Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.2 SAMPLE STATISTICS

188.7 204.7 193.2 169.0

168.1 189.8 166.3 200.0

Table 31.1 Experimental data giving eight measurements of the round trip

time in milliseconds for a computer ‘packet’ to travel from Cambridge UK to

Cambridge MA.

31.2.1 Averages

The simplest number used to characterise a sample is themean,whichforN

valuesxi,i=1, 2 ,...,N, is defined by

̄x=

1

N

∑N

i=1

xi. (31.2)

In words, thesample meanis the sum of the sample values divided by the number

of 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=^18 (188.7 + 204.7 + 193.2 + 169.0 + 168.1 + 189.8 + 166.3 + 200.0)

=

1479. 8

8

= 184. 975.

Since the sample values in table 31.1 are quoted to an accuracy of one decimal place, it is
usual to quote the mean to the same accuracy, i.e. asx ̄= 185.0.

Strictly speaking the mean given by (31.2) is thearithmetic meanand this is by

far the most common definition used for a mean. Other definitions of the mean

are possible, though less common, and include

(i) thegeometric mean,

̄xg=

(N

∏

i=1

xi

) 1 /N

, (31.3)

(ii) theharmonic mean,

̄xh=

N

∑N

i=1^1 /xi

, (31.4)

(iii) theroot mean square,

̄xrms=

(∑

N

i=1x

2

i

N

) 1 / 2

. (31.5)