1. Chapter I : Describing Data With Graphs
Kian Jahromi
May 31, 2012
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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
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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.
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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
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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.
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6. VARIABLES AND DATA
The following data set is a multivariate data set. Each column itself is a
Univariate data set.
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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.
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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:
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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:
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10. GRAPHS FOR CATEGORICAL DATA
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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:
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12. GRAPHS FOR CATEGORICAL DATA
and the pie chart is as follows:
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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
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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
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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.
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16. GRAPHS FOR QUANTITATIVE DATA
Example
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17. GRAPHS FOR QUANTITATIVE DATA
Example
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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.
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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.
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