LESSON 2: FREQUENCY DISTRIBUTION Outline Frequency distribution, histogram, frequency polygon Relative frequency histogram Cumulative relative frequency graph Stem-and-leaf plots Scatter diagram Pie charts, bar chart, line chart Some special frequency distribution forms

FREQUENCY DISTRIBUTION Consider the following data that shows days to maturity for 40 short-term investments

FREQUENCY DISTRIBUTION First, construct a frequency distribution An arrangement or table that groups data into non-overlapping intervals called classes and records the number of observations in each class Approximate number of classes Number of observation Number of classes Less than 50 5-7 50-200 7-9 200-500 9-10 500-1,000 10-11 1,000-5,000 11-13 5,000-50,000 13-17 More than 50,000 17-20

FREQUENCY DISTRIBUTION Approximate class width is obtained as follows:

FREQUENCY DISTRIBUTION Classes and counts for the days-to-maturity data

HISTOGRAM

FREQUENCY POLYGON A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes. The frequencies represent the heights of the midpoints.

FREQUENCY POLYGON

RELATIVE FREQUENCY HISTOGRAM

RELATIVE FREQUENCY HISTOGRAM Class relative frequency is obtained as follows:

RELATIVE FREQUENCY HISTOGRAM Relative-frequency distribution for the days-to-maturity data

OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH A cumulative relative frequency graph or ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH

OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH 1.000 1.000 0.900 0.800 0.725 0.600 Cumulative Frequency 0.550 0.400 0.300 0.200 0.100 0.075 0.000 40 50 60 70 80 90 100 Number of Days to Maturity

STEM-AND-LEAF DISPLAY When summarizing the data by a group frequency distribution, some information is lost. The actual values in the classes are unknown. A stem-and-leaf display offsets this loss of information. The stem is the leading digit. The leaf is the trailing digit.

STEM-AND-LEAF DISPLAY Diagrams for days-to-maturity data: (a) stem-and-leaf (b) ordered stem-and-leaf Stem Leaves Stem Leaves 3 3 4 4 5 5 6 6 7 7 8 8 9 9 (a) (b)

SCATTTER DIAGRAM Often, we are interested in two variables. For example, we may want to know the relationship between advertising and sales experience and time required to produce an unit of a product

SCATTTER DIAGRAM Scatter diagrams show how two variables are related to one another To draw a scatter diagram, we need a set of two variables Label one variable x and the other y Each pair of values of x and y constitute a point on the graph

SCATTTER DIAGRAM In some cases, the value of one variable may depend on the value of the other variable. For example, sales depend on advertising time required to produce an item of a product depend on the number of units produced before In such cases, the first variable is called dependent variable and the second variable is called independent variable. For example, Independent variable Dependent variable Advertising Sales Number of units produced Production time/unit

SCATTTER DIAGRAM Usually, independent variable is plotted on the horizontal axis (x axis) and the dependent variable on the vertical axis (y axis) Sometimes, two variables show some relationships positive relationship: two variables move together i.e., one variable increases (or decreases) whenever the other increases (or, decreases). Example: advertising and sales. negative relationship: one variable increases (or, decreases) whenever the other decreases (increases). Example: number of units produced and production time/unit

SCATTTER DIAGRAM Relationship between two variables may be linear or non-linear. For example, the relationship between advertising and sales may be linear. the relationship between number of units produced and the production time/unit may be nonlinear.

SCATTTER DIAGRAM (EXAMPLE)

SCATTTER DIAGRAM (EXAMPLE)

PIE CHART A pie chart is the most popular graphical method for summarizing quantitative/nominal data A pie chart is a circle is subdivided into a number of slices Each slice represents a category Angle allocated to a slice is proportional to the proportion of times the corresponding category is observed Since the entire circle corresponds to 3600, every 1% of the observations corresponds to 0.01  3600 = 3.60

PIE CHART (EXAMPLE)

BAR CHART Bar charts graphically represent the frequency or relative frequency of each category as a bar rising vertically The height of each bar is proportional to the frequency or the relative frequency All the bars must have the same width A space may be left between bars Bar charts may be used for qualitative data or categories that should be presented in a particular order such as years 1995, 1996, 1997, ...

BAR CHART (EXAMPLE)

LINE CHART Line charts are often used when the categories are points in time. Such a chart is called a time-series chart. For example, consider a graph that shows monthly or weekly sales data. Frequency of each category is represented by a point above and then points are joined by straight lines

LINE CHART (EXAMPLE)

CHOICE OF A CHART Pie chart Small / intermediate number of categories Cannot show order of categories Emphasizes relative values e.g., frequencies Bar chart Small / intermediate/large number of categories Can present categories in a particular order, if any

CHOICE OF A CHART Bar chart Small/intermediate/large number of categories Can present categories in a particular order, if any Emphasizes relative values e.g., frequencies Line chart Emphasizes trend, if any

READING AND EXERCISES Lesson 2 Reading: Section 2-1, pp. 22-33 2-1, 2-9, 2-13, 2-14