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Organizing & Visualizing Data
MATH 102
Contemporary Math
S. Rook
Overview
• Section 15.1 in the textbook:
– Populations & samples
– Frequency tables & relative frequency tables
– Histograms
– Stem-and-Leaf displays
Populations & Samples
Populations & Samples
• Statistics is the area of mathematics where we
collect data (information), analyze the data, and
predict outcomes based on the data
• Best case is when we can collect data from a
population
– i.e. ALL individuals or objects that satisfy certain
conditions
– e.g. Polling a class of 60 students to see who they
voted for Class President
Populations & Samples (Continued)
• Often infeasible to survey a population if it is
extremely large so we survey a sample
– i.e. A subset of the population
– e.g. Polling a sample of 5,000 prospective voters to
see who they voted for President out of all 1.2 million
prospective voters in a state
• An important aim of statistics involves taking data
acquired from a sample and using it to make
inferences about the larger population
– Thus extremely important to obtain a sample that has
the same makeup of the population
Populations & Samples (Example)
Ex 1: Identify whether the situation describes a
population or a sample:
a) By using the 10 students in the front row,
the instructor determined that the average of
the exam was 80 for the entire class of 30.
b) The full union voted 98 – 2 on the motion
to strike.
Frequency Tables & Relative
Frequency Tables
Frequency Tables & Relative
Frequency Tables
• Collected data is often arranged in a visual manner
so as to make it more understandable
• A frequency table lists options and the number of
objects that satisfy that option
• A relative frequency table lists options and the
proportion of objects that satisfy that option
– Proportion is calculated like probability
– The sum of all proportions MUST add to 1
Frequency Tables & Relative
Frequency Tables (Example)
Ex 2: Construct a frequency table for the data and
then extend the table to include relative frequency:
a) 10 rolls of a single six-sided die produced the
following results: 3, 5, 5, 6, 2, 4, 1, 5, 2, 1
b) A parking lot of 12 cars was observed and the
colors of the cars, (B)lue, (b)lack, (S)ilver, and
(R)ed were recorded: S, b, B, R, R, R, b, b, S, R, B,
B
Histograms
Histogram
• Another visual way to depict a collection of data is
through a histogram (i.e. bar graph)
– A graphical way to represent a frequency table
• Just like a Cartesian Plane a histogram has a
horizontal axis and a vertical axis
– Options are listed on the horizontal axis
– Frequencies are listed on the vertical axis
• Sometimes data is too spread out to see a pattern
– Group data into classes or bins
• The class width is the distance between the beginning
of one class and the beginning of the next class
Histograms (Example)
Ex 3: Construct a histogram to visually
represent the heights of athletes in a certain
sport. Use classes of width 2 starting at 64.5:
a) 69, 71, 74, 74, 78, 68, 68, 71, 75, 69, 76,
68, 69, 74, 65, 73
b) 72, 77, 71, 80, 69, 73, 69, 75, 66, 68, 77,
74, 76, 73, 72, 75
Stem-and-Leaf Displays
Stem-and-Leaf Display
• Another commonly used visual aid for
displaying data is a stem-and-leaf display
• View each data point as having a stem (usually
first digit) and a leaf (last digit)
• To create a stem-and-leaf display:
– List stems in the left column and draw a vertical
bar
– Write leaves with the same stem to the right in
ascending order
• Data with the same stem must be ordered
Stem-and-Leaf Displays (Example)
Ex 4: The following data are the ages of
students in a class. Represent the data by
using a stem-and-leaf display:
a) 29, 32, 34, 43, 47, 43, 22, 38, 42, 39, 37, 33,
42, 18, 22, 39, 21, 26, 18, 43
b) 32, 38, 22, 39, 21, 26, 28, 16, 13, 20, 21,
29, 22, 24, 33, 47, 23, 22, 18, 33
Summary
• After studying these slides, you should know how to
do the following:
– Differentiate between samples and populations
– Visually depict data by constructing a:
• Frequency table and/or relative frequency table
• Histogram
• Stem-and-leaf display
• Additional Practice:
– See problems in Section 15.1
• Next Lesson:
– Measures of Central Tendency (Section 15.2)