Basic Statistics

Frequency Distributions & Graphs

Structure of Research(The Scientific Method)

Identify the

Problem

Reviewing

Information

Collecting

Data

Analyzing

Data

Drawing

Conclusions

A Systematic Approach

STRUCTURE OF STATISTICS

STATISTICS

DESCRIPTIVE

INFERENTIAL

TABULAR

GRAPHICAL

NUMERICAL

CONFIDENCEINTERVALS

TESTS OF HYPOTHESIS

STRUCTURE OF STATISTICS

STATISTICS

DESCRIPTIVE

INFERENTIAL

NUMERICAL

CONFIDENCEINTERVALS

TESTS OF HYPOTHESIS

Now, we will look at the tabular and graphical approaches.

TABULAR

GRAPHICAL

QUESTIONNAIRE

A Self-Concept Scale

Scale: 1=Strongly Disagree4=Neither Agree nor Disagree7=Strongly Agree

ITEMS:

1. I usually achieve what I want when I work hard for it.

2. Once I make a plan, I am almost certain to make it work....

10. Almost anything is possible for me if I really want it.

Step 1

Step 2

Scores of 100 college students on the self-concept questionnaire.

A possible first step in organizing data for interpretation is to arrange the scores by size, usually from highest to lowest.

Step 3

The relative frequency relative frequency of a class is obtained by dividing the class frequency

by the total frequency.

RELATIVE FREQUENCY DISTRIBUTIONRELATIVE FREQUENCY DISTRIBUTION

Grouped Frequency Distribution

Use to present the data as a graph or as table

Grouping and Loss of Information

More usable/comprehensible information

Precise Information

AccuracyEase of communication

tradeoff

GRAPHIC PRESENTATION OF A GRAPHIC PRESENTATION OF A FREQUENCY DISTRIBUTIONFREQUENCY DISTRIBUTION

•Histogram vs. Bar Graph

•Polygons (Line Graphs)

•Frequency/Relative Freq

•Cumulative Distributions

•Percentiles

•Stem-and-Leaf Displays

HISTOGRAHISTOGRAMM

The Histogram is a series of column, each having as its base one class interval as its height the number of cases,

or frequency, in that class.

FOR WATER USAGE (1,000 GALLONS)FOR WATER USAGE (1,000 GALLONS)

Histogram is a graphing technique that is appropriate for quantitative data.

To avoid having the figure appear too flat or too steep, it is usually well to arrange the scales so that the height of the histogram is 2/3 to 3/4 of its width.

score

FrequencyPercent

5 %

10%

15%

20%

25%ordinate

abscissa

Percent

5 %

10%

15%

20%

25%

When one is comparing two distributions that are based on unequal numbers of observations,

percentages are preferable.

South North West

male female

FREQUENCY FREQUENCY POLYGONPOLYGON

In the polygon a point is located above the midpoint of each class interval to represent the frequency in that class. These points are then joined by straight lines.

10 15 20 25 30 35

15 10 5 0

5 40

The lowest class interval midpoints have zero frequencies. Frequency polygons are closed at both ends.

Describing Distributions

normal Positively skewed Negatively skewed

Rectangular BimodalThe Y-axis represents frequency, and the X-axis represents the numerical value of the observations

Y Y Y

Y Y

X X X

X X

THE BAR GRAPH

A Bar graph is used to present the frequencies of the categories of qualitative variable. A conventional bar graph looks exactly like a histogram except for the wider spaces between the bars.

A bar chart bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).

Source: Source: Dept. of Health & Human Services.Dept. of Health & Human Services.

Construct a bar chart for the number of persons with AIDS per 100,000 population for selected metropolitan areas of July 1990.

CityNumber with AIDS per

100,000 Population

Atlanta, GA 922Austin, TX 245Dallas, TX 711Houston, TX 1,245New York, NY 6,565San Francisco, CA 1,935Washington, DC 1,059West Palm Beach, FL 353

BAR CHART FOR THE AIDS DATABAR CHART FOR THE AIDS DATA

ATLANTAATLANTA

AUSTINAUSTIN

DALLASDALLAS

HOUSTONHOUSTON

NY, NY.NY, NY.

SAN. FRAN.SAN. FRAN.

WASH., D.CWASH., D.C..

W. P. BEACHW. P. BEACH

Cumulative Percentage Curve

Frequency and percentage polygons can be readily converted into cumulative percentage curve. The cumulative percentage or Ogive Curve is the most common type of cumulative distribution.

Cumulative percent

IQ score

Step 1: Percent (110--119) = 363/2200 = 0.165

Step 2: 0.165x100 = 16.50%

Step 3: 73.77% + 16.50%=90.27%

Y

X

Cumulative percentage

IQ score

P45=100

Percentile and percentile score

Percentiles are points in a distribution at or below which a given percent of the cases lie.P45 corresponds to an IQ 100 score of approximately 100; therefore 55 % of the IQ

scores exceed 100.

THE LINE GRAPHA line graph is used to show a picture of the relationship

between two variables.

A point on a line graph represents the value on the Y variable that goes with the corresponding value on the X variable.

Educational Level (years)

2120191817161514128

Cou

nt

200

100

0

When summarizing the data by a group frequency distribution, some information is lost since we would only have the classes and the frequency counts for the classes. We will not know what are the actual values in the classes.

A stem-and-leaf display offsets this loss of information. The stem is/are the leading digit(s). The leaf is the trailing digit (units digit). The stem is placed to the left of a vertical line and the

leaf to the right.

Stem-and-Leaf PlotsStem-and-Leaf Plots

The Dean of the College of Education reports the following number of students in the 15 sections of basic statistics offered this semester. Construct a stem-and-leaf chart for the data.• 27, 36, 29, 21, 24, 26, 32, 30, 36, 30, 28, 23, 17, 41, 19.

STEMSTEM LEAFLEAF

1234

7 91 3 4 6 7 8 9 0 0 2 6 6 1

Another advantage of a stem-and-leaf display is that it is easily reproduced with a line printer.

A pie chart pie chart is especially useful in displaying a relative frequency (percentage) distribution.

A circle is divided proportionally to the relative frequency (percentage) and portions of the circle are allocated for the the different groups.

PIE CHARTPIE CHART

EXAMPLEEXAMPLEA sample of 200 college students were asked to indicate their favorite soft drink. The results of the survey are given on the next slide. Draw pie chart for this information.

PIE CHART FOR THE TASTE TEST

Coca-Cola

Pepsi

Dr. Pepper Seven-Up

Others