Chapter 2: Organizing Quantitative and Qualitative Data PDF

Preview

Summary

Relative frequency, percent frequency, Kristofor Paulson...

Description

Chapter 2 Organizing Quantitative and Qualitative Data Organizing Qualitative Data When using qualitative data there are limitations to the ways to organize and interpret the data: Qualitative Data can be organized through tables and charts: Tables Frequency Distribution Relative Frequency Distribution Percent Distribution Charts Bar Chart Pie Chart Frequency The frequency of a data set is simply the number of observations in each non-overlapping class; for qualitative data the classes are simply labels or names or each category: Example: A manager is trying to figure out inventory needs for soda pop, by evaluating the last 75 purchases. A simple way to do this to simply use the “Countif” function in Excel: The result being the frequency of each class Relative Frequency Relative Frequency is simply the fraction of elements out of the total number of elements that belong to a certain class. The formula for the relative frequency is the number of items in a class divided by the total number of elements (observations) Percent Frequency The percent frequency is the percentage of observations out of the total that belong to a particular class; it is the relative frequency multiplied by 100 (Class Frequency/Total Observations)*100; Bar Chart A bar chart is simply a graphical representation of either the frequency, relative frequency, or percent frequency; a relative frequency chart is below Example: Warm or Cold the Spring Break Dilemma Summarizing Quantitative Data When summarizing and drawing conclusions using quantitative data (the goal being to illustrate the variability in the data), the number of classes, the class width, and the limits of each class have to be defined. The 3 step procedure in organizing quantitative data is as follows: 1. Determine the Number of Non-overlapping Classes – Using Sturge’s Rule as a guideline (2^K power Rule) 2. Determine the class width – making sure all classes have the same width 3. Determine where each class limits – where does the first class begin and where does it end Number of Classes and Sturge’s Rule 1. Sturge’s Rule uses 2^k power rule to determine the approximate number of classes. Larger data sets will have more classes Smaller data sets will have fewer classes For example if a data set had 220 observations, using Sturge’s Rule there should be approximately 8 classes; it is not an exact science, but more of a guideline

Determining the Class Width In determining the class width keep in mind a couple things: Each class should be the same width Class widths usually end in a 0 or 5 (like 2.5, 15, or 50) Classes do not overlap The formula for the approximate class width is: (Max Value – Min Value)/Number of Classes Maximum Value – the largest data value Minimum Value – smallest data value Number of Classes – determined in Step 1 Determining the Class Limits In determining the class limits make sure: 1. Each observation is assigned to a class 2. each of the classes is non-overlapping (an observation belongs to one and only one class) The Cumulative Distributions Cumulative Frequency – the number of items that take on a value of equal to or less than a certain value. Cumulative Relative Frequency – The fraction of items out of the total that take on a value less than or equal to a certain value; equal to the cumulative frequency/total number of observations Cumulative Percent Frequency – The percentage of observations that take on a value of less than or equal to certain value; cumulative relative frequency multipled by 100 Example 1. What is the Cumulative Percent Frequency of the $10-14 class? (Number of items that take on a value of less than or equal to $14/total number of observations )*100= 30/65 = 46.15% 2. What is the Cumulative Frequency of the $5-9 class? number of items that take on a value of $9 or less: 25 3. What is Cumulative Relative Frequency of the $15-19 class? Number of Items less than or equal to $19/ Total number of observations: 50/65 = 0.769 Pivot Table Example of Pivot Table and Quantitative Data: SAT scores, 281 observations 2^K: 2^8 = 256 Max Value: 1385; Min Value: 790 Approximate Class Width = (1385-790)/8 = 74.375 (use 75 or 80) First class starting point: minimum value is 790 Start at 790 or some value slightly lower Make sure all classes are non-overlapping All data is in a class or group Bar Chart of SAT Scores Pivot Table Practice Describing data using 2 Variables

Studying a variable and its attributes is important, but may lead to an imperfect picture; this is why statisticians investigate the relationship between two variables. Two methods of evaluating the relationship are: Crosstabulations Scatter Plots Crosstabulation Crosstabulation - is a tabular summary of two variables; the two variables are either both qualitative, both quantitative, or one of each. For example you might look at the relationship a student’s GPA in a semester and the number of credits the student is taking. Insights gained: 1. 120/320 =37.5% took 12 credits or less 2. 180/320 56.25 earned a GPA > 3.25 3. Of the students who took 21 credits or more (50), only 20% earned a GPA over 3.25 Scatter Plot Scatter Plot – graphical representation of two quantitative variables; one variable, the dependent variable, is located on the y-axis, the independent variable is located on the x-axis. Example: A researcher is interested in looking at the relationship between studying and a student’s grade on an exam. There should be a positive relationship (the more you study the better you do). The results are on the next slide Scatter Plot (Studying vs. Grade on Exam)...