Presenting Data in Tables & Charts Organizing Numerical Data

Data with 20 or more observations should be organized

The Ordered Array: arranges raw data in order from the smallest observation to the largest observation.

Raw Data Arranged in an Ordered Array

The Ordered Array makes it easy to identify:

extreme values typical values range where the majority of values are concentrated

Stem and Leaf Display: shows where raw data clusters over a range of observations.

EXAMPLE : the following data represent the weekly salary checks earned by a sample of eight secretaries: $555 $490 $648 $832 $710 $590 $576 $623

First, put the values in ascending order and then use the 100s column as the stems, use the 10s column as the leaves, and either ignore the units column or round the units column and then use the 10s column as the leaves.

$555 $490 $648 $832 $710 $590 $576 $623 4 | 9 5 | | 24 7 | 1 8 | 3

To further illustrate, how we can organize data to present, analyze and interpret findings,

we will study data from a previous QBA questionnaire: 1) USD students’ auto costs 2) USD students’ maximum auto speeds

Raw Data from student questionnaire (partial)

Stem & Leaf Auto Costs

Stem & Leaf MPH

And just for fun, let’s look at GPA GPA Stem unit: =

How Else Can We Organize our Data?

Numerical Data Frequency Distribution Relative Frequency Distribution Percentage Frequency Distribution Cumulative Frequency Distribution

Frequency Distribution

Frequency Distribution for Numerical Data (5. Auto Cost ($)) 0FrequencyPercentage % % % % % % %

Selecting the Number of Classes There is no “correct” number of classes (K) to use in a frequency distribution. However, the frequency distribution should have at least 5 classes, but no more than 20

Caution! If you have too “FEW” classes (K), a large portion of your data, lies in one class. However, if there are a number of empty classes, or too many classes with a frequency of 1 or 2, this may indicate too “MANY” classes (K).

Approximate Number of Classes in Frequency Distribution # Observations# Classes Less than 505 – 7 50 – 2007 – – – 1,00010– 11 1,000 – 5, ,000 – 50,00013 – 17 More than 50,

What do you gain by organizing your data in a Frequency Distribution?

Hint! From pages of raw data

Answer Reduce large numbers of data points to a workable number of classes and frequencies. Study the frequency distribution and learn a great deal about the shape of the data set.

Raw Data from student questionnaire (partial)

Frequency Distribution

Frequency Distribution for Numerical Data (5. Auto Cost ($)) 0FrequencyPercentage % % % % % % %

Constructing a Frequency Distribution Gather the sample data Arrange data in an Ordered Array Select the number of classes to be used Determine class width: range/ # of classes Determine the class limits for each class so that the distribution is easy to interpret Count the number of data values in each class (the raw frequencies) Determine the Relative Frequencies

Relative Frequency = Raw frequency count in each class Total number of observations (n)

Relative Frequency is essential for comparing the relationship between two data sets.

To Convert Relative Frequency to Percent Frequency: Multiply Relative Frequency X 100

Example 15. A doctor's office staff has studied the waiting times for patients who arrive at the office with a request for emergency service. The following data were collected over a one-month period (the waiting times are in minutes) Use classes of 0 - 4, 5- 9, and so on. a. Show the frequency distribution. b. Show the relative frequency distribution. c. Show the cumulative frequency distribution. d. Show the relative cumulative frequency distribution.

How Else Can We Organize our Data?

Graphic Techniques to Describe Numerical Data 1) Histogram (continuous data) 2) Polygon 3) Ogive 4) Scattergram

Histogram Uni-modal Bi-modal Skewed: i) right or positively skewed ii) left or negatively skewed

Histogram Auto Costs

Histogram MPH

Negative or Left Skewed

Positive or Right Skewed

Quiz Would incomes of employees in large firms tend to be positively or negatively skewed? Why?

Quiz Do exam grades tend to be positively or negatively skewed? Why?

A Scatter Diagram Graphs bivariate data to examine whether a relationship exists between two numerical variables.

Is there a relationship between the price of their auto and the maximum MPH a USD student has driven?

Is there a relationship between the number of alcoholic beverages consumed per week and the number of hours studied per week?

Alch bev / wk (#)mph

Tables & Charts for Categorical Data 1)Summary Table: similar to Frequency Distribution. 2)Contingency Table for Crosstabulation of Bivariate Categorical Data. 3)Bar Chart: graphical representation of frequency of occurrence. 4)Pie Chart: graphical emphasis of proportion 5)Pareto Diagram 6)Side-by-Side Bar Charts: for bivariate categorical data.

Summary Table

Soft drinkFreq.Relative Freq Coke Classic19.38 or 38% Diet Coke 8.16 or 16% Dr. Pepper 5.10 or 10% Pepsi-Cola or 26% Sprite 5.10 or 10% Total or 100%

Contingency Table for Crosstabulation of Bivariate Categorical Data

Gender vs. Number of Alcoholic Drinks per Week

Contingency Table for Gender vs. Auto Costs

Contingency Table of Gender vs. MPH Cross Tabulation of Categorical DataCross Tabulation of Categorical Data 29. The following data are for 30 observations on two qualitative variables, X, and Y. The categories for x are A, B, and C; the categories for y are 1 and 2. Cross Tabulation of Categorical DataCross Tabulation of Categorical Data Cross Tabulation of Categorical DataCross Tabulation of Categorical Data

Contingency Table of Live on/off Campus by Gender

The following data are for 30 observations on two qualitative variables, X and Y. The categories for X are A, B, and C; the categories for Y are 1 and 2.

Categories for X are A,B, and C. The categories for Y are 1 and 2.

Side-by-side Bar Charts

Pareto Diagram Separates the “vital few” from the “trivial many”.