MAT 209 - Statistics Chapter 2 - Summarizing Data

Listing – Rearranging sets of data to simplify their use.

Grouping – Sorting data into categories or classes.

2.1 Dot Diagram

·      Dot diagram can be arranged with categories on the x-axis and dots along the y-axis

·      OR categories on the y-axis with dots along the x-axis.
Pareto diagram - A graphical overview of data in ranking order from the most frequent, down to the least frequent.

2.2 Stem and Leaf Displays

 A special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit).

 Data is broken into categories of numbers.

Stem – Represents each row.

Lead – Represents each digit to the right of the vertical line.

Example:

9	63
10	05	27	29	34	35	57	67	70	79	79
11	00	04	60	73
12	04	72	81
13	47
14	39
15

2.3 Frequency Distribution

Frequency distribution – A list, table or graph that displays the frequency of various outcomes in a sample.

Process: 1^st – Choose classes, 2^nd – Sorting/tallying data into classes, 3^rd – Counting number of items in each class, 4^th – Displaying results in form of chart or table.

Things to keep note of during the process of converting data into frequency distribution table:

1.     Rarely use < 6 classes, or > 15 classes.

2.     Make sure each item goes into only one class.

3.     Use classes covering equal ranges (or intervals of values). Open classes such as less than or more than should generally be avoided.

Example of frequency distribution table:

Interval	Frequency
[900,1000)	1
[1000,1100)	10
[1100,1200)	4
[1200,1300)	3
[1300,1400)	1
[1400,1500)	1

Right hand column is called class frequency.

Class limits: Smallest and largest value that can go into any given class (Subdivided into lower class limits and upper class limits).

Class marks: Midpoints of classes, obtained by = (upper limit + lower limit)/2.

Class interval: Length of classes, obtained by = (upper limit – lower limit).

Class interval of the distribution: Common class intervals are all of equal length.

Frequency distribution can also be further modified in two ways:

Percent distribution = Divide each class frequency/total number items group * 100%.

Cumulative distribution = Add up the class frequencies starting from top or bottom of distribution.

In similar fashion there are also cumulative percentage distributions = Adding up the percent distributions.

Categorical distribution – Probability distribution that describes the possible results of a random variable that can take on one of K possible categories, with the probability of each category separately specified.

Box and Whiskers Plot

Groups data into quartiles with the median being the second quartile.

Data presented is the 5 number summary: Min, Q1, Median, Q3, Max.

2.4 Graphical Presentations

Histogram - A graphical representation of the distribution of numerical data, group numbers into ranges.

Bar chart - A chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally. 

Frequency Polygon - Another type of frequency distribution graph. In a frequency polygon, the number of observations is marked with a single point at the midpoint of an interval. A straight line then connects each set of points. (Note that classes with 0 frequencies are added to tie down the graph. An example of such can be found below.

Ogive – Cumulative frequency for frequency polygon, cumulative frequencies are plotted at corresponding class limits instead of class marks.

Pictograms – Pictogramme, pictograph, or simply picto, and in computer usage an icon, is an ideogram that conveys its meaning through its pictorial resemblance to a physical object. 

Pie chart - A circular statistical graphic, which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice is proportional to the quantity it represents.