Probability & statistics - part 3

Stem and leaf display

A basic stem-and-leaf display contains two columns separated by a vertical line. The left column contains the stems and the right column contains the leaves. The first step in constructing a stem and leaf display is to decide how to split each observation (weight) into two parts: a stem and a leaf.

Example

The weights in pounds of a group of workers are as follows:

173	183	162	168	154
165	177	179	158	180
171	160	145	186	164
175	151	171	182	166
188	169	175	162	157

For this example, define the first two digits of an observation to be its stem and the third digit to be its leaf. Thus, the weights are split into a stem and a leaf and can be displayed as follows:

Stem	Leaf
14	5
15 16 17	4 8 1 7 2 8 5 0 4 6 9 2 3 7 9 1 5 1 5
18	3 0 6 2 8

Frequency Tables

Frequency distributions are used to organize and present frequency counts in a summary form so that the information can be interpreted more easily. A frequency distribution of data can be shown in a table or graph. Some common methods of showing frequency distributions include frequency tables, histograms or bar charts.

The hardest, and most important, step in constructing a frequency distribution is choosing the number and width of the classes. Constructing a stem and leaf display first is often helpful. For the above example, the stem and leaf display suggests using five classes, each with a width of 10 pounds. The number (or frequency) of weights falling into each class is then recorded as shown in the table that follows. Care must be taken to define the classes in such a way that each measurement belongs to exactly one class.

Class Limits		Frequency
140 up to 150		1
150 up to 160		4
160 up to 170		8
170 up to 180		7
180 up to 190		5
Total		25

Histogram

Histograms are another useful graphical tool for quantitative data. Usually a histogram would be drawn with following features.

·         The height of the column shows the frequency for a specific range of values.

·         Columns are usually of equal width; however a histogram may show data using unequal ranges (intervals) and therefore have columns of unequal width.

·         The values represented by each column must be mutually exclusive and exhaustive. Therefore, there are no spaces between columns and each observation can only ever belong in one column.

Histogram can be used to comment on the shape of the distribution as well. Following images represent different types of histograms with different distribution shapes.

The following figure represents a histogram for the above weight data. Notice that if a statistical package is used to draw a histogram, it automatically decides the width of the classes, but some of them have options to change the widths as needed by the user.

Scatter Diagrams

A scatter diagram is a tool for analyzing relationships between two quantitative variables. The plot displays as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis drawing a point for each pair. Scatter diagrams will generally show the possible correlations between the variables as displayed below. 

So far in the descriptive methods, the graphical and tabular methods are introduced and then the focus would be on the numerical summaries of the data.

next lesson we will discuss most important one : - Box plot

Laahiru.C.fernando.