be reasonable as compared to the case of no adjustment: 

We obtain very similar standard deviations: 3.35 Vs 4.00.

 Now we can experience the difficulties of achieving an effective balancing:

It is difficult to predict when adjustment is required and how much to adjust. 

Real Life: Unstable Source of Variation

So far we have assumed that the source of variation was random with an average of zero.

Now let us try something closer to reality:

Download Inertia variation.xlsm

Try to achieve balance now. 

With automatic Adjust = - Tilt we can achieve:

You can try other automatic adjustment formulas as a function of tilt by modifying the formula in A6.

Continuous Process Balancing by Accumulation in a Tank

By accumulating a fluid in a tank we achieve the autocorrelation effect which is useful to reduce the standard deviation of a critical metric.

Factories often need to drain water to a river or to the sea in which case they have to comply with regulations about its pH. 

Pure water has a pH of 7. Some local regulations require that water pH should be between 5.5 and 9.5 before it can be drained into a river.

In the following example accumulation in a tank has been used in order to reduce the pH standard deviation and meet the required specs. We have done a capability analysis of both the input to the tank and the output drained to the river:

Ppk has increased from 0.26 (totally unacceptable) to 1.15. Total ppms are reduced from 342,594 to 285.

Looking at autocorrelation:

We can see significant autocorrelation produced by the tank accumulation.

Looking at pH stability:

We notice clustering and trends but this happens well within the spec limits.

Body Weight Autocorrelation

 Looking at the data used in


Let's check it for autocorrelation:

We confirm that body weight is indeed autocorrelated: our weight today is correlated to the weight we had in the previous days: it has inertia. 


  • Process inertia shows with autocorrelation: metric values are dependent of previous values
  • Inertia causes a delay between cause and effect
  • The standard deviation of the sources of variability is reduced in the effects
  • Mechanical inertia is proportional to the mass of the object
  • Fluid inertia is proportional to the volume of its storage in a tank or pond
  • Mechanical inertia is used by flywheels to smooth rotation speed
  • Fluid accumulation in a tank is used to reduce the standard deviation of its parameters