Appendix 7

Standard Deviation

If more than 5 measurements are taken, then you can use statistical analysis to obtain the statistical uncertainty known as the standard deviation of the sample (denoted by the symbol s_s)where

where x_i is the i^th measurement of the quantity,

µ is the mean value of the n measurements and

∑ is the standard mathematical symbol for summation.

Or, the standard deviation of the mean (a somehow better quality measure of uncertainty, denoted by the symbol s)where s = s_s/√n .

Calculation of uncertainty s using statistical analysis.

You can use this method with five measurements, but theoretically it is only valid for twenty or more values.)

Note: What follows is a description of how the standard deviation is obtained. However, we will use Excel or calculators to obtain this value, not the following method.

Example: Suppose nine measurements for a length were made. Here is how to calculate the uncertainty s, with µ = (1/n)(x₁ + x₂ + x₃ +...+ x₉)

Deviation Deviation

Trial # Length (x_i) from mean (x_i- µ) squared (x_i-µ)²

1 14.24 .02 .0004

2 14.21 - .01 .0001

3 14.23 .01 .0001

4 14.23 .01 .0001

5 14.22 .00 0

6 14.23 .01 .0001

7 14.21 -.01 .0001

8 14.24 .02 .0004

9 14.21 -.01 .0001

------- ------

sum 128.02 .0014

µ = 128.02/9 = 14.224 and s = (1/9)(.0014)^1/2 = .004

so the length would be stated as µ + s = 14.224 + .004

Yeah but, just what is the standard deviation? Well, it's sorta the average distance of any answer from the mean, but it's actually more than that.

Confidence Intervals and Reporting Uncertainty

Standard Deviation of the Mean

To get a good estimate of some quantity you need several measurements, and you really want to know how uncertain the average of those several measurements is, since it is the average that you will write down (as a best estimate). This uncertainty in the average is known as the standard deviation of the mean or S.D. M. (s) for short.

It is this quantity that answers the question, "If I repeat the entire series of N measurements and get a second average, within what distance from the original mean would I be willing to bet that this second average would lie 68 % of the time, (i.e. when do I have a 68% confidence that this second average would come close to the first one)?" The answer is that you should expect a second average (that results from redoing the set of measurements) to have a 68% probability of lying within one S.D.M. of the first average you determined. Thus, the S.D.M. is sometimes referred to as a 68% confidence interval.

Once you know the Sample Standard Deviation (S.D.) it is simple to calculate an estimate for the standard deviation of the mean or S.D.M. This is simply the sample standard deviation of the sample of N measurements divided by the square root of N.

It is also referred to at times as the standard error. Since the S.D.M. is actually a measure of uncertainty rather than of an error (in the sense of a mistake), we prefer not to use this term.

The 95% Confidence Interval or S(95)

Suppose we wanted to be 95% sure rather than 68% sure that another average was in a certain range of an average. In fact, often when you are asked to report data based on measurements, we would like to have you report the mean or average along with a 95% confidence interval with a "± " (plus or minus) sign in front of it. We will use the notation S(95) for this quantity.

If you were to take a several hundred or more of data points, S(95) would be very close to twice the standard deviation of the mean. For a limited number of measurements, S(95) and twice the S.D.M. are not the same, but the tend to be similar to each other.