quantitative variables
(numerical): yield numerical information.
- classified as either discrete or continuous.
qualitative variables
(categorical): does not assume a numerical value but rather, is classifiable into 2 or more non-numeric, distinct, categorical values,
- i.e. gender (male, female, other),
species (chinstrap, gentoo, adelie), smell (stinky, neutral, fresh, Old Spice fresh), etc. - When there is ordering among the levels of a categorical variable, the variable is said to be ordinal; when no such ordering exists, the variable is called nominal.
Variable
A variable is a characteristic/condition that varies for different individuals/observations,
i.e. height, weight, g.p.a., gender, number of times visited by aliens, etc.
discrete variable
is a variable whose possible values can be listed, even though the list may continue indefinitely.
-involves a count of something
continuous variable
is a variable whose possible values form some interval of numbers.
-involves a measurement of something
Data
Values of a variable.
- information, measurements, or observations.
- In order to obtain data, one must first identify variables of interest. The data then
correspond to particular realizations of said variables.
Observation
Each individual piece of date is called this.
Data set
A collection of all observations is called a data set
frequency distribution
A frequency distribution of qualitative data is a listing of the distinct values and their frequencies.
- Frequency distributions are typically displayed in a table.
Frequency or count
In a qualitative dataset, the number of times a particular, distinct value occurs is referred to as its frequency or count.
To Construct a Frequency Distribution of Qualitative Data
Step 1 List the distinct values of the observations in the data set in the first column of a table.
Step 2 For each observation, place a tally mark in the second column of the table in the row of the appropriate distinct value.
Step 3 Count the tallies for each distinct value and record the totals in the third column of the table.
Relative-Frequency Distribution of Qualitative Data (or proportions)
A relative-frequency distribution of qualitative data is a listing of the distinct values and their relative frequencies.
- provides a table of the values of the observations and (relatively) how often they occur.
-to obtain we first find a frequency distribution and then divide each freq by the total number of observations.
To Construct a Relative-Frequency Distribution of Qualitative Data
Step 1 Obtain a frequency distribution of the data.
Step 2 Divide each frequency by the total number of observations.
Pie chart
Is a disk divided into wedge-shaped pieces proportional to the relative frequencies of the qualitative data
- Look for: values that form large and small proportions of the data set.
To construct a pie chart
Step 1 Obtain a relative-frequency distribution of the data by applying
Procedure 2.2.
Step 2 Divide a disk into wedge-shaped pieces proportional to the relative frequencies
Step 3 Label the slices with the distinct values and their relative frequencies.
Bar chart
- displays the distinct values of the qualitative data on a horizontal axis and the relative frequencies (or frequencies or percents) of those values on a vertical axis.
- The relative frequency of each distinct value is represented by a vertical bar whose height is equal to the relative frequency of that value.
- The bars should be positioned so that they do not touch each other.
- Another graphical display for qualitative data is the bar chart. Frequencies, relative frequencies, or percents can be used to label a bar chart. Although we primarily use relative frequencies, some of our applications employ frequencies or percents.
- Look for: frequently and infrequently occurring values.
To construct a bar chart
Step 1 Obtain a relative-frequency distribution of the data by applying
Procedure 2.2.
Step 2 Draw a horizontal axis on which to place the bars and a vertical axis on which to display the relative frequencies.
Step 3 For each distinct value, construct a vertical bar whose height equals the relative frequency of that value.
Step 4 Label the bars with the distinct values, the horizontal axis with the name of the variable, and the vertical axis with “Relative frequency.”
Classes/categories/bins
To organize quantitative data, we first group the observations into classes (also known as categories or bins) and then treat the classes as the distinct values of qualitative data.
- Consequently, once we group the quantitative data into classes, construct frequency and relative-frequency distributions of the data in exactly the same way as we did for qualitative data.
- classes or bins, which are non-overlapping intervals of equal width that, collectively, span the
entire range of the data
3 common sense and important guidelines for grouping quantitative data into classes are:
- The number of classes should be small enough to provide an effective summary but large enough to display the relevant characteristics of the data. A rule of thumb is that the number of classes should be between 5 and 20.
- Each observation must belong to one, and only one, class. That is, each observation should belong to some class and no observation should belong to more than one class.
- Whenever feasible, all classes should have the same width. Roughly speaking, this guideline means that, if possible, all classes should cover the same number of possible values. We’ll make this guideline more precise later in this section.
Single-value grouping
we use the distinct values of the observations as the classes, a method completely analogous to that used for qualitative data.
- Single-value grouping is particularly suitable for discrete data in which there are only a small number of distinct values.
Limit grouping
A second way to group quantitative data is to use class limits. With this method, each class consists of a range of values.
- The smallest value that could go in a class is called the lower limit of the class, and the largest value that could go in the class is called the upper limit of the class.
- This method of grouping quantitative data is called limit grouping.
- It is particularly useful when the data are expressed as whole numbers and there are too many distinct values to employ single-value grouping.
Terms used in limit grouping
Lower class limit: The smallest value that could go in a class.
Upper class limit: The largest value that could go in a class
Class width: The difference between the lower limit of a class and the lower limit of the next-higher class.
Class mark: The average of the two class limits of a class.
Cutpoint grouping
A third way to group quantitative data is to use class cutpoints. As with limit grouping, each class consists of a range of values.
- The smallest value that could go in a class is called the lower cutpoint of the class, and the smallest value that could go in the next-higher class is called the upper cutpoint of the class.
- Note that the lower cutpoint of a class is the same as its lower limit and that the upper cutpoint of a class is the same as the lower limit of the next higher class.
- The method of grouping quantitative data by using cutpoints is called cutpoint grouping.
- This method is particularly useful when the data are continuous and are expressed with decimals.
Terms used in cutpoint grouping
Lower class cutpoint: The smallest value that could go in a class.
Upper class cutpoint: The smallest value that could go in the next-higher class (equivalent to the lower cutpoint of the next-higher class).
Class width: The difference between the cutpoints of a class.
Class midpoint: The average of the two cutpoints of a class.
