displaying and summarizing quantitative data€¦ · title: deveaux intro stats, 4e author: deveaux...

1 Copyright © 2014, 2012, 2009 Pearson Education, Inc.

Chapter 3

Displaying and

Summarizing

Quantitative Data


3.1

Displaying

Quantitative

Variables


Histograms

• Histogram: A chart that

displays quantitative data

• Great for seeing the distribution of the data

• Most earthquake generating tsunamis have magnitudes

between 6.5 and 8.

• Japan and Sumatra quakes (9.0 and 9.1) are rare.

• Quakes under 5 rarely cause tsunamis.

• Quakes between 7.0 and 7.5 most common for

causing tsunamis

A histogram of tsunami generating earthquakes


Choosing the Bin Width

• Different bin widths tell different

stories.

• Choose the width that best shows

the important features.

• Presentations can feature two

histograms that present the same

data in different ways.

• A gap in the histogram means that

there were no occurrences in that

range.


Relative Frequency Histograms

• Relative Frequency Histogram

• The vertical axis represents

the relative frequency, the

frequency divided by the total.

• The horizontal axis is the same

as the horizontal axis for the frequency histogram.

• The shape of the relative frequency histogram is the

same as the frequency histogram.

• Only the scale of the y-axis is different.


Histograms and StatCrunch

• Enter Data.

• Graphics →

Histogram

• Click on the data

variable and Next.

• Select Frequency or

Relative Frequency.

• Put in starting value

and/or Binwidth if

desired.

• Click Next twice, and type in

labels. Click Create Graph.


Stem-and-Leaf Displays

• Stem-and-Leaf: Shows both the

shape of the distribution and all

of the individual values

• Not as visually pleasing as a

histogram; more technical looking

• Can only be used for small collections of data

• The first column (stems) represents the leftmost digit.

• The second column (leaves) shows the remaining digit(s).


Stem and Leaf with StatCrunch

• Enter Data

• Graphics → Stem and Leaf

• Click on the variable name

and Next

• Select Outlier Trimming

Type and Create Graph!


Dotplots

• Dotplot: Displays dots to describe

the shape of the distribution

• There were 30 races with a winning

time of 122 seconds.

• Good for smaller data sets

• Visually more appealing than

stem-and-leaf

• In StatCrunch:

Graphics → Dotplot


Think Before you Draw

• Is the variable quantitative? Is the answer to the survey

question or result of the experiment a number whose

units are known?

• Histograms, stem-and-leaf diagrams, and dotplots

can only display quantitative data.

• Bar and pie charts display categorical data.


3.2

Shape


Modes

• A Mode of a histogram is a hump or high-frequency bin.

• One mode → Unimodal

• Two modes → Bimodal

• 3 or more → Multimodal

Unimodal Multimodal Bimodal


Uniform Distributions

• Uniform Distribution: All the bins have the same

frequency, or at least close to the same frequency.

• The histogram for a uniform distribution will be flat.


Symmetry

• The histogram for a symmetric distribution will look the

same on the left and the right of its center.

Symmetric Not

Symmetric Symmetric


Skew

• A histogram is skewed right if the longer tail is on the

right side of the mode.

• A histogram is skewed left if the longer tail is on the left

side of the mode.

Skewed Left Skewed Right


Outliers

• An Outlier is a data value that is far above or far below

the rest of the data values.

• An outlier is sometimes just

an error in the data collection.

• An outlier can also be the

most important data value.

• Income of a CEO

• Temperature of a person with

a high fever

• Elevation at Death Valley


Example

• The histogram shows the amount

of money spent by a credit card

company’s customers. Describe

and interpret the distribution.

• The distribution is unimodal. Customers most

commonly spent a small amount of money.

• The distribution is skewed right. Many customers

spent only a small amount and a few were spread out

at the high end.

• There is an outlier at around $7000. One customer

spent much more than the rest of the customers.


3.3

Center


The Median

• Median: The center of the

data values

• Half of the data values are to

the left of the median and half

are to the right of the median.

• For symmetric distributions, the median is directly

in the middle.


Calculating the Median: Odd Sample Size

• First order the numbers.

• If there are an odd number of numbers, n, the median is

at position .

• Find the median of the numbers: 2, 4, 5, 6, 7, 9, 9.

•

• The median is the fourth number: 6

• Note that there are 3 numbers to the left of 6 and 3 to

the right.

1

2

n

1 7 14

2 2

n


Calculating the Median: Even Sample Size

• First order the numbers.

• If there are an even number of numbers, n, the median

is the average of the two middle numbers: .

• Find the median of the numbers: 2, 2, 4, 6, 7, 8.

•

• The median is the average of the third and the fourth

numbers:

6

32 2

n

, 12 2

n n

4 6Median 5

2


3.4

Spread


Spread

• Locating the center is only part of the story

• Are the data all near the center or are they spread out?

• Is the highest value much higher than the lowest value?

• To describe data, we must discuss both the center and

the spread.


Range

• The range is the difference between the maximum and

minimum values.

Range = Maximum – Minimum

• The ages of the guests at your dinner party are:

16, 18, 23, 23, 27, 35, 74

• The range is: 74 – 16 = 58

• The range is sensitive to outliers. A single high or low

value will affect the range significantly.


Percentiles and Quartiles

• Percentiles divide the data in one hundred groups.

• The nth percentile is the data value such that n percent

of the data lies below that value.

• For large data sets, the median is the 50th percentile.

• The median of the lower half of the data is the 25th

percentile and is called the first quartile (Q1).

• The median of the upper half of the data is the 75th

percentile and is called the third quartile (Q3).


StatCrunch, Q1, Median, and Q3

• Enter the data.

• Stat → Summary Stats

→ Columns

• Click on the variable and

then Calculate.


The Interquartile Range

• The Interquartile Range (IQR) is the difference between

the upper quartile and the lower quartile

IQR = Q3 – Q1

• The IQR measures the range of the middle half of the

data.

• Example: If Q1 = 23 and Q3 = 44 then

IQR = 44 – 23 = 21


The Interquartile Range

• The Interquartile Range for earthquake causing

tsunamis is 0.9.

• The picture below shows the meaning of the IQR.


Benefits and Drawbacks of the IQR

• The Interquartile Range is not sensitive to outliers.

• The IQR provides a reasonable summary of the spread

of the distribution.

• The IQR shows where typical values are, except for the

case of a bimodal distribution.

• The IQR is not great for a general audience since most

people do not know what it is.


3.5

Boxplots and

5-Number

Summaries


5-Number Summary

• The 5-Number Summary provides a numerical

description of the data. It consists of

• Minimum

• First Quartile (Q1)

• Median

• Third Quartile (Q3)

• Maximum

• The list to the right shows the

5-Number Summary for the

tsunami data.


Interpreting the 5-Number Summary

• The smallest tsunami-causing earthquake

had magnitude 3.7.

• The largest tsunami-causing earthquake

had magnitude 9.1.

• The middle half of tsunami-causing

earthquakes is between 6.7 and 7.6.

• Half of tsunami-causing earthquakes have

magnitudes below 7.2 and half are above 7.2.

• A tsunami-causing earthquake less than 6.7 is small.

• A tsunami-causing earthquake more than 7.6 is small.


Boxplots

• A Boxplot is a chart that displays the

5-Point Summary and the outliers.

• The Box shows the Interquartile Range.

• The dashed lines are called fences,

outside the fences lie the outliers.

• Above and below the box are the whiskers

that display the most extreme data values

within the fences.

• The line inside the box shows the median.


Finding the Fences

• The lower fence is defined by

Lower Fence = Q1 – 1.5 × IQR

• The upper fence is defined by

Upper Fence = Q3 + 1.5 × IQR

• Tsunami Example: Q1 = 6.7, Q3 = 7.6

IQR = 7.6 – 6.7 = 0.9

• Lower Fence = 6.7 – 1.5 × 0.9 = 5.35

• Upper Fence = 7.6 + 1.5 × 0.9 = 8.95


StatCrunch and Boxplots

• Enter data and go to

Graphics → Boxplot.

• Click on the variable and

Next.

• Check “Use fences to

identify outliers.” Then

Next

• Type in labels and click on

Create Graph.


Step-by-Step Example of Shape, Center,

Spread: Flight Cancellations • Question: How often are flights cancelled?

• Who? Months

• What? Percentage of Flights Cancelled at U.S. Airports

• When? 1995 – 2011

• Where? United States

• How? Bureau of Transportation Statistics Data


Flight Cancellations: Think

• Identify the Variable

• Percent of flight cancellations at U.S. airports

• Quantitative: Units are percentages.

• How will be data be summarized?

• Histogram

• Numerical Summary

• Boxplot


Flight Cancellations: Show

• Use StatCrunch to create the histogram,

boxplot, and numerical summary.


Flight Cancellations: Tell

• Describe the shape, center, and spread of the

distribution. Report on the symmetry, number of modes,

and any gaps or outliers. You should also mention any

concerns you may have about the data.

• Skewed to the Right: Can’t be a negative percent.

Bad weather and other airport troubles can cause

extreme cancellations.

• IQR is small: 1.23%. Consistency among cancellation

percents

• Extraordinary outlier at 20.2%: September 2001


3.6

The Center of

Symmetric

Distributions:

The Mean


The Mean

• The Mean is what most people think of as the average.

• Add up all the numbers and divide by the number of

numbers.

• Recall that S means “Add them all.”

• In StatCrunch, the mean is listed in the

Summary Statistics.

yy

n


The Mean is the “Balancing Point”

• If you put your finger

on the mean, the

histogram will

balance perfectly.


Mean Vs. Median

• For symmetric distributions, the mean and the median

are equal.

• The balancing point is at the center.

• The tail “pulls” the mean towards it more than it does to

the median.

• The mean is more sensitive to outliers than the median.


The Mean Is Attracted to the Outlier

• The mean is larger

than the median

since it is “pulled”

to the right by the

outlier.

• The median is a better

measure of the center

for data that is skewed.


Why Use the Mean?

• Although the median is a better measure of the center,

the mean weighs in large and small values better.

• The mean is easier to work with.

• For symmetric data, statisticians would rather use the

mean.

• It is always ok to report both the mean and the median.


3.7

The Spread of

Symmetric

Distributions:

The Standard

Deviation


The Variance

• The variance is a measure of how far the data is spread

out from the mean.

• The difference from the mean is: .

• To make it positive, square it.

• Then find the average of all of these distances, except

instead of dividing by n, divide by n – 1.

• Use s2 to represent the variance.

• The variance will mostly be used to find the standard

deviation s which is the square root of the variance.

y y

2

2

1

y ys

n


Standard Deviation

• The variance’s units are the square of the original units.

• Taking the square root of the variance gives the

standard deviation, which will have the same units as y.

• The standard deviation is a number that is close to the

average distances that the y values are from the mean.

• If data values are close to the mean (less spread out),

then the standard deviation will be small.

• If data values are far from the mean (more spread out),

then the standard deviation will be large.

2

1

y ys

n


The Standard Deviation and Histograms

A B C

Answer: C, A, B

Order the histograms below from smallest standard deviation to largest standard deviation.


3.8

Summary—What

to Tell About a

Quantitative

Variable


What to Tell

• Histogram, Stem-and-Leaf, Boxplot

• Describe modality, symmetry, outliers

• Center and Spread

• Median and IQR if not symmetric

• Mean and Standard Deviation if symmetric.

• Unimodal symmetric data: IQR > s. Check for errors.

• Unusual Features

• For multiple modes, possibly split the data into groups.

• When there are outliers, report the mean and standard

deviation with and without the outliers.


Example: Fuel Efficiency

• The car owner has checked the fuel efficiency each time

he filled the tank. How would you describe the fuel

efficiency?

• Plan: Summarize the distribution of the car’s fuel

efficiency.

• Variable: mpg for 100 fill ups, Quantitative

• Mechanics: show a histogram

• Fairly symmetric

• Low outlier


Fuel Efficiency Continued

• Which to report?

• The mean and median are close.

• Report the mean and standard deviation.

• Conclusion

• Distribution is unimodal and symmetric.

• Mean is 22.4 mpg.

• Low outlier may be investigated, but limited effect on

the mean

• s = 2.45; from one filling to the next, fuel efficiency

differs from the mean by an average of about 2.45 mpg.


What Can Go Wrong?

• Don’t make a histogram for categorical data.

• Don’t look for shape, center,

and spread for a bar chart.

• Choose a bin width appropriate

for the data.


What Can Go Wrong? Continued

• Do a reality check

• Don’t blindly trust your calculator. For example, a

mean student age of 193 years old is nonsense.

• Sort before finding the median and percentiles.

• 315, 8, 2, 49, 97 does not have median of 2.

• Don’t worry about small differences in the quartile

calculation.

• Don’t compute numerical summaries for a categorical

variable.

• The mean Social Security number is meaningless.


What Can Go Wrong? Continued

• Don’t report too many decimal places.

• Citing the mean fuel efficiency as 22.417822453 is

going overboard.

• Don’t round in the middle of a calculation.

• For multiple modes, think about separating groups.

• Heights of people → Separate men and women

• Beware of outliers, the mean and standard deviation are

sensitive to outliers.

• Use a histogram or dotplot to ensure that the mean

and standard deviation really do describe the data.

displaying and summarizing quantitative data€¦ · title: deveaux intro stats, 4e author: deveaux...

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