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140 CHAPTER 8. POLYALPHABETIC CIPHERS
Examples: Compute the mean and M.R. for each of the following sets.
(1) {5, 9, 2, 10, 8}.
The sum of the five numbers is 5 + 9 + 2 + 10 + 8 = 34, so their mean is
34/5 = 6.8. The M.R. is
(5 − 6.8) 2 + (9 − 6.8) 2 + (2 − 6.8) 2 + (10 − 6.8) 2 + (8 − 6.8) 2
34 2 = 42.8
1156
or .037. (Our formulation of roughness will generally lead to small numbers.
That’s ok: we are currently worried about relative degrees of roughness,
rather than the meaning of the roughness measurement.)
(2) {4, 11, 13, 8, 5, 7}. 6
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
With roughness now understood, let us return to our goal: understanding
frequency counts. What is the roughness of a frequency count Using the same
notation as before, we have
M.R. = (#A − x)2 + (#B − x) 2 + · · · + (#Z − x) 2
N 2 ,
where x = #A+#B+···+#Z
26
= N 26
is the usual mathematical symbol for mean.
Notice that if we multiply out the term contributed by A, we have
(#A − x) 2 = #A 2 − 2#Ax + x 2 .
Concentrating on the numerator of roughness, we have 26 similar terms, each
with three parts. Summing the first parts gives
#A 2 + #B 2 + · · · + #Z 2 .
Summing the last part gives 26 copies of x 2 . But x = N/26, so these terms
contribute
( 2 N
26x 2 = 26 =
26) N 2
26
to the numerator. The middle terms are a bit more complicated, so we saved
them for last:
−2#Ax − 2#Bx − · · · − 2#Zx = −2x(#A + #B + · · · + #Z)
= −2x(N)
= −2 N 26 N
= −2 N 2
26
6 The mean is 8 and M.R. = .026.