What Are We Summarizing? Lecture 11 Sections 4.1 – 4.2 Tue, Feb 7, 2006

What Are We Summarizing? There are various types of data. There are various types of data. How the data are summarized depends on the type of data. How the data are summarized depends on the type of data. See Data Set 1, p See Data Set 1, p How best to summarize Gender? How best to summarize Gender? How best to summarize Age? How best to summarize Age? How best to summarize Blood Pressure? How best to summarize Blood Pressure?

Qualitative Variables Qualitative variable – A variable whose values are not numerical, but can be divided into categories. Qualitative variable – A variable whose values are not numerical, but can be divided into categories. The values of a qualitative variable may or may not have a natural order. The values of a qualitative variable may or may not have a natural order. Examples: Examples: Gender. Gender. Questionnaire response, from strongly agree to strongly disagree. Questionnaire response, from strongly agree to strongly disagree.

Summarizing Qualitative Variables Typically, we use percentages or proportions to summarize qualitative variables. Typically, we use percentages or proportions to summarize qualitative variables. 40% of the subjects are female. 40% of the subjects are female. 70% of the responses were “agree” or “strongly agree.” 70% of the responses were “agree” or “strongly agree.”

Quantitative Variables Quantitative variable – A variable whose values are numerical. Quantitative variable – A variable whose values are numerical. The values of a quantitative variable always have a natural order. The values of a quantitative variable always have a natural order. Examples: Examples: Number of members in a household. Number of members in a household. The distance of an employee’s daily commute. The distance of an employee’s daily commute. A quantitative variable may be continuous or discrete. A quantitative variable may be continuous or discrete.

Summarizing Quantitative Variables Typically, we use averages to summarize quantitative variables. Typically, we use averages to summarize quantitative variables. The average household has 3.2 members. The average household has 3.2 members. The company’s employees commute an average distance of 17.2 miles. The company’s employees commute an average distance of 17.2 miles.

Quantitative Variables A quantitative variable may be continuous or discrete. A quantitative variable may be continuous or discrete.

Continuous Variables Continuous variable – The set of theoretically possible values of the variable forms a continuous set of real numbers. Continuous variable – The set of theoretically possible values of the variable forms a continuous set of real numbers. Typically these are measured quantities: length, time, area, weight, etc. Typically these are measured quantities: length, time, area, weight, etc. Example: The length of time a student takes to complete a test. Example: The length of time a student takes to complete a test. Usually the noun does not have a plural form (“time”). Usually the noun does not have a plural form (“time”).

Discrete Variables Discrete variable – The set of theoretically possible values of the variable forms a set of isolated points on the number line. Discrete variable – The set of theoretically possible values of the variable forms a set of isolated points on the number line. Typically this is count data; a verbal description usually contains the phrase “the number of.” Typically this is count data; a verbal description usually contains the phrase “the number of.” Example: The number of students who completed the test within 40 minutes. Example: The number of students who completed the test within 40 minutes. Usually the noun has a plural form (“students”) Usually the noun has a plural form (“students”)

Discrete vs. Continuous Some data may be considered to be either discrete or continuous. Some data may be considered to be either discrete or continuous. Example: Time vs. Minutes. Example: Time vs. Minutes. How much time do I have for the test? How much time do I have for the test? How many minutes do I have for the test? How many minutes do I have for the test? Example: Money vs. Dollars. Example: Money vs. Dollars. How much money is in your wallet? How much money is in your wallet? How many dollars are in your wallet? How many dollars are in your wallet? In such cases, consider it to be continuous. In such cases, consider it to be continuous.

Discrete vs. Continuous Some data may be considered to be either discrete or continuous. Some data may be considered to be either discrete or continuous. Example: Time vs. Minutes. Example: Time vs. Minutes. How much time do I have for the test? How much time do I have for the test? How many minutes do I have for the test? How many minutes do I have for the test? Example: Money vs. Dollars. Example: Money vs. Dollars. How much money is in your wallet? How much money is in your wallet? How many dollars are in your wallet? How many dollars are in your wallet? In such cases, consider it to be continuous. In such cases, consider it to be continuous.

Discrete vs. Continuous The distinction is based on the nature of the variable, not the manner in which it is measured or recorded. The distinction is based on the nature of the variable, not the manner in which it is measured or recorded. Example: Measure the time it takes each student to finish a test, to the nearest minute. Example: Measure the time it takes each student to finish a test, to the nearest minute. The possible times are 0, 1, 2, 3, … minutes. The possible times are 0, 1, 2, 3, … minutes. Is that discrete or continuous? Is that discrete or continuous?

Let’s Do It! Let’s do it! 4.1, p. 216 – What Type of Variable? Let’s do it! 4.1, p. 216 – What Type of Variable? Think about it, p Think about it, p. 217.

Parameters and Statistics For quantitative variables (discrete or continuous), the most commonly used statistic is the average of the numbers. For quantitative variables (discrete or continuous), the most commonly used statistic is the average of the numbers. Average weight of the postal packages. Average weight of the postal packages. For qualitative variables, the most commonly used statistic is the proportion of values in a specific category. For qualitative variables, the most commonly used statistic is the proportion of values in a specific category. Proportion of packages that are in the light category. Proportion of packages that are in the light category.

Qualitative or Quantitative? Caution: Sometimes numbers are used merely as labels on the categories. That alone will not make the data quantitative. Caution: Sometimes numbers are used merely as labels on the categories. That alone will not make the data quantitative.

Qualitative or Quantitative? On an opinion survey: On an opinion survey: 1 = strongly disagree 1 = strongly disagree 2 = disagree 2 = disagree 3 = neutral 3 = neutral 4 = agree 4 = agree 5 = strongly agree 5 = strongly agree Is it legitimate to average the responses? Is it legitimate to average the responses?