# Use of σ in quoting measurement accuracy

1. Oct 2, 2008

### cepheid

Staff Emeritus
I know that σ is the symbol typically used for standard deviation, but what does the use of σ mean in these contexts?

ex. 1: "[our fancy new instrument] ... allows for the identification of > 500 sources at greater than 10-sigma"

ex. 2: "These estimates assume ... a 1σ polarization uncertainty P = 1%.

Please note that if the symbol doesn't show up for you, it is supposed to be the letter sigma.

2. Oct 2, 2008

### Staff: Mentor

It's the standard deviation of a hypothetical ensemble of measurements of the same quantity, which differ only in random experimental errors, and are assumed to be distributed according to a Gaussian probability distribution.

An interval of $\pm \sigma$ around the ideal mean value contains about 63% of the hypothetical ensemble, and $\pm 5 \sigma$ gets you up to about 95%, if I remember correctly.

3. Oct 2, 2008

### brewnog

Indeed, so in terms of manufacturing defects, one sigma corresponds to the equivalent of 690,000 parts per million failures, four sigma 6,210 ppm, and six sigma 3.4 ppm.

4. Oct 2, 2008

### stewartcs

It's more like 99.99994% for $\pm 5 \sigma$.

http://en.wikipedia.org/wiki/68-95-99.7_rule

CS

5. Oct 2, 2008

### cepheid

Staff Emeritus
I'm still not sure I understand. How do you construct this hypothetical ensemble? And are you saying that each measurement in the ensemble is a single number that differs from the others only *due to* random experimental errors? Most importantly, HOW does this apply to the examples above? I have another one that says:

" [The instrument] has detected > 80 sources per square degree (at 5-sigma)." What does this mean? Or does this have nothing to do with the specific number 80 and more to do with the definition of what a source is as compared to the background (this is in the context of submillimetre astronomy)?

6. Oct 2, 2008

### stewartcs

In this example I believe the statement means that your fancy new instrument will detect source outliers up to 10 sigma from the mean (i.e. sources with a z-score of $\pm 10$). Hard to say without knowing more about the application though.

CS

7. Oct 2, 2008

### stewartcs

I would interpret that as meaning that instrument detected >80 sources per square degree that were within $\pm 5 \sigma$ of the mean. In other words the detected value, y, was within 5 sigma of the mean value. This y value was detected >80 times.

CS