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5.2.3 STATISTICS
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Not all statistics are appropriate for all scales. The following indicates which common statistical methods are appropriate: Nominal scale: Only mode, median, and percentiles Ordinal scale: The above and Spearman correlations Interval scale: The above and mean, standard deviation, and Pearson correlations Ratio scale: All statistics Absolute scale: All statistics
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EXAMPLE 5.4 AVERAGES
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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.
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Software Metrics
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Product Metrics
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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.
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EXAMPLE 5.5 LINES OF CODE
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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.
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McCABE S CYCLOMATIC NUMBER
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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)
EXAMPLE 5.6
Determine the cyclomatic number from the control flow graph shown in Fig. 5-1.
a b c d
Fig. 5-1.
Control ow graph.
There are 8 nodes, so n 8. There are 11 arcs, so e 11. The cyclomatic number is C 11 8 2 5:
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
CHAPTER 5 Software Metrics
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.
EXAMPLE 5.7
Label the regions in the control flow graph from Example 5.6 with Roman numerals. As shown in Fig. 5-2, there are five regions. Region I is the outside of the graph.
a I e II
b III f
c IV g V
Fig. 5-2.
Control ow graph with roman numerals.
Calculating the cyclomatic number from control ow graphs is time-consuming. Constructing a control ow graph from a large program would be prohibitively time-consuming. McCabe found a more direct method of calculating his measure. He found that the number of regions is usually equal to one more than the number of decisions in a program, C  1, where  is the number of decisions. In source code, an IF statement, a WHILE loop, or a FOR loop is considered one decision. A CASE statement or other multiple branch is counted as one less decision than the number of possible branches. Control ow graphs are required to have a distinct starting node and a distinct stopping node. If this is violated, the number of decisions will not be one less than the number of regions.
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