# 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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