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These results agree exactly with those obtained previously. Analogous to the sample variance s2 , the variability in the population is de ned by the population variance ( 2). As in earlier chapters, the positive square root of 2 , or , will denote the population standard deviation. When the population is nite and consists of N values, we may de ne the population variance as a 1xi N We observed previously that the sample mean could be used as an estimate of the population mean. Similarly, the sample variance is an estimate of the population variance. In 7, we will discuss estimation of parameters more formally. Note that the divisor for the sample variance is the sample size minus one 1n 12, while for the population variance it is the population size N. If we knew the true value of the population mean , we could nd the sample variance as the average squared deviation of the sample observations about . In practice, the value of is almost never known, and so the sum of

( L ORo). ,

(6-5)

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the squared deviations about the sample average x must be used instead. However, the observations xi tend to be closer to their average, x, than to the population mean, . Therefore, to compensate for this we use n 1 as the divisor rather than n. If we used n as the divisor in the sample variance, we would obtain a measure of variability that is, on the average, consistently smaller than the true population variance 2 . Another way to think about this is to consider the sample variance s2 as being based on n 1 degrees of freedom. The term degrees of freedom results from the fact that the n deviations x1 x, x2 x, p , xn x always sum to zero, and so specifying the values of any n 1 of these quantities automatically determines the remaining one. This was illustrated in Table 6-1. Thus, only n 1 of the n deviations, xi x, are freely determined. In addition to the sample variance and sample standard deviation, the sample range, or the difference between the largest and smallest observations, is a useful measure of variability. The sample range is de ned as follows. De nition If the n observations in a sample are denoted by x1, x2, p , xn, the sample range is r max1xi 2 min1xi 2 (6-6)

Mathematically, any round function F that outputs n/2 bits will work However, it is also clear that the security of a Feistel Cipher depends on F , and not every F will result in a secure cipher For example, F(Ri-1, K i ) = Ri-1 will yield an insecure cipher One nice feature of a Feistel Cipher is that all questions of security boil down to questions about the round function F The round function may be simple or complex, but at least the analyst knows to focus on F Many well-known block ciphers are Feistel Ciphers For example, the most famous block cipher in history, the Data Encryption Standard (DES), is a Feistel Cipher Even today, many block ciphers follow Feistel s approach, and many others (such as TEA [158]) vary only slightly from the strict definition given here.

For the pull-off force data, the sample range is r 13.6 12.3 1.3. Generally, as the variability in sample data increases, the sample range increases. The sample range is easy to calculate, but it ignores all of the information in the sample data between the largest and smallest values. For example, the two samples 1, 3, 5, 8, and 9 and 1, 5, 5, 5, and 9, both have the same range (r 8). However, the standard deviation of the rst sample is s1 3.35, while the standard deviation of the second sample is s2 2.83. The variability is actually less in the second sample. Sometimes, when the sample size is small, say n 8 or 10, the information loss associated with the range is not too serious. For example, the range is used widely in statistical quality control where sample sizes of 4 or 5 are fairly common. We will discuss some of these applications in 16. EXERCISES FOR SECTIONS 6-1 AND 6-2

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