Discrete Graphs Andrew Samuels

Data Set – a collection of data values Data Points – individual values within a data set (can consist of many numbers) N – the size of the data set

A professor presented his class the results of their Statistics 101 Midterm exam:

Frequency table – gives the frequency of the occurrence Here the professor presented the same data in a frequency table

Bar graph – presents information from the Frequency table in a graph Here the professor presented the same data in a Bar graph

Outliers – extreme data points that do not fit the overall pattern of the data Relative frequencies – frequencies given in terms of percentages of the total population Here the professor presented the same data in a Relative frequency Bar graph

Pictogram – Frequency charts that use icons or pictures instead of bars Here the professor presented the same data in a Pictogram

Evaluate the following Pictograms that represent the same data:

Tricks to mislead in presenting Bar graph data 1)Stretching the scale of the vertical axis 2)Cheating on the starting value on the vertical axis Look for objectivity verse propaganda

Variables

Variable – any characteristic that varies with the members of a population i.e. Scores, time spent studying

Two Types of variables: 1.) Quantitative (numerical) variable – a variable that represents a measurable quantity. Can be either continuous or discrete:

Continuous – when the difference between the values of the quantitative variable can be arbitrarily small ex.) height, weight, foot size, time to run a 5k

Discrete - when the difference between the values of the quantitative variable change by minimum increments ex.) IQ, SAT scores, shoe size, points in a game

Sometimes the lines between continuous and discrete can become blurred like rounding weight to the nearest pound would make this discrete.

2.) Qualitative (categorical) variables – cannot be measured numerically Ex. Nationality, gender, hair color

Here is data about the enrollment at the five schools at Tasmania State University. Other includes undeclared students, interdisciplinary majors, ect.

Here are two Bar graphs showing the same data in different ways. Some prefer the later graph when presented with qualitative (categorical) variables.

Pie chart – the entire population is the pie (100%) - each slice is proportional to the relative frequency of the corresponding category.

360⁰/100 = 3.6⁰ x% is given by an angle measure of 3.6x degrees Here is a Pie chart composed of data gathered by Nielsen. Prime time is 8 – 11pm

Children make up 15% of the population at large Teens make up 8% Using absolute percentages can be misleading. *When comparing characteristics of a population that is broken up into categories, it is essential to take into account the relative sizes of the various categories

Class Intervals

At times there may be too many categories to represent data clearly using a Bar graph or Pie chart. Such is the case with SAT scores. They range from 200 – 800 in increments of 10 points. The data presented like this would produce 61 different possible categories, too many for an effective Bar graph.

Class Intervals – aggregating (grouping together) data points into categories. Rule of thumb – you should have between 5 and 20 class intervals

Class Intervals are needed to convert test scores (quantitative/numerical variable) into grades (qualitative/categorical variable). The professor could convert using an absolute scale (like a ten or seven point system), or a relative scale (fit class intervals to class performance on that test “curve”).

Histogram – variation of a bar graph used to display a continuous quantitative variable.

Starting salaries of last year’s graduating class TSU N = 3258 Salary range $40,350 - $74,800 Class intervals must be set up

Since the data is continuous, a histogram has no gaps between class intervals.

Endpoint Convention – where does the data fall if it is exactly on the boundary between two classes? Simple Histograms are designed with class intervals of equal length.