Nuts & Volts - May 2004

by Gerard Fonte

In The Trenches

The Business of Electronics Through Practical Design and Lessons Learned

In The Trenches

Statistics — Part 1

Statistical analysis is an extremely powerful tool. It is important for an engineer to be familiar with techniques and methods of statistical analysis. Statistical procedures are often used to define reliability, but they are also very useful in signal processing.

Statistics Equates to Boring

I don't think I've met anyone (including myself) who liked any course in statistics. The professor lectures on and on about Type I and Type II errors and the "Null Hypothesis." It wasn't until years after my classwork that I understood the fundamental principles of statistics. At that point, statistical analysis became almost intuitive. It also became easy and I was able to apply it to many areas — including playing poker.

Statistics can be used to improve the signal-to-noise (S/N) ratio by an arbitrary amount. You can use statistics to make an eight-bit analog-to-digital (A/D) converter into one with 10 to 12 bits of resolution. Statistical process control allows manufacturers to make products with one in a million failure rates. You can use statistical modeling to see if your circuit will fail.

A Different Definition

My practical definition of statistics is: methods for extracting a signal from noise. When you take an average (the heights of people, for

MAY 2004

example), you are pulling a common value from a group. You can view an average as a signal that represents the group. More specifically, the individuals of the group are not identical, so there is some variation. This variation is typically defined as a "normal" distribution. The probability function of this distribution has a more familiar name; it's called a "Gaussian" distribution. Ever hear of Gaussian noise?

Let's look at a more concrete example. We say that a signal has a frequency of 1 MHz, but take a close look at a 1 MHz signal with a spectrum analyzer. As you narrow the bandwidth of the spectrum analyzer, the 1 MHz signal changes from a line into a bell-shaped curve. This is a Gaussian function — or a normal distribution! The "1 MHz" signal is really an average of many signals. The "cleaner" the signal, the narrower the distribution and vice versa. (This example is rather simplified.)

Now, suppose you have two signals — one is 1 MHz and the other is 1.00001 MHz. Using the spectrum analyzer, can you see if there are two different signals or only one? It's clear that you will need a very narrow bandwidth — about 10 Hz — to determine this because there is only 10 Hz between the two signals. Suppose you didn't have a 10 Hz bandwidth; you wouldn't see the second signal and you'd think only one signal was present. (This is also known as a "Type II" error.) In other words, you couldn't find the signal because of the noise (or the other signal). Perhaps you can now start to see why statistics is important.

Basic Rules

The spectrum analyzer performed all of the number-crunching for you. You simply turned it on and used it. Many statistical procedures require you to do the math yourself. In order to do this properly, there are rules that have to be followed.

The first is the concept of significant digits which is also called resolution or rounding error. You cannot increase the precision of measurements beyond what was measured (except for a special procedure, detailed later). My desk is 33 inches tall. That's two significant digits. It really means that the desk is between 32. 5 and 33. 5 inches high — or ±0.5 inches. Now I convert that into centimeters and get 83. 82 centimeters — but that's four significant digits. I clearly didn't measure my desk to 0.01 centimeters or 0.004 inches. So I have to say that my desk is 84 centimeters high, maintaining two significant digits. Obviously, many calculations will exceed the precision of the measurement, but the final number must not do so. If I wanted to show that I really did measure my desk precisely, I would add the appropriate number of digits to the measurement — or 33.000 inches.

Number Classes

Not all numbers are created equal. Some are more useful than others; these are divided into four classes: Nominal, Ordinal, Interval, and Ratio (NOIR, which is French for black).