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Chapter I : Describing Data With Graphs

                          Kian Jahromi


                          May 31, 2012




Kian Jahromi ()   Chapter I : Describing Data With Graphs   May 31, 2012   1 / 19
Table of contents



1   VARIABLES AND DATA
      TYPES OF VARIABLES


2   GRAPHS FOR CATEGORICAL DATA


3   GRAPHS FOR QUANTITATIVE DATA


4   Interpreting Graphs with a Critical Eye




       Kian Jahromi ()    Chapter I : Describing Data With Graphs   May 31, 2012   2 / 19
VARIABLES AND DATA


Definitions
Definition
A Variable is a characteristic that changes or varies over time and/or for
different individuals or objects under consideration.

Definition
An experimental unit is the individual or object on which a variable is
measured. A single measurement or data value results when a variable is
actually measured on an experimental unit.

Definition
A population is the set of all measurements of interest to the investigator.

Definition
A sample is a subset of measurements selected from the population of
interest.
      Kian Jahromi ()       Chapter I : Describing Data With Graphs   May 31, 2012   3 / 19
VARIABLES AND DATA




Example
Identify the experimental units on which the following variables are
measured:
a. Gender of a student
The student
b. Number of errors on a midterm exam
The midterm exam
c. Age of a cancer patient
The patient
e. Colour of a car entering a parking lot
The Car




      Kian Jahromi ()       Chapter I : Describing Data With Graphs   May 31, 2012   4 / 19
VARIABLES AND DATA




Definition
Univariate data result when a single variable is measured on a single
experimental unit.

Definition
Bivariate data result when two variables are measured on a single
experimental unit. Multivariate data result when more than two variables
are measured.




     Kian Jahromi ()       Chapter I : Describing Data With Graphs   May 31, 2012   5 / 19
VARIABLES AND DATA




The following data set is a multivariate data set. Each column itself is a
Univariate data set.




      Kian Jahromi ()       Chapter I : Describing Data With Graphs   May 31, 2012   6 / 19
VARIABLES AND DATA       TYPES OF VARIABLES


Definition
Qualitative variables measure a quality or characteristic on each
experimental unit. Quantitative variables measure a numerical quantity
or amount on each experimental unit.

Definition
Definition A discrete variable can assume only a finite or countable
number of values. A continuous variable can assume the infinitely many
values corresponding to the points on a line interval.




     Kian Jahromi ()       Chapter I : Describing Data With Graphs   May 31, 2012   7 / 19
GRAPHS FOR CATEGORICAL DATA


Graphs for Categorical Data

After the data have been collected, they can be consolidated and
summarized to show the following information:
           (i) What values of the variable have been measured
          (ii) How often each value has occurred For this purpose, you can
               construct a statistical table that can be used to display the

Example
A bag contains 25 candies:




     Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   8 / 19
GRAPHS FOR CATEGORICAL DATA

So, the Statistical table for last page example is as follows:




Also, it is possible to express the frequency of each categories using
following formulas:
            (i) Relative frequency= frequency (n is the total number of
                                        n
                measurements)
           (ii) Percent= 100 × Relative frequency
The following table contain the relative frequency and percent for each
categories of last example:




      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   9 / 19
GRAPHS FOR CATEGORICAL DATA




Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   10 / 19
GRAPHS FOR CATEGORICAL DATA


Example
Fifty people are grouped into four categories A, B, C, and D and the
number of people who fall into each category is shown in the table:




The following figure is the bar chart for upper table:




      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   11 / 19
GRAPHS FOR CATEGORICAL DATA




and the pie chart is as follows:




      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   12 / 19
GRAPHS FOR QUANTITATIVE DATA


GRAPHS FOR QUANTITATIVE DATA
Line Charts
When a quantitative variable is recorded over time at equally spaced
intervals (such as daily, weekly, monthly, quarterly, or yearly), the data set
forms a time series. Time series data are most effectively presented on a
line chart with time as the horizontal axis. The idea is to try to discern a
pattern or trend that will likely continue into the future, and then to use
that pattern to make accurate predictions for the immediate future.
Example




      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   13 / 19
GRAPHS FOR QUANTITATIVE DATA




Dotplots
Many sets of quantitative data consist of numbers that cannot easily be
separated into categories or intervals of time. You need a different way to
graph this type of data! The simplest graph for quantitative data is the
dotplot. For a small set of measurements for example, the set 2, 6, 9, 3, 7,
6 you can simply plot the measurements as points on a horizontal axis.




Example




      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   14 / 19
GRAPHS FOR QUANTITATIVE DATA




Stem and Leaf Plots
Another simple way to display the distribution of a quantitative data set is
the stem and leaf plot. This plot presents a graphical display of the data
using the actual numerical values of each data point.
How Do I Construct a Stem and Leaf Plot?
             1. Divide each measurement into two parts: the stem and the
                leaf .
             2. List the stems in a column, with a vertical line to their right.
             3. For each measurement, record the leaf portion in the same
                row as its corresponding stem.
             4. Order the leaves from lowest to highest in each stem.
             5. Provide a key to your stem and leaf coding so that the
                reader can recreate the actual measurements if necessary.
      Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   15 / 19
GRAPHS FOR QUANTITATIVE DATA




Example




    Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   16 / 19
GRAPHS FOR QUANTITATIVE DATA




Example




    Kian Jahromi ()          Chapter I : Describing Data With Graphs   May 31, 2012   17 / 19
Interpreting Graphs with a Critical Eye


Definition
A distribution is symmetric if the left and right sides of the distribution,
when divided at the middle value, form mirror images.




Definition
A distribution is skewed to the right if a greater proportion of the
measurements lie to the right of the peak value. Distributions that are
skewed right contain a few unusually large measurements.




      Kian Jahromi ()                Chapter I : Describing Data With Graphs   May 31, 2012   18 / 19
Interpreting Graphs with a Critical Eye




Definition
A distribution is skewed to the left if a greater proportion of the
measurements lie to the left of the peak value. Distributions that are
skewed left contain a few unusually small measurements.




Definition
A distribution is unimodal if it has one peak; a bimodal distribution has
two peaks.Bimodal distributions often represent a mixture of two different
populations in the data set.


      Kian Jahromi ()                Chapter I : Describing Data With Graphs   May 31, 2012   19 / 19

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Chapter One (STAT 160)

  • 1. Chapter I : Describing Data With Graphs Kian Jahromi May 31, 2012 Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 1 / 19
  • 2. Table of contents 1 VARIABLES AND DATA TYPES OF VARIABLES 2 GRAPHS FOR CATEGORICAL DATA 3 GRAPHS FOR QUANTITATIVE DATA 4 Interpreting Graphs with a Critical Eye Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 2 / 19
  • 3. VARIABLES AND DATA Definitions Definition A Variable is a characteristic that changes or varies over time and/or for different individuals or objects under consideration. Definition An experimental unit is the individual or object on which a variable is measured. A single measurement or data value results when a variable is actually measured on an experimental unit. Definition A population is the set of all measurements of interest to the investigator. Definition A sample is a subset of measurements selected from the population of interest. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 3 / 19
  • 4. VARIABLES AND DATA Example Identify the experimental units on which the following variables are measured: a. Gender of a student The student b. Number of errors on a midterm exam The midterm exam c. Age of a cancer patient The patient e. Colour of a car entering a parking lot The Car Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 4 / 19
  • 5. VARIABLES AND DATA Definition Univariate data result when a single variable is measured on a single experimental unit. Definition Bivariate data result when two variables are measured on a single experimental unit. Multivariate data result when more than two variables are measured. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 5 / 19
  • 6. VARIABLES AND DATA The following data set is a multivariate data set. Each column itself is a Univariate data set. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 6 / 19
  • 7. VARIABLES AND DATA TYPES OF VARIABLES Definition Qualitative variables measure a quality or characteristic on each experimental unit. Quantitative variables measure a numerical quantity or amount on each experimental unit. Definition Definition A discrete variable can assume only a finite or countable number of values. A continuous variable can assume the infinitely many values corresponding to the points on a line interval. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 7 / 19
  • 8. GRAPHS FOR CATEGORICAL DATA Graphs for Categorical Data After the data have been collected, they can be consolidated and summarized to show the following information: (i) What values of the variable have been measured (ii) How often each value has occurred For this purpose, you can construct a statistical table that can be used to display the Example A bag contains 25 candies: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 8 / 19
  • 9. GRAPHS FOR CATEGORICAL DATA So, the Statistical table for last page example is as follows: Also, it is possible to express the frequency of each categories using following formulas: (i) Relative frequency= frequency (n is the total number of n measurements) (ii) Percent= 100 × Relative frequency The following table contain the relative frequency and percent for each categories of last example: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 9 / 19
  • 10. GRAPHS FOR CATEGORICAL DATA Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 10 / 19
  • 11. GRAPHS FOR CATEGORICAL DATA Example Fifty people are grouped into four categories A, B, C, and D and the number of people who fall into each category is shown in the table: The following figure is the bar chart for upper table: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 11 / 19
  • 12. GRAPHS FOR CATEGORICAL DATA and the pie chart is as follows: Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 12 / 19
  • 13. GRAPHS FOR QUANTITATIVE DATA GRAPHS FOR QUANTITATIVE DATA Line Charts When a quantitative variable is recorded over time at equally spaced intervals (such as daily, weekly, monthly, quarterly, or yearly), the data set forms a time series. Time series data are most effectively presented on a line chart with time as the horizontal axis. The idea is to try to discern a pattern or trend that will likely continue into the future, and then to use that pattern to make accurate predictions for the immediate future. Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 13 / 19
  • 14. GRAPHS FOR QUANTITATIVE DATA Dotplots Many sets of quantitative data consist of numbers that cannot easily be separated into categories or intervals of time. You need a different way to graph this type of data! The simplest graph for quantitative data is the dotplot. For a small set of measurements for example, the set 2, 6, 9, 3, 7, 6 you can simply plot the measurements as points on a horizontal axis. Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 14 / 19
  • 15. GRAPHS FOR QUANTITATIVE DATA Stem and Leaf Plots Another simple way to display the distribution of a quantitative data set is the stem and leaf plot. This plot presents a graphical display of the data using the actual numerical values of each data point. How Do I Construct a Stem and Leaf Plot? 1. Divide each measurement into two parts: the stem and the leaf . 2. List the stems in a column, with a vertical line to their right. 3. For each measurement, record the leaf portion in the same row as its corresponding stem. 4. Order the leaves from lowest to highest in each stem. 5. Provide a key to your stem and leaf coding so that the reader can recreate the actual measurements if necessary. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 15 / 19
  • 16. GRAPHS FOR QUANTITATIVE DATA Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 16 / 19
  • 17. GRAPHS FOR QUANTITATIVE DATA Example Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 17 / 19
  • 18. Interpreting Graphs with a Critical Eye Definition A distribution is symmetric if the left and right sides of the distribution, when divided at the middle value, form mirror images. Definition A distribution is skewed to the right if a greater proportion of the measurements lie to the right of the peak value. Distributions that are skewed right contain a few unusually large measurements. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 18 / 19
  • 19. Interpreting Graphs with a Critical Eye Definition A distribution is skewed to the left if a greater proportion of the measurements lie to the left of the peak value. Distributions that are skewed left contain a few unusually small measurements. Definition A distribution is unimodal if it has one peak; a bimodal distribution has two peaks.Bimodal distributions often represent a mixture of two different populations in the data set. Kian Jahromi () Chapter I : Describing Data With Graphs May 31, 2012 19 / 19