Name:
Ms. D’Amato
Date:
Block:
Chapter 4: Displaying Quantitative Data
Could we have looked at Enron stock prices over time and seen any hints of trouble??
A table full of numbers can be confusing so let's
HOW???
- Bar charts and pie charts are for
- We need a way to create a picture for quantitative data.
.
variables.
Histograms: Displaying the Distribution of Price Changes
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We can’t list all the quantitative variables since there are too many of them.
First, slice up the entire span of values covered by the quantitative variable into equal-width
piles called
.
The bins and counts in each bin give the
of the quantitative variable.
Once we have the bins and counts, we can then display them by creating a
.
Like a bar chart, a histogram plots the bin counts as the heights of
.
Let’s again look at the data for the changes in Enron’s stock prices from 1997 – 2001 as a
histogram:
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The cases are months, so the height of each bar
shows the number of months that have price
changes falling into that bin.
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The Who are months and the What are price
changes.
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Displays the distribution of prices changes by
showing the number of months that have prices
changes in each of the bins.
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Does the distribution of Price Change look as
you expected?
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A bar chart has
order.
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In a histogram, there are no spaces because the bins slice up
of a quantitative variable.
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A gap indicates that no values fell in that bin.
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Both kinds of displays satisfy the area principle.
A
between the bars because the categories could appear in any
, replaces the counts on the vertical axis with the
of the total number of cases falling in each bin.
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Here is a relative frequency histogram of the monthly price changes in Enron stock:
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Should be the same shape, just different labels.
Creating Histograms:
1) Title your graph.
2) Label the axes. The vertical axis is the frequency and the horizontal axis is the quantitative
variable.
3) Choose a scale for your vertical axis. It must be consistent!
4) Determine the intervals of your bins for your quantitative variable. To do this, look at the
lowest and highest data value.
5) Choose what to do with values on the edges (example: does $10 go into the $5-10 bin or the
$10-15 bin). Typically, you put these values into the next higher bin.
6) Count the number values that fall into each bin, and then draw the bins. Remember, no spaces
between bins!!! Draw a bar extending from the lower value of each interval to the lower value
of the next interval. The height of each bar should be equal to the frequency of its
corresponding interval.
Stem-and-Leaf Displays
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Like a histogram but shows the
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Easier to draw than a histogram.
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Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health
clinic. Which graphical display do you prefer?
data values.
Creating A Stem-and-Leaf Display:
1) Cut each value into leading digits (stem) and trailing digit (leaf).
2) Use the stems to create labels for each bin.
3) List the bins down the left side of the chart.
4) Place all of the trailing digits in each bin, from least to greatest
5) Label your stem-and-leaf display and create a key!
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Contains all the information found in a histogram and satisfies the
and shows the distribution.
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Preserves the individual data values.
Dotplots
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A simple display. It just places a dot along an axis for each case in the data.
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Like a stem-and-leaf display, but with dots instead of digits for all the leaves.
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The dotplot below shows Kentucky Derby winning times (in seconds), plotting each race as its
own dot:
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Can be displayed
or
.
Think Before You Draw, Again
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Remember the “Make a picture” rule?
Now that we have options for data displays, you need to Think carefully about which type of
display to make.
Before making a stem-and-leaf display, a histogram, or a dotplot, check the:
o Quantitative Data Condition: The data are values of a quantitative variable whose units are
known.
Shape, Center, and Spread
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When describing a distribution, always tell about three things:
, and
.
,
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With histograms, stem-and-leaf displays, and dotplots, you can describe the
of the data in ways you couldn't for bar charts and pie charts
What Is the Shape of the Distribution?
1. Does the histogram have a single, central hump (peak) or several separated bumps?
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Humps/peaks in a histogram are called
If there is one main peak, it’s
If there are two main peaks, it’s
If there are three or more peaks, it’s
No peaks?! It’s
.
.
.
.
.
Which is which?
2. Is the histogram symmetric?
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If you can fold the histogram along a
through the middle
and have the edges match pretty closely, the histogram is symmetric.
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The (usually) thinner ends of a distribution are called the
. If one tail
stretches out farther than the other, the histogram is said to be
to the
side of the longer tail.
In the figure below, the histogram on the left is said to be skewed left, while the
histogram on the right is said to be skewed right.
3. Do any unusual features stick out?
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Sometimes it’s the unusual features that tell us something interesting or exciting about
the data.
You should always mention any stragglers, or
, that stand off away
from the body of the distribution.
Outliers can affect almost every method we discuss in this course. They can be
informative or might just be an error.
Are there any gaps in the distribution? If so, we might have data from more than one
group.
The following histogram has outliers — there are three cities in the leftmost bar:
Where is the Center of the Distribution?
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If you had to pick a single number to describe all the data what would you pick?
It’s easy to find the center when a histogram is unimodal and symmetric — it’s right in the
middle.
On the other hand, it’s not so easy to find the center of a skewed histogram or a histogram with
more than one mode.
For now, we will “eyeball” the center of the distribution. In the next chapter we will find the
center numerically.
How Spread Out is the Distribution?
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Variation matters, and Statistics is about variation.
Are the values of the distribution tightly clustered around the center or more spread out?
Would you rather invest in a stock whose price gyrates wildly or one the grows steadily?
In the next two chapters, we will talk about spread…
Comparing Distributions
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Often we would like to compare two or more distributions instead of looking at one
distribution by itself.
When looking at two or more distributions, it is very important that the histograms have been
put on the
. Otherwise, we cannot really compare the two distributions.
When we compare distributions, we talk about the shape, center, and spread of each
distribution.
Compare the following distributions of ages for female and male heart attack patients:
What can we tell from these histograms? Look
at shape, center, and spread.
Timeplots: Order, Please!
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For some data sets, we are interested in how the data behave over time. In these cases, we
construct timeplots of the data.
Re-expressing Skewed Data to Improve Symmetry
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One way to make a skewed distribution more symmetric is to re-express or
the data by applying a simple function (example, logarithmic function).
Note the change in skewness from the raw data (Figure 4.11) to the transformed data:
What Can Go Wrong?
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Don’t make a histogram of a categorical variable — bar charts or pie charts should be used for
categorical data.
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Don’t look for shape, center, and spread of a bar chart.
o Bar charts displays categorical data and can be arranged in any order left to right.
o Concepts like symmetry, center and spread make sense only for quantitative data.
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Don’t use bars in every display—save them for histograms and bar charts.
o The bars in bar charts indicate how many cases of a categorical variable there are while
bars in histograms indicate the number of cases piled in each interval of a quantitative
variable.
o Below is a badly drawn timeplot and the proper histogram for the number of eagles
sighted in a collection of weeks:
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Choose a bin width appropriate to the data.
o Changing the bin width changes the appearance of the histogram:
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Avoid inconsistent scales, either within the display or when comparing two displays.
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Label clearly so a reader knows what the plot displays.
o Good intentions, bad plot:
What have we learned?
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We’ve learned how to make a picture for quantitative data to help us see the story the data have
to Tell.
We can display the distribution of quantitative data with a histogram, stem-and-leaf display,
or dotplot.
Tell about a distribution by talking about shape, center, spread, and any unusual features.
We can compare two quantitative distributions by looking at side-by-side displays (plotted on
the same scale).
Trends in a quantitative variable can be displayed in a timeplot.