# Standard deviation as uncertainty of a single measurement

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?

• How could you know it's 2 stds away if you'd only taken one measurement before it??? Sep 26, 2017 at 4:42
• 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. Sep 26, 2017 at 7:44