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Dealing with Uncertainties: Means, Standard Deviation and Standard Error Measured values are always subject to Error, or Uncertainty. To work with errors we can use some statistical techniques; these will help us understand how confident we can be in our data. 1. The mean is widely used and easily calculated. It is simply the sum of all the readings divided by the number of readings. It is sometimes just called the average; but mean is a better term as there are different types of average. (Note β the word mean has other meanings in English as well, so watch out for this) The mean is often given the symbol µ (Greek letter βmuβ). This is used in Loughborough University. π₯Μ and <x> may also be seen in books. Mathematically, π 1 π = β π₯π π π=1 2. The Standard Deviation is also useful; for a large population of readings, 68% of the readings would be expected to lie within one standard deviation of the mean. This is also called the RMS or Root Mean Square deviation; it is calculated by ο· finding the difference (deviation) from the mean of each reading: = Deviation ο· squaring each deviation = Square Deviation ο· finding the total of these square values ο· dividing this by the number of values = Mean Square Deviation ο· taking the root = Root Mean Square Deviation This is generally given the symbol Οn (βSigma nβ), and is properly called the βpopulation standard deviationβ. In symbols this is 1 n ο¨xi ο ο ο©2 ο³n ο½ ο₯ n i ο½1 3. Slightly more useful is the Sample Standard Deviation Οn-1 which is almost identical but you divide by n-1 not n. This is used in Loughborough University, and is the one given on the formula sheet, as π ππβ1 1 = β β(π₯π β π)2 πβ1 π=1 Finally, to get to standard error, which has the symbol s , simply divide the sample standard deviation by root of n, i.e. π = ππβ1 βπ Quote your answer to an appropriate number of decimal places - look at the data you have to get an idea. Normally only one or two significant figures can be justified. How do I use these? Itβs easier than it sounds, but it does need practice. You can do them manually or use the built in statistical functions on your calculator. These are very good but you look up how to do it! Your calculator will offer you xΟn and xΟn-1. Choose xΟn-1 as it is the sample standard deviation. Practice: For these numbers: 11.47, 11.31, 11.12, 11.06 and 11.10. Calculate the ο· ο· ο· ο· Mean, Population Sample Deviation Sample Standard Deviation and Standard Error Try it out manually (draw a table), then try to get the same results on your calculator. You should get: Mean = 11.212 (round to 11.21) Population standard Deviation: Sample Standard Deviation = 0.173; round to 0.17 Standard Error = 0.07755 β round to 0.08. So the best answer is 11.21 ± 0.08. It is meaningless to add more decimal places. deviation deviation Number (number minus squared mean) 11.47 11.31 11.12 11.06 11.1 total 0.258 0.098 -0.092 -0.152 -0.112 0.066564 0.009604 0.008464 0.023104 0.012544 56.06 There are 5 values so n = 5 mean 11.212 sum of squared deviations 0.12028 sum of squared deviations / (n-1) 0.03007 root 0.173407 this is the Sample Standard Deviation = 0.17 (two 2 s.f.) = 0.08 (rounded) divide this by root n to get standard error: (s) 0.07755