Introduction To Statistics

Lecture 04

Dr. MUMTAZ AHMEDMTH 161: Introduction To StatisticsReview of Previous LectureGraphical Methods of Data PresentationsGraphs for qualitative dataBar ChartsSimple Bar ChartMultiple Bar ChartComponent Bar ChartPie Charts22Objectives of Current LectureGraphical Methods of Data PresentationsGraphs for quantitative dataHistogramsFrequency PolygonCumulative Frequency Polygon (Frequency Ogive)33Graphs For Quantitative DataCommon methods for graphing quantitative data are:

HistogramFrequency PolygonFrequency OgiveHistograms For Quantitative DataA histogram is a graph that consists of a set of adjacent bars with heights proportional to the frequencies (or relative frequencies or percentages) and bars are marked off by class boundaries (NOT class limits).It displays the classes on the horizontal axis and the frequencies (or relative frequencies or percentages) 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.It is similar to a bar graph. However, a histogram utilizes classes or intervals and frequencies while a bar graph utilizes categories and frequencies.Histograms For Quantitative DataExample: Construct a Histogram for ages of telephone operators.

Age (years)No of Operators11-151016-20521-25726-301231-356Total40Histograms For Quantitative DataMethod: First construct Class Boundaries (CB).

Age (years)No of Operators11-151016-20521-25726-301231-356Total40Histograms For Quantitative DataMethod: First construct Class Boundaries (CB).

Age (years)Class BoundariesNo of Operators11-1510.5-15.51016-20521-25726-301231-356Total40Histograms For Quantitative DataMethod: First construct Class Boundaries (CB).

Age (years)Class BoundariesNo of Operators11-1510.5-15.51016-2015.5-20.5521-2520.5-25.5726-3025.5-30.51231-3530.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Age (years)Class BoundariesNo of Operators11-1510.5-15.51016-2015.5-20.5521-2520.5-25.5726-3025.5-30.51231-3530.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Histograms For Quantitative DataMethod: Construct Histogram by taking CB along X-axis and frequencies along Y-axis.

Class BoundariesNo of Operators (f)10.5-15.51015.5-20.5520.5-25.5725.5-30.51230.5-35.56Total40Frequency Polygon For Quantitative DataGraph of frequencies of each class against its mid point (also called class marks, denoted by X).

Class Mark (X) or Mid point: It is calculated by taking average of lower and upper class limits.Example: (Ages of Telephone Operators)

Frequency Polygon For Quantitative DataGraph of frequencies of each class against its mid point (also called class marks, denoted by X).

Class Mark (X) or Mid point: It is calculated by taking average of lower and upper class limits.Example: (Ages of Telephone Operators)

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Total40Frequency Polygon For Quantitative DataMethod: Take Mid Points along X-axis and Frequency along Y-axis.

Frequency Polygon For Quantitative DataMethod: Take Mid Points along X-axis and Frequency along Y-axis.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Frequency Polygon For Quantitative DataMethod: Construct Bars with height proportional to the corresponding freq.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Frequency Polygon For Quantitative DataMethod: Construct Bars with height proportional to the corresponding freq.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Frequency Polygon For Quantitative DataMethod: Join Mid points to get Frequency Polygon.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Frequency Polygon For Quantitative DataMethod: Join Mid points to get Frequency Polygon.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Frequency Polygon For Quantitative DataMethod: Join Mid points to get Frequency Polygon.

Age (years)No of OperatorsMid Point (X)11-1510(11+15)/2=1316-2051821-2572326-30122831-35633Cumulative Frequency Polygon (called Ogive) For Quantitative DataOgive is pronounced as OJive (rhymes with alive).Cumulative Frequency Polygon is a graph obtained by plotting the cumulative frequencies against the upper or lower class boundaries depending upon whether the cumulative is of less than or more than type.Cumulative Frequency Polygon (called Ogive) For Quantitative DataOgive is pronounced as OJive (rhymes with alive).Cumulative Frequency Polygon is a graph obtained by plotting the cumulative frequencies against the upper or lower class boundaries depending upon whether the cumulative is of less than or more than type.Less than Cumulative Frequency Age (years)Class BoundariesNo of Operators (f)Cumulative Frequency11-15Less than 15.5101016-20Less than 20.551521-25Less than 25.572226-30Less than 30.5123431-35Less than 35.5640Total40Cumulative Frequency Polygon (Ogive) For Quantitative DataMethod: Take Upper Class Boundaries along X-axis and Cumulative Frequency along Y-axis.

Cumulative Frequency Polygon (Ogive) For Quantitative DataMethod: Take Upper Class Boundaries along X-axis and Cumulative Frequency along Y-axis.

Class BoundariesCumulative FrequencyLess than 15.510Less than 20.515Less than 25.522Less than 30.534Less than 35.540Cumulative Frequency Polygon (Ogive) For Quantitative DataMethod: Take Upper Class Boundaries along X-axis and Cumulative Frequency along Y-axis.

Class BoundariesCumulative FrequencyLess than 15.510Less than 20.515Less than 25.522Less than 30.534Less than 35.540Cumulative Frequency Polygon (Ogive) For Quantitative DataMethod: Join less than Class Boundaries with corresponding Cumulative Frequencies.Class BoundariesCumulative FrequencyLess than 15.510Less than 20.515Less than 25.522Less than 30.534Less than 35.540Cumulative Frequency Polygon (Ogive) For Quantitative DataMethod: Join less than Class Boundaries with corresponding Cumulative Frequencies.

Class BoundariesCumulative FrequencyLess than 15.510Less than 20.515Less than 25.522Less than 30.534Less than 35.540Distributional ShapeDistribution of a Data SetA table, a graph, or a formula that provides the values of the data set and how often they occur.An important aspect of the distribution of a quantitative data is its shape. The shape of a distribution frequently plays a role in determining the appropriate method of statistical analysis.

To identify the shape of a distribution, the best approach usually is to use a smooth curve that approximates the overall shape.Distributional ShapeFigure displays a relative-frequency histogram for the heights of the 3000 female students.It also includes a smooth curve that approximates the overall shape of the distribution.Note: Both the histogram and the smooth curve show that this distribution of heights is bell shaped, but the smooth curve makes seeing the shape a little easier.

Advantage of smooth curves:It skips minor differences in shapeand concentrate on overall patterns.

Frequency Distributions in PracticeCommon Type of Frequency Distribution:Symmetric DistributionNormal Distribution (or Bell Shaped)Triangular DistributionUniform Distribution (or Rectangular)

Frequency Distributions in PracticeCommon Type of Frequency Distribution: Asymmetric or skewed DistributionRight Skewed DistributionLeft Skewed DistributionReverse J-Shaped (or Extremely Right Skewed) J-Shaped (or Extremely Left Skewed)

Frequency Distributions in PracticeCommon Type of Frequency Distribution: Bi-Modal DistributionMultimodal DistributionU-Shaped Distribution

Identifying DistributionExample: (Household Size): The relative-frequency histogram for household size in the United States is shown in figure.Identify the distribution shape for sizes of U.S. households.

Identifying DistributionTo identify the distributional shape, Draw a smooth curve through the histogram.

Identifying DistributionTo identify the distributional shape, Draw a smooth curve through the histogram.

Identifying DistributionTo identify the distributional shape, Draw a smooth curve through the histogram.

Decision:

ReviewLets review the main concepts:

Graphical Methods of Data PresentationsGraphs for quantitative dataHistogramsFrequency PolygonCumulative Frequency Polygon (Frequency Ogive)42Next LectureIn next lecture, we will study:Introduction To MS-ExcelConstructing Frequency Table in MS-ExcelConstructing Graphs in MS-Excel43