In The Trenches
point. Since all normal distributions
have the same shape, you can define
any particular distribution with a single
number that identifies how much it
varies. This term is called the "variance."
The square root of the variance is
more useful from a practical
standpoint and is called the "standard
deviation" or "sigma." Given a normal
curve, 70% of the resistors will be
within (±) one standard deviation of
the average. Over 95% of the resistors
will be within two standard deviations.
That doesn't mean that 5% of the
resistors will be outside of their
tolerance rating. The manufacturers
make certain that virtually all of the
resistors meet their rating. They used
to do this by measuring them. Of
course, that's too expensive to do
today. Instead, they employ very
stringent standards so that the manufacturing process guarantees that the
resistors meet the specifications.
They do this by defining a very
narrow distribution that is so tight that
the actual specified resistor tolerance
limit is often six standard deviations
away from the average. A name for
this is "six-sigma manufacturing" and
it uses "statistical quality control."
This means that there are only a few
chances in a million that any resistor
is outside the tolerance rating.
Non-normal
Distributions
Non-normal distributions are
unusual, but, when they occur, the
results can be quite peculiar. Let's
examine 1K resistors from way back
when the 20% tolerance was the standard. If you bought a bag of 200 and
measured them, the average would
be 1,000 Ω. That's what you'd expect.
While measuring them, however,
you'd notice that there weren't any
resistors between 900 Ω and 1, 100 Ω.
Instead of one distribution, you had
two — one from 800 to 900 Ω and
another from 1, 100 to 1,200 Ω.
Worse yet, these distributions weren't
normal, either. They were lop-sided.
The reason for this is very simple.
The resistor manufacturers measured
MAY 2004
every resistor and sorted them out
according to their tolerance. Better
tolerance resistors sold for more
money. It was very cost effective for
manufacturers to skim the lots for
precision parts.
However, this meant that circuit
analysis was compromised. A
nominal 20%, 1,000 Ω resistor would
never be 1,000 Ω. It would either be
between 800 and 900 Ω or between
1, 100 and 1,200 Ω. I think you can
see the problems this could cause
when resistor ratios needed to be
made (as in a simple voltage divider).
Modern day components may
also have non-normal distributions.
This is especially true for tape and
reel components.
Here's a real story (no pun
intended): I had designed a very
low-power transmitter that needed a
small chip capacitor of 6 to 10 pF for
peak tuning. I knew that the final
value depended on manufacturing
processes. I was surprised when the
manufacturer said it couldn't be
peaked. I visited the plant to determine
the problem. It turned out that the " 10
pF" capacitor had a tolerance of 20%
and was actually 9 pF. They also tried
using two " 4 pF" capacitors in
parallel, but these had a tolerance of
±0.75 pF and were actually 4. 5 pF
each. Two of these capacitors, in
parallel, had a total of 9 pF, too. It's
easy to see why they had a problem.
Everything they tried was actually
exactly the same. All the capacitors
tested in the 10 pF reel were 9 pF and
all the 4 pF were 4. 5 pF. Why?
The manufacturing process for
the capacitors doesn't vary much
from piece to piece. Instead, there are
drifts and changes over time. This
means that capacitors that are
manufactured at virtually the same
time will have very similar values.
Tape and reel packaging forces
components made at the same time to
be in close physical proximity until
they are used. In this case, the normal
distribution refers to millions of components over a long period of time. The
statistical problem here was that the
"sample size was too small."
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