Probability and Statistics Dr. Saeid Moloudzadeh Frequency table and graphs 1 Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions

Probability and Statistics Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions 2

Chapter 0: Descriptive Statistics Contents 0.1 Fundamental Concepts 0.2 Frequency table and graphs 0.3 Measures of center 0.4 Measures of variation 3

Chapter 0: Descriptive Statistics Contents 0.1 Fundamental Concepts 0.2 Frequency table and graphs 0.3 Measures of center 0.4 Measures of variation 4

Section 2: Frequency table and graphs Ordinary frequency table The information in a discrete data set can be summarized in a frequency table, or frequency distribution that includes frequency, relative frequencies, cumulative frequency and cumulative relative frequency. The number of observations that fall into particular class of the quantitative variable is called the frequency of that class. A table listing all classes and their frequencies is called a frequency distribution. 5

Section 2: Frequency table and graphs In addition of the frequencies, we are often interested in the percentage of a class. We find the percentage by dividing the frequency of the class by the total number of observations and multiplying the result by 100. The percentage of the class, expressed as a decimal, is usually referred to as the relative frequency of the class. A table listing all classes and their relative frequencies is called a relative frequency distribution 6

Section 2: Frequency table and graphs 7 A cumulative frequency (cumulative relative frequency) is obtained by summing the frequencies (relative frequencies) of all classes up to the specific class.

Section 2: Frequency table and graphs Example 0.3: The following data represent the number of days of sick leave taken by each of 50 workers of a given company over the last 6 weeks: 2, 2, 0, 0, 5, 8, 3, 4, 1, 0, 0, 7, 1, 7, 1, 5, 4, 0, 4, 0, 1, 8, 9, 7, 0, 1, 7, 2, 5, 5, 4, 3, 3, 0, 0, 2, 5, 1, 3, 0, 1, 0, 2, 4, 5, 0, 5, 7, 5, 1 Summarizing data in a frequency table. 8

Section 2: Frequency table and graphs 9

Section 2: Frequency table and graphs Line Graphs, Bar Graphs and Frequency Polygons Data from a frequency table can be graphically pictured by a line graph, which plots the successive values on the horizontal axis and indicates the corresponding frequency by the height of a vertical line. A line graph for the data of example is shown in the following figure. 10

Section 2: Frequency table and graphs 11

Section 2: Frequency table and graphs Sometimes the frequencies are represented not by lines but rather by bars having some thickness. These graphs, called bar graphs, are often utilized. Following figure presents a bar graph for the data of the example

Section 2: Frequency table and graphs 13

Section 2: Frequency table and graphs Another type of graph used to represent a frequency table is the frequency polygon, which plots the frequencies of the different data values and then connects the plotted points with straight lines. Following figure presents the frequency polygon of the data of the example

Section 2: Frequency table and graphs 15

Section 2: Frequency table and graphs A polygon plot of these relative frequencies is presented in the following figure. A plot of the relative frequencies looks exactly like a plot of the absolute frequencies, except that the labels on the vertical axis are the old labels divided by the total number of observations in the data set. 16

Section 2: Frequency table and graphs 17

Section 2: Frequency table and graphs Frequency distribution for grouped data 18

Section 2: Frequency table and graphs 19

Section 2: Frequency table and graphs 20

Section 2: Frequency table and graphs 21

Section 2: Frequency table and graphs 22

Section 2: Frequency table and graphs 23

Section 2: Frequency table and graphs frequency histogram A bar graph plot of the data, with the bars placed adjacent to each other, is called a histogram. The vertical axis of a histogram can represent either the class frequency or the relative class frequency. In the former case, the histogram is called a frequency histogram and in the latter a relative frequency histogram. The following figure presents a frequency histogram of the data of the example

Section 2: Frequency table and graphs To Construct a Histogram from a Data Set 1. Arrange the data in increasing order. 2. Choose class intervals so that all data points are covered. 3. Construct a frequency table. 4. Draw adjacent bars having heights determined by the frequencies in step 3. 25

Section 2: Frequency table and graphs 26