Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 1 of 37 Chapter 2 Section 2 Organizing Quantitative Data

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 2 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 3 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 4 of 37 Chapter 2 – Section 2 ●Raw quantitative data comes as a list of values … each value is a measurement, either discrete or continuous ●Comparisons (one value being more than or less than another) can be performed on the data values ●Mathematical operations (addition, subtraction, …) can be performed on the data values

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 5 of 37 Chapter 2 – Section 2 ●Discrete quantitative data can be presented in tables in several of the same ways as qualitative data  Values listed in a table  By a frequency table  By a relative frequency table ●We use the discrete values instead of the category names

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 6 of 37 Chapter 2 – Section 2 ●Consider the following data ●We would like to compute the frequencies and the relative frequencies

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 7 of 37 Chapter 2 – Section 2 ●The resulting frequencies and the relative frequencies

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 8 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 9 of 37 Chapter 2 – Section 2 ●Discrete quantitative data can be presented in bar graphs in several of the same ways as qualitative data ●We use the discrete values instead of the category names ●We arrange the values in ascending order ●For discrete data, these are called histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 10 of 37 Chapter 2 – Section 2 ●Example of histograms for discrete data  Frequencies  Relative frequencies

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 11 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 12 of 37 Chapter 2 – Section 2 ●Continuous data cannot be put directly into frequency tables since they do not have any obvious categories ●Categories are created using classes, or intervals of numbers ●The continuous data is then put into the classes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 13 of 37 Chapter 2 – Section 2 ●For ages of adults, a possible set of classes is 20 – – – – and older ●For the class 30 – 39  30 is the lower class limit  39 is the upper class limit

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 14 of 37 Chapter 2 – Section 2 ●The class width is the difference between the upper class limit and the lower class limit ●For the class 30 – 39, the class width is 40 – 30 = 10 ●The class width is the difference between the upper class limit and the lower class limit ●For the class 30 – 39, the class width is 40 – 30 = 10 ●Why isn’t the class width 39 – 30 = 9?  The class 30 – 39 years old actually is 30 years to 39 years 364 days old … or 30 years to just less than 40 years old  The class width is 10 years, all adults in their 30’s

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 15 of 37 Chapter 2 – Section 2 ●All the classes (20 – 29, 30 – 39, 40 – 49, 50 – 59) all have the same widths, except for the last class ●The class “60 and above” is an open-ended class because it has no upper limit ●All the classes (20 – 29, 30 – 39, 40 – 49, 50 – 59) all have the same widths, except for the last class ●The class “60 and above” is an open-ended class because it has no upper limit ●Classes with no lower limits are also called open-ended classes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 16 of 37 Chapter 2 – Section 2 ●The classes and the number of values in each can be put into a frequency table ●In this table, there are 1147 subjects between 30 and 39 years old AgeNumber 20 – – – – and older110

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 17 of 37 Chapter 2 – Section 2 ●Good practices for constructing tables for continuous variables  The classes should not overlap ●Good practices for constructing tables for continuous variables  The classes should not overlap  The classes should not have any gaps between them ●Good practices for constructing tables for continuous variables  The classes should not overlap  The classes should not have any gaps between them  The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends) ●Good practices for constructing tables for continuous variables  The classes should not overlap  The classes should not have any gaps between them  The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends)  The class boundaries should be “reasonable” numbers ●Good practices for constructing tables for continuous variables  The classes should not overlap  The classes should not have any gaps between them  The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends)  The class boundaries should be “reasonable” numbers  The class width should be a “reasonable” number

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 18 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 19 of 37 Chapter 2 – Section 2 ●Just as for discrete data, a histogram can be created from the frequency table ●Instead of individual data values, the categories are the classes – the intervals of data

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 20 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 21 of 37 Chapter 2 – Section 2 ●A stem-and-leaf plot is a different way to represent data that is similar to a histogram ●To draw a stem-and-leaf plot, each data value must be broken up into two components  The stem consists of all the digits except for the right most one  The leaf consists of the right most digit  For the number 173, for example, the stem would be “17” and the leaf would be “3”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 22 of 37 Chapter 2 – Section 2 ●In the stem-and-leaf plot below  The smallest value is 56  The largest value is 180  The second largest value is 178

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 23 of 37 Chapter 2 – Section 2 ●To read a stem-and-leaf plot  Read the stem first  Attach the leaf as the last digit of the stem  The result is the original data value ●To read a stem-and-leaf plot  Read the stem first  Attach the leaf as the last digit of the stem  The result is the original data value ●Stem-and-leaf plots  Display the same visual patterns as histograms  Contain more information than histograms  Could be more difficult to interpret (including getting a sore neck)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 24 of 37 Chapter 2 – Section 2 ●To draw a stem-and-leaf plot  Write all the values in ascending order ●To draw a stem-and-leaf plot  Write all the values in ascending order  Find the stems and write them vertically in ascending order ●To draw a stem-and-leaf plot  Write all the values in ascending order  Find the stems and write them vertically in ascending order  For each data value, write its leaf in the row next to its stem ●To draw a stem-and-leaf plot  Write all the values in ascending order  Find the stems and write them vertically in ascending order  For each data value, write its leaf in the row next to its stem  The resulting leaves will also be in ascending order ●To draw a stem-and-leaf plot  Write all the values in ascending order  Find the stems and write them vertically in ascending order  For each data value, write its leaf in the row next to its stem  The resulting leaves will also be in ascending order ●The list of stems with their corresponding leaves is the stem-and-leaf plot

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 25 of 37 Chapter 2 – Section 2 ●Modifications to stem-and-leaf plots  Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another) ●Modifications to stem-and-leaf plots  Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another)  If we wanted to compare two sets of data, we could draw two stem-and-leaf plots using the same stem, with leaves going left (for one set of data) and right (for the other set) ●Modifications to stem-and-leaf plots  Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another)  If we wanted to compare two sets of data, we could draw two stem-and-leaf plots using the same stem, with leaves going left (for one set of data) and right (for the other set)  There are cases where constructing a descending stem-and-leaf plot could also be appropriate (for test scores, for example)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 26 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 27 of 37 Chapter 2 – Section 2 ●A dot plot is a graph where a dot is placed over the observation each time it is observed ●The following is an example of a dot plot

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 28 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 29 of 37 Chapter 2 – Section 2 ●A useful way to describe a variable is by the shape of its distribution ●Some common distribution shapes are  Uniform  Bell-shaped (or normal)  Skewed right  Skewed left

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 30 of 37 Chapter 2 – Section 2 ●A variable has a uniform distribution when  Each of the values tends to occur with the same frequency  The histogram looks flat

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 31 of 37 Chapter 2 – Section 2 ●A variable has a bell-shaped distribution when  Most of the values fall in the middle  The frequencies tail off to the left and to the right  It is symmetric

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 32 of 37 Chapter 2 – Section 2 ●A variable has a skewed right distribution when  The distribution is not symmetric  The tail to the right is longer than the tail to the left  The arrow from the middle to the long tail points right Right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 33 of 37 Chapter 2 – Section 2 ●A variable has a skewed left distribution when  The distribution is not symmetric  The tail to the left is longer than the tail to the right  The arrow from the middle to the long tail points left Left

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 34 of 37 Chapter 2 – Section 2 ●Learning objectives  Organize discrete data in tables  Construct histograms of discrete data  Organize continuous data in tables  Construct histograms of continuous data  Draw stem-and-leaf plots  Draw dot plots  Identify the shape of a distribution  Draw time-series graphs

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 35 of 37 Chapter 2 – Section 2 ●When the variable is measured at different points in time, the data is time-series data ●It is natural to plot time-series data against time ●Such a plot is a time-series plot ●Time series plots are used to  Identify long term trends  Identify regularly occurring trends (“seasonality”)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 36 of 37 Chapter 2 – Section 2 ●The following is an example of a time-series graph ●The horizontal axis shows the passage of time

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 2 Section 2 – Slide 37 of 37 Summary: Chapter 2 – Section 2 ●Quantitative data can be organized in several ways  Histograms based on data values are good for discrete data  Histograms based on classes (intervals) are good for continuous data  The shape of a distribution describes a variable … histograms are useful for identifying the shapes  Time series graphs are useful for showing trends and patterns over time