1. 1 Describing Quantitative Data Here we study ways of describing a variable that is quantitative.

2. 2 Quantitative variables have values that are numbers. Remember that qualitative variables may use numbers, but the variable really has values that represent groups. Example of a qualitative variable: eye color 1 = blue, 2 = green, 3 = red (especially on Friday morning ). Our initial method of describing a quantitative variable will be basically the same as with a qualitative variable, with some modification in our understanding. Let’s consider the variable age. Consider the first 20 people you see today. Consider yourself if you look in the mirror, but just count yourself once. The age of these folks could be 1 day to 110 years in Nebraska, right?

3. 3 Remember, a frequency distribution is a tabular summary of data showing the number, or frequency, of items in each of several nonoverlapping classes. With a variable like eye color (qualitative), we typically make each color a class. But with a variable like age (quantitative), if we make each age a class then we could have so many classes that the distribution is hard to interpret. The authors suggest grouping the ages into classes and having anywhere from 5 to 20 classes. Let’s digress for a minute and think about a data set. Say I have data on people. Say I have social security number, eye color, age and blood alcohol level last Thursday night at 11:30. On the next screen I have what the data might look like in Excel, or other computer programs. Note each column is a variable. Each row represents a person in this example. Thus in each row we see the values of the variables for each person.

4. 4 SS#Eye colorageBlood alcohol level 123456789Blue22.00 987654321Blue21.016 567891234Green19.010 345678912Blue27.00 654321987Brown20.00 000000000Red22.023

5. 5 The reason for my digression was to have you begin to think about data sets. (Typically) A variable is in a column. The values down the column are for different people (or what ever the subject might be). I believe it is useful to think about data as you consider statistical ideas. Here we are looking at how to describe a column of data, one variable. Now, when we have a quantitative variable like age we have to think about how many classes to have. We want each class to have more than a few people in it. For now, let’s not worry too much about how many classes to have. The “width” of each class should be equal. Using age as an example, we might have classes that have 5 consecutive ages included. The first class might be 10-14 year olds, then 15-19 year olds and so on.

6. 6 Class “limits” need to be considered. Each person should be in only one class. Each class has a lower limit and an upper limit and these limits are exclusive to the class. On the next screen I have an example of the frequency, relative frequency and percent frequency distributions for the variable age for 50 people. The frequency column is just the counting of the number of people in each class. The relative frequency is the frequency of each class divided by the total number of people in the data set. The percent frequency is the relative frequency times 100. (Look back at the distributions we had for the qualitative variable. Does it look the same?)

7. 7 AgeFrequencyRelative Frequency Percent Frequency 10-14190.3838 15-1980.1616 20-2450.110 25-29130.2626 30-3450.110 Total501.00100

8. 8 Do you know why we put information in columns? Because then we can call’um as we see’um. Sorry:) So, the frequency, relative frequency and percent frequency distributions are different ways of summarizing information about a numerical variable. Notes about our table. 1) The total, or sum, of the frequency column is equal to the number of observations, n. 2) The total, or sum, of the relative frequency column is equal to 1. 3) The total, or sum, of the percent frequency column is equal to 100.

9. 9 Bar graphs are used for qualitative variables. What amounts to the same thing for quantitative variables are called histograms. Histograms just put the the frequency, relative frequency and percent frequency distributions into visual form. The form is a graph with certain properties. The variable of interest is put along the horizontal axis. We would have the variable age on the axis.

10. 10 Imagine you have a piece of construction paper that is blue. Do you remember way back when in school you would cut strips of paper and then curl the paper with the scissors? Well, we will not need to curl the paper here! I mention this silly example because I want you to think about cutting strips that are of the same with and are as wide as the class width (remember class widths are equal). The height of each strip would then represent the frequency, relative frequency or percent frequency on the variable. You would tape each strip onto the graph above each category. So the vertical axis, or height, in the bar graphs is either the frequency, relative frequency or percent frequency distributions. In constructing the histogram on a quantitative variable THERE IS NO SPACE between each bar to help us remember we have a quantitative variable.

11. 11 Pie Charts The authors do not mention it, but pie charts could be made in a similar fashion to what we saw before. Cumulative Distributions Have you every accumulated a bunch of junk in your room? Yea, me to. Each day more stuff just shows up. So tomorrow I will have all the stuff I have today and more. Cumulative distributions are kind of like my story. When you look at the frequency distribution we just saw, a slight modification can make then into cumulative distributions. For the cumulative frequency, start with the first class in the first row. The cumulative value for this row is the frequency.

12. 12 But the cumulative value for the second row is the frequency for the first row plus the frequency for the second row. So to get the cumulative frequency for a given row, add up the frequencies for that row and all previous rows. The cumulative relative frequency and cumulative percent frequency are found as before: cumulative relative frequency is cumulative frequency divided by total and the cumulative percent frequency is the cumulative relative frequency times 100. What’s a henway? About 4 or 5 pounds! What’s an Ogive? It is what we call a graph of a cumulative frequency distribution. The horizontal axis has values of the variable and the vertical axis has the appropriate cumulative frequency.

13. 13 What is the most frequently occurring age group in this example? How many times does it occur? (the group is 30- 39 and the frequency 17)

14. 14 This is a frequency Ogive (or polygon).