Drawback of Using Frequency Distribution

• Frequency of occurrence is the foundation for much of statistical analysis because it provides a meaningful arrangement of observed data.

• But there are some drawbacks to basing evaluations directly on the number of data points found in each class interval.

• It is clumsy to compare groups of different size in terms of straight tallies.

Drawback of Using Frequency Distribution

• For example, knowing that there are 254 electrical engineering majors at one university and only 154 at a second says little about relative importance of that concentration in the two engineering schools unless the respective total enrollments are also included in the comparison.

• The first school has 1542 engineering students, so that the proportion, or relative frequency, of electrical engineers is 254/1547 = .164.

• The second university has 655 engineering students, and the relative frequency of the electrical concentration is much higher 154/655 = .235.

• Thus electrical engineering is more dominant in the second institution.

• The foregoing suggests that it may be helpful to divide each of the original frequencies by the sample size, expressing the distribution in terms of relative frequencies.

Relative Frequency DistributionRelative 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.

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.

HISTOGRAM

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

70 64 99 55 64 89 87 6562 38 67 70 60 69 78 3975 56 71 51 99 68 95 8657 53 47 50 55 81 80 9851 31 63 66 85 79 83 70

RELATIVE FREQUENCY HISTOGRAM

Relative-frequency distribution for the days-to-maturity data

Days to Maturity

Relative Frequency

RELATIVE FREQUENCY HISTOGRAM

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

40 50 60 70 80 90 100

Number of Days to Maturity

Rel

ativ

e F

req

uen

cy

CUMULATIVE RELATIVE FREQUENCY GRAPH

• When plotted on a graph, the cumulative frequency distribution gives another visual summary of the sample.

• A cumulative relative frequency graph is a graph that represents the cumulative frequencies for the classes in a frequency distribution.

• Each dot is plotted directly above the upper class limit at a height equal to the cumulative frequency for that interval.

CUMULATIVE RELATIVE FREQUENCY GRAPH

Class Frequency RelativeFrequency

CumulativeRelative

Frequency

OGIVE

CUMULATIVE RELATIVE FREQUENCY GRAPH

0.075 0.100

0.300

0.550

0.725

0.9001.000

0.000

0.200

0.400

0.600

0.800

1.000

40 50 60 70 80 90 100

Number of Days to Maturity

Cu

mu

lati

ve F

req

uen

cy

PIE CHARTS

• A pie chart is the most popular graphical method for summarizing quantitative 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

Code Area Number Proportion Angles on aof Area of Graduates of Graduates Pie Chart

1 Accounting 732 Finance 523 General Mgmnt 364 Marketing 645 Other 28

PIE CHARTS (EXAMPLE)

PIE CHART

129%

221%

314%

425%

511%

More0%

12345More

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

• Line chart– Small/intermediate/large number of categories– Can present categories in a particular order, if any – Emphasizes trend, if any

CHOICE OF A CHART

SCATTTER DIAGRAMS

• 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 DIAGRAMS

• 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 DIAGRAMS

• 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 DIAGRAMS

• 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 DIAGRAMS

• 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 DIAGRAMS (EXAMPLE)

Advertizing Sales1,000 of dollars 1,000 of dollars

1 303 405 404 502 355 503 352 25

SCATTER DIAGRAM

0

5

10

0 2000 4000 6000 8000 10000 12000

Number of units produced

Pro

du

ctio

n t

ime

(ho

urs

)/u

nit

Problems to be solved in class2.16 (text book) The cumulative

relative frequency distribution for the depth (in feet) of oil well shafts in a particular region is given here:

Altogether there are 700 wells in the region.

a) Make a table for relative frequency distribution

b) Determine the original frequency distribution

c) Using the answer of part b, determine the cumulative frequency distribution

Depth of well

Cumulative proportion

0-1000 0.09

1000-2000 0.35

2000-3000 0.72

3000-4000 0.88

4000-5000 0.95

5000-6000 0.98

6000-7000 0.99

7000-8000 1.00