Mathematical Methods for Physics and Engineering - Matematica.NET

31.2 SAMPLE STATISTICS188.7 204.7 193.2 169.0168.1 189.8 166.3 200.0Table 31.1 Experimental data giving eight measurements of the round triptime in milliseconds for a computer ‘packet’ to travel from Cambridge UK toCambridge MA.31.2.1 AveragesThe simplest number used to characterise a sample is the mean, whichforNvalues x i , i =1, 2,...,N, is defined by¯x = 1 NN∑x i . (31.2)In words, the sample mean is the sum of the sample values divided by the numberof values in the sample.◮Table 31.1 gives eight values for the round trip time in milliseconds for a computer ‘packet’to travel from Cambridge UK to Cambridge MA. Find the sample mean.Using (31.2) the sample mean in milliseconds is given by¯x = 1 (188.7 + 204.7 + 193.2 + 169.0 + 168.1 + 189.8 + 166.3 + 200.0)8= 1479.8 = 184.975.8Since the sample values in table 31.1 are quoted to an accuracy of one decimal place, it isusual to quote the mean to the same accuracy, i.e. as ¯x = 185.0. ◭Strictly speaking the mean given by (31.2) is the arithmetic mean and this is byfar the most common definition used for a mean. Other definitions of the meanare possible, though less common, and include(i) the geometric mean,i=1(ii) the harmonic mean,( N) 1/N∏¯x g = x i , (31.3)¯x h =i=1N∑ Ni=1 1/x , (31.4)i(iii) the root mean square,¯x rms =( ∑Ni=1 x2 iN) 1/2. (31.5)1223