Signal and Noise Examples

Pure Signal Without Noise

Each observed value is assumed to be the sum of two components: a 'signal' component and a 'noise' component. We can never observe these separately, we only know their sum.
This graph is an example of pure signal with no noise.
Values start at $18 million and increase every year by $815,000.
This is what the model says we expect the school appropriation trajectory to be.

Signal with $180,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $180,000.
The noise contributes nothing on average because its mean is zero.
We can never observe the signal and noise separately, only combined.
Up to a point, statistical analysis can give us an estimate of the signal.
As the noise standard deviation gets larger and larger, eventually the signal is overwhelmed and becomes undetectable.

Signal with $280,000 Noise

Signal with $380,000 Noise

Signal with $480,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $480,000.
These are the actual values we see in the School Appropriation data.
At this noise level, the signal is clear and easily detectable, provided you have the right model.
If you are only looking at percentage changes, you have the wrong model and you will most likely miss the signal completely or misenterpret it.
The school appropriation data clearly exhibits the characteristics of linear growth, and does not have the characteristics we expect of constant percentage growth.

Signal with $1,000,000 Noise

Signal with $5,000,000 Noise

Signal with $10,000,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $10,000,000.

Signal with $20,000,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $20,000,000.
The signal is still statistically detectable.

Signal with $50,000,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $50,000,000.
The signal is no longer significantly different from zero.

Signal with $100,000,000 Noise

Signal with noise.
The noise is random with a mean of zero and a standard deviation of $100,000,000.
The signal is not statistically detectable with this much noise.

The School Appropriation Signal with Varying Levels of Noise