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If we have made the following several measurements of the quantity x that follow a normal distribution:

86, 85, 84, 89, 85, 89, 87, 85, 82, 85

The mean is 85.7 and the standard deviation(std) is about 2.

My textbook says that if we take a second measurement, then its uncertainty is the standard deviation. Could someone explain to me the logic behind this. I know there is a 68% confidence for the measurement to be 1 std away from the correct value so if my measured value is actually 2 stds away from the correct value then using the value of 1 std as an uncertainty doesn't even give range that includes the correct value in fact it is way far from the correct value. So could someone explain why we use the standard deviation as the uncertainty of individual measurements?

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  • $\begingroup$ How could you know it's 2 stds away if you'd only taken one measurement before it??? $\endgroup$ – Señor O Sep 26 '17 at 4:42
  • $\begingroup$ It is common to assume that any measurement can be decomposed as the true value plus a random variable following a normal distribution centered in zero. Therefore, having only two measurements of a quantity gives you an estimator for the true value, which is the mean of your values, and the std of the underlying (assumed) distribution, the error. So I think the answer lies in reversing the question: the error is used as an estimator for stds. $\endgroup$ – G.Clavier Sep 26 '17 at 7:44
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if my measured value is actually 2 stds away from the correct value then using the value of 1 std as an uncertainty doesn't even give range that includes the correct value

That's absolutely true. But the 1-sigma range, as we call it - that is, the range of one standard deviation surrounding the measurement - won't always contain the true value. Actually, under normal circumstances (obscure pun intended), the 1-sigma range will include the true value only 68% of the time.

In theory, any individual measurement could be arbitrarily far away from the true value, even 57 standard deviations. So you can't give an absolute upper and lower bound for how far off a measurement will be. You have to use some other way of characterizing how much the measurements are spread away from the true value, and the standard deviation is just that.

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  • $\begingroup$ Thank you for the answer. My textbook uses measuring a length as an example of normally distributed measurement. However last time I check, every time I measure a length I get the same best estimate and then I can estimate the uncertainty without having to take the measurement many times. Is the use of length as an example is just for simplification or is taking many measurement of a length truly done in lab. $\endgroup$ – NegativeTension Sep 27 '17 at 2:43
  • $\begingroup$ That sounds like a followup question that may be best discussed in Physics Chat. I'm not sure I could do justice to it in the comments. $\endgroup$ – David Z Sep 27 '17 at 2:50

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