LESSON 2: FREQUENCY DISTRIBUTION

Transcription

1 LESSON : 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 1 FREQUENCY DISTRIBUTION Consider the following data that shows days to maturity for short-term investments

2 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 , -11 1,-5, ,-5, More than 5, 17-3 FREQUENCY DISTRIBUTION Approximate class width is obtained as follows: Largest value - Smallest value Approximat e class width = Number of classes

3 FREQUENCY DISTRIBUTION Classes and counts for the days-to-maturity data Days to Maturity TALLY Number of Investments 5 HISTOGRAM 1 Frequency Number of Days to Maturity 6 3

4 HISTOGRAM Classes: Categories for grouping data. Frequency: The number of observations that fall in a class. Frequency distribution: A listing of all classes along with their frequencies. Relative frequency: The ratio of the frequency of a class to the total number of observations. Relative-frequency distribution: A listing of all classes along with their relative frequencies. Lower cutpoint: The smallest value that can go in a class. Upper cutpoint: The smallest value that can go in the next higher class. The upper cutpoint of a class is the same as the lower cutpoint of the next higher class. Midpoint: The middle of a class, obtained by taking the average of its lower and upper cutpoints. Width: The difference between the upper and lower cutpoints of a class. 7 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. 8

5 FREQUENCY POLYGON Classes Mid-value Frequency 9 FREQUENCY POLYGON Frequency Number of Days to Maturity 5

6 RELATIVE FREQUENCY HISTOGRAM Frequency histogram: A graph that displays the classes on the horizontal axis and the frequencies of the classes on the vertical axis. The frequency of each class is represented by a vertical bar whose height is equal to the frequency of the class. Relative-frequency histogram: A graph that displays the classes on the horizontal axis and the relative frequencies of the classes on the vertical axis. The relative frequency of each class is represented by a vertical bar whose height is equal to the relative frequency of the class. 11 RELATIVE FREQUENCY HISTOGRAM Class relative frequency is obtained as follows: Class relative frequency = Class frequency Total number of observatio ns 1 6

7 RELATIVE FREQUENCY HISTOGRAM Relative-frequency distribution for the days-to-maturity data Days to Maturity Relative Frequency 13 3.% RELATIVE FREQUENCY HISTOGRAM Relative Frequency 5.%.% 15.%.% 5.%.% Number of Days to Maturity 1 7

8 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. 15 OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH Class Frequency Relative Frequency Cumulative Relative Frequency 16 8

9 OGIVE CUMULATIVE RELATIVE FREQUENCY GRAPH Cumulative Frequency Number of Days to Maturity 17 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. 18 9

10 STEM-AND-LEAF DISPLAY Diagrams for days-to-maturity data: (a) stem-and-leaf (b) ordered stem-and-leaf Stem Leaves Stem Leaves (a) (b) 19 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

11 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 1 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 11

12 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 3 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. 1

13 SCATTTER DIAGRAM (EXAMPLE) Advertizing Sales 1, of dollars 1, of dollars SCATTER DIAGRAM 6 Sales 6 Advertising 6 13

14 SCATTTER DIAGRAM (EXAMPLE) Number of units Production time produced hours/unit SCATTER DIAGRAM Production time (hours)/unit Number of units produced 8 1

15 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 36, every 1% of the observations corresponds to.1 36 = PIE CHART (EXAMPLE) Code Area Number Proportion Angles on a of Area of Graduates of Graduates Pie Chart 1 Accounting 73 Finance 5 3 General Mgmnt 36 Marketing 6 5 Other

16 PIE CHART 5% 5 More 11% % 3 1% 1 9% 1% More 31 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,

17 BAR CHART (EXAMPLE) Code Area Number of Area of Graduates 1 Accounting 73 Finance 5 3 General Mgmnt 36 Marketing 6 5 Other 8 33 BAR CHART Count of Area Area 3 17

18 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 35 LINE CHART (EXAMPLE) Year Fatal Accidents Number of Passengers Per 1, Departures Millions

19 Line Chart Fatal Accidents Per 1, Departures Year 37 Line Chart Number of Passengers (Millions) Year 38 19

20 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 Emphasizes relative values e.g., frequencies 39 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 Small/intermediate/large number of categories Can present categories in a particular order, if any Emphasizes trend, if any

21 SYMMETRIC HISTOGRAM Frequency Number of Units Sold 1 SYMMETRIC HISTOGRAM Frequency Number of Units Sold 1

22 POSITIVELY SKEWED HISTOGRAM Frequency Number of Units Sold 3 NEGATIVELY SKEWED HISTOGRAM Frequency Number of Units Sold

23 BIMODAL HISTOGRAM Frequency Number of Units Sold 5 NORMAL DISTRIBUTION HISTOGRAM Frequency Number of Units Sold 6 3

24 EXPONENTIAL DISTRIBUTION HISTOGRAM Frequency Number of Units Sold 7 UNIFORM DISTRIBUTION HISTOGRAM Frequency Number of Units Sold 8

25 READING AND EXERCISES Lesson Reading: Section -1, pp. -33 Exercises: -1, -9, -13,