STA 205 Lecture Notes - Lecture 7: Box Plot, Standard Deviation, Popeyes Louisiana Kitchen
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Packet 7: Summarizing Quantitative Data Textbook pages: 48 – 50; 53 – 72
After completing this material, you should be able to:
• describe the distribution of a quantitative variable by discussing its shape, center, spread, and unusual
characteristics.
• calculate (using StatCrunch) measures of center and measures of spread.
• apply the Empirical Rule or Chebyshev’s Rule to a distribution when discussing the standard deviation.
• compare distributions using boxplots.
Recall: What is a quantitative variable?
To summarize a quantitative variable, we need a new graphical display – a bar graph cannot be used. We will first look at
histograms for graphically summarizing quantitative data. What exactly is a histogram?
Example: Money magazine undertook a study in 2009 to estimate the average cost for a visit to a hospital emergency
room. A random sample of 175 emergency room visits in a certain urban area was taken, and the out-of-pocket costs
associated with that visit were recorded. A histogram for the collected data is given below:
When summarizing or describing a distribution, the following four characteristics must be discussed:
1.
2.
3.
4.
When asked to
describe a
distribution, make
sure you address
these four
characteristics in
context and in
complete sentences.
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Aquantitative variable measures observations and
results in numeric responses with units
Packet 7: Summarizing Quantitative Data page 2
STA 205 Notes Buckley Fall 2018
Let’s consider each of these four characteristics individually.
Shape of the distribution
When describing a distribution, the first think we should consider is what shape the distribution has. We will
consider five common shapes (shown below):
Shape
Histogram
Description
Measures of Center
Once we know the shape of a distribution, it is common to summarize it by finding a “typical” value of the
distribution – these values are generally referred to as measures of center. There are two common measures of
center which are used:
Measure of Center
Notation
Description
The calculation of these measures, while not difficult, can be tedious. Instead of calculating these summary
statistics by hand, we will rely on the use of StatCrunch.
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Packet 7: Summarizing Quantitative Data page 3
STA 205 Notes Buckley Fall 2018
How does the shape of the distribution affect measures of center?
Measures of Spread
Unfortunately, a measure of center doesn’t adequately describe a distribution. We also must have some idea
how the values in the distribution vary. This requires a measure of spread. There are three common measures of
spread which are used:
Measure of Spread
Notation
Description
The calculation of these measures can be quite difficult – the formula for standard deviation is quite tedious.
Instead of calculating these summary statistics by hand, we will rely on the use of StatCrunch.
Unusual Observations
Unusual observations are often referred to as outliers. When determining if unusual observations are present in
the data, look for observations which do not follow the overall patter of the data. These will generally be
observations which are in the tail of the distribution – either very large or very small.
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