Use of σ in quoting measurement accuracy

[cepheid]

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.

 

[jtbell]

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.

 

[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.

 

[stewartcs]

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.

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

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

CS

 

[cepheid]

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)?

 

[stewartcs]

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

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

 

[stewartcs]

" [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)?

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