Exercise 8 - Data Central Tendency

Suppose that the data for analysis includes the attribute age. The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

(a) What is the mean of the data?

(b) What is the median?

(d) What is the midrange of the data?

(e) Can you find (roughly) the first quartile (Q1) and the third quartile (Q3) of the data?

(f) Give the five-number summary of the data.

Solution:

a) Total No of age values given = 27

Mean of the data = 809/27 = 29.96

The mean age value is 29.96

b) Median, center of the data is 14th value.

The median age value is 25

c) mode, the value that occurs most frequently in the set.

The data set is multimodal. The mode are 25 and 35.

d) Midrange, average of the smallest and largest value in the data set.

smallest value = 13

largest value = 70

midrange = 83/2 = 41.5

The midrange age value is 41.5

e) First quartile (Q1) & Third quartile (Q3)

Median (Q2) = 25

First Quartile (Q1) = 20

Third Quartile (Q3) = 35

The first quartile (Q1) age value is 20 and third quartile (Q3) age value is 35

f) Five number summary

Minimum, Q1, Median, Q3, Maximum.

The five number summary of age value is 13, 20, 25, 35, 70

Decision Tree - Gini Index

The Gini index is used in CART. The Gini index measures the impurity of the data set, where p i - probability that data in the data set, D belong to class, C i and pi = |C i,D |/|D| There are 2 v - 2 possible ways to form two partitions of the data set, D based on a binary split on a attribute. Each of the possible binary splits of the attribute is considered. The subset that gives the minimum Gini index is selected as the splitting subset for discrete valued attribute. The degree of Gini index varies between 0 and 1. The value 0 denotes that all elements belong to a certain class or if there exists only one class, and the value 1 denotes that the elements are randomly distributed across various classes. A Gini Index of 0.5 denotes equally distributed elements into some classes. The Gini index is biased toward multivalued attributes and has difficulty when the number of classes is large.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Exercise 1 - Amdahl's Law

A programmer is given the job to write a program on a computer with processor having speedup factor 3.8 on 4 processors. He makes it 95% parallel and goes home dreaming of a big pay raise. Using Amdahl’s law, and assuming the problem size is the same as the serial version, and ignoring communication costs, what is the speedup factor that the programmer will get? Solution Speedup formula as per Amdahl's Law, N - no of processor = 4 f - % of parallel operation = 95% Speedup = 1 / (1 - 0.95) + (0.95/4) = 1 / 0.5 + (0.95/4) Speedup = 3.478 The programmer gets 3.478 as t he speedup factor.

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