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Temperature is an interval scale measure. Thus, it makes statistical sense to give an average temperature. However, the numbers on baseball players uniforms are a nominal scale measure. It does not make sense to give the average of the numbers on a team s uniforms. Similarly, the average ranking of the students in a class or the average of a student s rankings in a number of classes is not appropriate.

Fig. 4-6

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In any triangle ABC, the square of any side is equal to the sum of the squares of the other two sides diminished by twice the product of these sides and the cosine of the included angle; i.e., a2 b2 c2 b2 a2 a2 c2 c2 b2 2bc cos A 2ac cos B 2ab cos C

Product metrics are metrics that can be calculated from the document independent of how it was produced. Generally, these are concerned with the structure of the source code. Product metrics could be de ned for other documents. For example, the number of paragraphs in a requirements speci cation would be a product metric.

Y(z) = X ( z ) H ( z )

a ( z - a ) ( z - I/a)

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The most basic metric for size is the lines of code metric. There are many different ways to count lines of code. The definition may be a simple as the number of NEW LINE characters in the file. Often comments are excluded from the count of lines. Sometimes blank lines or lines with only delimiters are excluded. Sometimes statements are counted instead of lines.

where Thus,

When three parts of a triangle, not all angles, are known, the triangle is uniquely determined, except in one case noted below. The five cases of oblique triangles are

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McCabe s cyclomatic number, introduced in 1976, is, after lines of code, one of the most commonly used metrics in software development. Also called McCabe s complexity measure from the title of the original journal article, it is based on graph theory s cyclomatic number. McCabe tries to measure the complexity of a program. The premise is that complexity is related to the control ow of the program. Graph theory uses a formula, C e n 1 to calculate the cyclomatic number. McCabe uses the slightly modi ed formula: C e n 2p where: e Number of edges n Number of nodes p Number of strongly connected components (which is normally 1)

C, =

Determine the cyclomatic number from the control flow graph shown in Fig. 5-1.

- --

Case I: Given two angles and the side opposite one of them Case II: Given two angles and the included side Case III: Given two sides and the angle opposite one of them Case IV: Given two sides and the included angle Case V: Given the three sides

a (z-a)(z-a)

Fig. 5-1.

-+"-)

There are 8 nodes, so n 8. There are 11 arcs, so e 11. The cyclomatic number is C 11 8 2 5:

Case I II III IV V Use Law of Sines Sines Sines Cosines Cosines First Part to Find Side opposite second given angle Third angle, then either of remaining sides Angle opposite second given side Third side Any angle can be found

which is the same as Eq. (2.137).

A planar graph is a graph that can be drawn without lines crossing. The Swiss mathematician Leonhard Euler (1707 1783) proved for planar graphs that

From Fig. 2-23 and definition ( 4 . 3 )

In Case III there is not always a unique solution. It is possible to have no solution for the angle, one solution for the angle, or two solutions an angle and its supplement. See Example 11.3 and Prob. 11.2 for a complete discussion of this case.

~ [ n=] ( l , l , l , l ] - X ( z ) h [ n ] = ( l , l , 1)

2 n e r, where r number of regions, e number of edges, and n number of nodes. A region is an area enclosed (or de ned) by arcs. Using algebra, this can be converted to r e n 2. Therefore, the number of regions on a planar graph equals the cyclomatic number.

-H(z)

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