Distribution of component values with a given tolerance?
You can make no assumptions about distribution within the specified range.  50 &Omega; &plusmn;1% means exactly that.  Since 1% of 50 &Omega; is 500 m&Omega;, the manufacturer is saying that any one resistor you get will be from 49.5 &Omega; to 50.5 &Omega;.  You can't read into it or assume more than that.

Added:

Some people have pointed out that they have gotten tightly clumped values from a batch.  I have seen that too.  However, that changes nothing.

Depending on the type of part and the manufacturing, testing, and binning processes, you might get a tight distribution within a batch.  But the most important word is "might".  There is no guarantee, and just because one batch was tight you can't make assumptions about the next batch.

Consider a few different manufacturing scenarios:

The production process has good tolerance, so parts are made to specific values.  Each part is tested, and the rare outlyer is discarded.  In this case you probably do get something like a normal distribution.  The center might not be in the center of the range though, depending on the temperature, phase of moon, and species of dead fish waved over the equipment during the run.

The manufacturer sells different tolerance grades, with the high tolerance having a higher price.  Let's say the equipment can make 1% resistors reliably enough, but not as tight as .1% reliably.  In this case the manufacturer measures each unit and the ones within .1% are labeled and sold as such and the rest are labeled and sold as 1%.

In this scenario, the .1% parts probably have a fairly even distribution accross their range.  The 1% parts have more of a normal distribution, except that there is a gap within .1% of the ideal value.

The production process has wide variation.  Each part is tested and sold as the value it happens to fall within.  In this case you'd get a fairly even distribution within each tolerance band, but it may take a large number of parts to see that distribution.



When you don't know anything about the manufacturing process, there is nothing you can assume other than each part will be somewhere within the specified range.  You have to consider the value of each part as a separate uncorrellated random event.  Sometimes there might in fact be some correlation between sequential parts, but since you don't know when that is, you're still back to having to assume there isn't.  Even if you measure one batch and find a correlation, the next batch is a separate random event for which the data from the previous batch is unrelated.  Again you can't assume anything.

In summary, if you need to know more than the accuracy specified by the manufacturer, you have to measure each part individually.

Each time you flip a coin the result is random and uncorrelated to other times, but you can still get 3 heads in a row often enough to look like a pattern if you don't think about it carefully.

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