Download Chapter 12: Statistics and Probability

Chapter 12: Statistics
and Probability
Section 12.3: Distributions of Data
Quote of the Day
• “Be kind, for everyone you meet is fighting a hard battle.”
• Plato
Then/Now
You calculated measures of central tendency and
variation.
• Describe the shape of a distribution.
• Use the shapes of distributions to select
appropriate statistics.
Vocabulary
• A Distribution of data shows the observed or theoretical
frequency of each possible data value.
• A Histogram is a type of bar graph used to display data that
have been organized into equal intervals.
• A histogram is useful when viewing the overall distribution of the
data within a set over its range.
• You can see the shape of the distribution by drawing a curve over
the histogram
Concept
Example 1
Distribution Using a Histogram
Use the following data to construct a histogram and use
it to describe the shape of the distribution. Group the
data by intervals of 4. (0-4, 5-8, 9-12, etc….)
9, 18, 22, 12, 24, 25, 19, 25, 2
5, 28, 12, 22, 19, 28, 15, 23, 6
8, 27, 17, 14, 22, 21, 13, 24, 21
9, 25, 16, 24, 16, 25, 27, 21, 10
Example 1
Distribution Using a Histogram
Answer: The distribution is negatively skewed.
Notes
• A box-and-whisker plot displays the spread of a data set by
dividing it into four quartiles.
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The lowest data value is represented by the farthest left point
The highest data value is represented by the farthest right point
The median is the middle of the box
The median of the lower half of the data is the left side of the box
The median of the upper half of the data is the right side of the
box
• A box-and-whisker plot can also be used to identify the shape
of a distribution.
Concept
Example 2
Distribution Using a Box-and-Whisker Plot
Use the data below to construct a box-and-whisker
plot and use it to determine the shape of the
distribution.
9, 18, 22, 12, 24, 25, 19, 25, 2
5, 28, 12, 22, 19, 28, 15, 23, 6
8, 27, 17, 14, 22, 21, 13, 24, 21
9, 25, 16, 24, 16, 25, 27, 21, 10
Example 2
Distribution Using a Box-and-Whisker Plot
Answer: The right whisker is longer than the left, and
the median is to the left of the plot. Thus, the
distribution is positively skewed.
Notes
• You now have two methods that you can use to describe data
using statistics.
• The mean and median describe the center.
• The Standard Deviation and quartiles describe the spread.
• You can use the shape of the distribution to choose the most
appropriate statistics that describe the center and spread of a
set of data.
Notes
• Outliers can have a strong effect on the mean of a data set,
while the median is less affected.
• So, when the distribution is skewed, the mean lies away from
the majority of the data toward the tail.
• The median is less affected and stays near the majority of the
data.
• When choosing appropriate statistics to represent a set of
data, first determine the shape of the distribution
• If the distribution is relatively symmetric, the mean and standard
deviation can be used
• If the distribution is skewed or has outliers, use the five-number
summary.
Example 3
Choose Appropriate Statistics
Describe the center and spread of the data using
either the mean and standard deviation or the fivenumber summary. Justify your choice by
constructing a histogram for the data.
78, 68, 72, 71, 79, 67, 71, 78, 70
80, 76, 82, 82, 70, 84, 72, 71, 85
67, 86, 74, 86, 73, 72, 77, 87, 70
66, 88, 75, 72, 76, 71, 90, 69, 94
Example 3
Choose Appropriate Statistics
Answer: The range is 64 – 96 or 28. The median is
74.5, and half of the data are between 71
and 82.