Transcript Example - Cengage

Chapter 2
Turning Data
Into
Information
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
2.1 Raw Data
• Raw data are for numbers and category labels
that have been collected but have not yet been
processed in any way.
• When measurements are taken from a subset of a
population, they represent sample data.
• When all individuals in a population are measured,
the measurements represent population data.
• Descriptive statistics: summary numbers
for either population or a sample.
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2.2 Types of Data
• Raw data from categorical variables consist of
group or category names that don’t necessarily
have a logical ordering. Examples: eye color,
country of residence.
• Categorical variables for which the categories
have a logical ordering are called ordinal
variables. Examples: highest educational degree
earned, tee shirt size (S, M, L, XL).
• Raw data from quantitative variables consist
of numerical values taken on each individual.
Examples: height, number of siblings.
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Asking the Right Questions
One Categorical Variable
Question 1a: How many and what percentage of
individuals fall into each category?
Example: What percentage of college students favor the
legalization of marijuana, and what percentage of
college students oppose legalization of marijuana?
Question 1b: Are individuals equally divided across
categories, or do the percentages across categories
follow some other interesting pattern?
Example: When individuals are asked to choose a
number from 1 to 10, are all numbers equally likely
to be chosen?
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Asking the Right Questions
Two Categorical Variables
Question 2a: Is there a relationship between the two
variables, so that the category into which individuals fall
for one variable seems to depend on which category they
are in for the other variable?
Example: In Case Study 1.6, we asked if the risk of having a
heart attack was different for the physicians who took
aspirin than for those who took a placebo.
Question 2b: Do some combinations of categories stand out
because they provide information that is not found by
examining the categories separately?
Example: The relationship between smoking and lung cancer
was detected, in part, because someone noticed that the
combination of being a nonsmoker and having lung
cancer is unusual.
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Asking the Right Questions
One Quantitative Variable
Question 3a: What are the interesting summary measures,
like the average or the range of values, that help us
understand the collection of individuals who were
measured?
Example: What is the average handspan measurement, and
how much variability is there in handspan measurements?
Question 3b: Are there individual data values that provide
interesting information because they are unique or stand
out in some way?
Example: What is the oldest recorded age of death for a
human? Are there many people who have lived nearly
that long, or is the oldest recorded age a unique case?
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Asking the Right Questions
One Categorical and One Quantitative Variable
Question 4a: Are the measurements similar across
categories?
Example: Do men and women drive at the same
“fastest speeds” on average?
Question 4b: When the categories have a natural
ordering (an ordinal variable), does the measurement
variable increase or decrease, on average, in that
same order?
Example: Do high school dropouts, high school
graduates, college dropouts, and college graduates
have increasingly higher average incomes?
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Asking the Right Questions
Two Quantitative Variables
Question 5a: If the measurement on one variable is high
(or low), does the other one also tend to be high (or low)?
Example: Do taller people also tend to have larger
handspans?
Question 5b: Are there individuals whose combination of
data values provides interesting information because that
combination is unusual?
Example: An individual who has a very low IQ score but can
perform complicated arithmetic operations very quickly
may shed light on how the brain works. Neither the IQ
nor the arithmetic ability may stand out as uniquely low
or high, but it is the combination that is interesting.
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Explanatory and Response Variables
Many questions are about the relationship
between two variables.
It is useful to identify one variable as the
explanatory variable and the other variable
as the response variable.
In general, the value of the explanatory variable
for an individual is thought to partially explain the
value of the response variable for that individual.
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2.3 Summarizing One or Two
Categorical Variables
Numerical Summaries
• Count how many fall into each category.
• Calculate the percent in each category.
• If two variables, have the categories of
the explanatory variable define the rows
and compute row percentages.
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Example 2.1 Importance of Order
Survey of n = 190 college students.
About half (92) given the question:
“Randomly pick a letter --- S or Q.”
Note: 66% picked the first choice of S.
Other half (98) given the question:
“Randomly pick a letter --- Q or S.”
Note: 54% picked the first choice of Q.
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Example 2.2 Lighting the Way
to Nearsightedness
Survey of n = 479 children.
Those who slept with nightlight or in fully lit
room before age 2 had higher incidence of
nearsightedness (myopia) later in childhood.
Note: Study does not prove sleeping with light
actually caused myopia in more children.
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Visual Summaries
for Categorical Variables
• Pie Charts: useful for summarizing
a single categorical variable if not
too many categories.
• Bar Graphs: useful for summarizing
one or two categorical variables and
particularly useful for making comparisons
when there are two categorical variables.
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Example 2.3 Humans Are Not
Good Randomizers
Survey of n = 190 college students.
“Randomly pick a number between 1 and 10.”
Results: Most chose 7, very few chose 1 or 10.
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Example 2.4 Revisiting Nightlights
and Nearsightedness
Survey of
n = 479 children.
Response:
Degree
of Myopia
Explanatory:
Amount of
Sleeptime
Lighting
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2.4 Finding Information
in Quantitative Data
Long list of numbers – needs to be organized
to obtain answers to questions of interest.
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Five-Number Summaries
• Find extremes (high, low),
the median, and the quartiles
(medians of lower and upper
halves of the values).
• Quick overview of the data values.
• Information about the center,
spread, and shape of data.
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Example 2.5 Right Handspans
About 25% of handspans of females are
between 12.5 and 19.0 centimeters,
about 25% are between 19 and 20 cm,
about 25% are between 20 and 21 cm, and
about 25% are between 21 and 23.25 cm.
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Interesting Features of
Quantitative Variables
• Location: center or average.
e.g. median
• Spread: variability
e.g. difference between two
extremes or two quartiles.
• Shape: (later in Section 2.5)
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Outliers and How to Handle Them
Outlier: a data point that is not
consistent with the bulk of the data.
• Look for them via graphs.
• Can have big influence on conclusions.
• Can cause complications in some
statistical analyses.
• Cannot discard without justification.
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Example 2.6 Ages of Death
of U.S. First Ladies
Partial Data Listing and five-number summary:
Extremes are more interesting here:
Who died at 34? Martha Jefferson
Who lived to be 97? Bess Truman
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Possible Reasons for Outliers
and Reasonable Actions
• Mistake made while taking measurement or entering it
into computer. If verified, should be discarded/corrected.
• Individual in question belongs to a different group than
bulk of individuals measured. Values may be discarded if
summary is desired and reported for the majority group
only.
• Outlier is legitimate data value and represents natural
variability for the group and variable(s) measured.
Values may not be discarded — they provide important
information about location and spread.
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Example 2.7 Tiny Boatsmen
Weights (in pounds) of 18 men on crew team:
Cambridge:188.5, 183.0, 194.5, 185.0, 214.0,
203.5, 186.0, 178.5, 109.0
Oxford:
186.0, 184.5, 204.0, 184.5, 195.5,
202.5, 174.0, 183.0, 109.5
Note: last weight in each list is unusually small.
They are the coxswains for their teams,
while others are rowers.
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2.5 Pictures for
Quantitative Data
• Histograms: similar to bar graphs, used
for any number of data values.
• Stem-and-leaf plots and dotplots:
present all individual values, useful for
small to moderate sized data sets.
• Boxplot or box-and-whisker plot:
useful summary for comparing two
or more groups.
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Interpreting Histograms, Stemplots,
and Dotplots
• Values are centered around 20 cm.
• Two possible low outliers.
• Apart from outliers, spans range from about 16 to 23 cm.
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Describing Shape
• Symmetric, bell-shaped
• Symmetric, not bell-shaped
• Skewed Right: values trail off
to the right
• Skewed Left: values trail off
to the left
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Creating a Histogram
Step 1: Decide how many equally spaced (same
width) intervals to use for the horizontal axis.
Between 6 and 15 intervals is a good number.
Step 2: Decide to use frequencies (count) or relative
frequencies (proportion) on the vertical axis.
Step 3: Draw equally spaced intervals on horizontal
axis covering entire range of the data. Determine
frequency or relative frequency of data values in
each interval and draw a bar with corresponding
height. Decide rule to use for values that fall on
the border between two intervals.
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Creating a Dotplot
“A dotplot displays a dot for each
observation along a number line.
If there are multiple occurrences of an
observation, or if observations are too close
together, then dots will be stacked vertically.
If there are too many points to fit vertically
in the graph, then each dot may represent
more than one point.”
(Minitab, Release 12.1, 1998)
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Creating a Stem-and-Leaf Plot
Step 1: Determine stem values. The “stem”
contains all but the last of the displayed digits
of a number. Stems should define equally
spaced intervals.
Step 2: For each individual, attach a “leaf”
to the appropriate stem. A “leaf” is the last
of the displayed digits of a number. Often
leaves are ordered on each stem.
Note: More than one way to define stems.
Can use split-stems or truncate/round values first.
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Example 2.8 Big Music Collection
About how many CDs do you own?
Stem is ‘100s’ and leaf unit is ‘10s’.
Final digit is truncated.
Numbers ranged from 0 to about 450,
with 450 being a clear outlier and
most values ranging from 0 to 99.
The shape is skewed right.
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2.6 Numerical Summaries
of Quantitative Data
Notation for Raw Data:
n = number of individuals in a data set
x1, x2 , x3,…, xn represent individual raw data values
Example: A data set consists of handspan
values in centimeters for six females;
the values are 21, 19, 20, 20, 22, and 19.
Then, n = 6
x1= 21, x2 = 19, x3 = 20, x4 = 20, x5 = 22, and x6 = 19
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Describing the Location
of a Data Set
• Mean: the numerical average
• Median: the middle value (if n odd)
or the average of the middle two
values (n even)
Symmetric: mean = median
Skewed Left: mean < median
Skewed Right: mean > median
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Determining the Mean and Median
x

x
i
The Mean
where
x
i
n
means “add together all the values”
The Median
If n is odd: M = middle of ordered values.
Count (n + 1)/2 down from top of ordered list.
If n is even: M = average of middle two ordered values.
Average values that are (n/2) and (n/2) + 1
down from top of ordered list.
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Example 2.9 Will “Normal” Rainfall
Get Rid of Those Odors?
Data: Average rainfall (inches)
for Davis, California for 47 years
Mean = 18.69 inches
Median = 16.72 inches
In 1997-98, a company
with odor problem blamed
it on excessive rain.
That year rainfall was
29.69 inches. More rain
occurred in 4 other years.
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The Influence of Outliers
on the Mean and Median
Larger influence on mean than median.
High outliers will increase the mean.
Low outliers will decrease the mean.
If ages at death are: 70, 72, 74, 76, and 78
then mean = median = 74 years.
If ages at death are: 35, 72, 74, 76, and 78
then median = 74 but mean = 67 years.
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Describing Spread: Range
and Interquartile Range
• Range = high value – low value
• Interquartile Range (IQR) =
upper quartile – lower quartile
• Standard Deviation
(covered later in Section 2.7)
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Example 2.10 Fastest Speeds Ever Driven
Five-Number
Summary
for 87 males
•
•
•
Median = 110 mph measures the center of the data
Two extremes describe spread over 100% of data
Range = 150 – 55 = 95 mph
Two quartiles describe spread over middle 50% of data
Interquartile Range = 120 – 95 = 25 mph
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Notation and Finding the Quartiles
Split the ordered values into the half
that is below the median and the half
that is above the median.
Q1 = lower quartile
= median of data values
that are below the median
Q3 = upper quartile
= median of data values
that are above the median
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Example 2.10 Fastest Speeds (cont)
Ordered Data
(in rows of
10 values)
for the 87 males:
55 60 80 80 80 80 85 85 85 85
90 90 90 90 90 92 94 95 95 95
95 95 95 100 100 100 100 100 100 100
100 100 101 102 105 105 105 105 105 105
105 105 109 110 110 110 110 110 110 110
110 110 110 110 110 112 115 115 115 115
115 115 120 120 120 120 120 120 120 120
120 120 124 125 125 125 125 125 125 130
130 140 140 140 140 145 150
• Median = (87+1)/2 = 44th value in the list = 110 mph
• Q1 = median of the 43 values below the median =
(43+1)/2 = 22nd value from the start of the list = 95 mph
• Q3 = median of the 43 values above the median =
(43+1)/2 = 22nd value from the end of the list = 120 mph
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Percentiles
The kth percentile is a number that has
k% of the data values at or below it and
(100 – k)% of the data values at or above it.
• Lower quartile = 25th percentile
• Median = 50th percentile
• Upper quartile = 75th percentile
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Picturing Location
and Spread with Boxplots
Boxplots for right handspans
of males and females.
• Box covers the middle
50% of the data
• Line within box marks
the median value
• Possible outliers are
marked with asterisk
• Apart from outliers, lines
extending from box reach
to min and max values.
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How to Draw a Boxplot
of a Quantitative Variable
Step 1: Label either a vertical axis or a horizontal axis
with numbers from min to max of the data.
Step 2: Draw box with lower end at Q1 and upper end at Q3.
Step 3: Draw a line through the box at the median M.
Step 4: Draw a line from Q1 end of box to smallest data
value that is not further than 1.5  IQR from Q1.
Draw a line from Q3 end of box to largest data value
that is not further than 1.5  IQR from Q3.
Step 5: Mark data points further than 1.5  IQR from either
edge of the box with an asterisk. Points represented
with asterisks are considered to be outliers.
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2.7 Bell-Shaped Distributions
of Numbers
Many measurements follow a predictable pattern:
• Most individuals are clumped around the center
• The greater the distance a value is from the
center, the fewer individuals have that value.
Variables that follow such a pattern are said
to be “bell-shaped”. A special case is called
a normal distribution or normal curve.
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Example 2.11 Bell-Shaped
British Women’s Heights
Data: representative sample of 199 married British couples.
Below shows a histogram of the wives’ heights with a normal
curve superimposed. The mean height = 1602 millimeters.
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Describing Spread
with Standard Deviation
Standard deviation measures variability
by summarizing how far individual
data values are from the mean.
Think of the standard deviation as
roughly the average distance
values fall from the mean.
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Describing Spread
with Standard Deviation
Both sets have same mean of 100.
Set 1: all values are equal to the mean so there is
no variability at all.
Set 2: one value equals the mean and other four values
are 10 points away from the mean, so the average
distance away from the mean is about 10.
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Calculating the Standard Deviation
Formula for the (sample) standard deviation:
 x  x 
2
s
i
n 1
The value of s2 is called the (sample) variance.
An equivalent formula, easier to compute, is:
s
x
2
i
 nx
2
n 1
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Calculating the Standard Deviation
Step 1: Calculate x, the sample mean.
Step 2: For each observation, calculate the
difference between the data value
and the mean.
Step 3: Square each difference in step 2.
Step 4: Sum the squared differences in step 3,
and then divide this sum by n – 1.
Step 5: Take the square root of the value
in step 4.
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Calculating the Standard Deviation
Consider four pulse rates: 62, 68, 74, 76
Step 1:
62  68  74  76 280
x

 70
4
4
Steps 2 and 3:
120
Step 4: s 
 40
4 1
2
Step 5: s  40  6.3
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Population Standard Deviation
Data sets usually represent a sample from a larger
population. If the data set includes measurements for
an entire population, the notations for the mean and
standard deviation are different, and the formula for
the standard deviation is also slightly different.
A population mean is represented by the symbol m
(“mu”), and the population standard deviation is
 x  m 
2

i
n
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Interpreting the Standard Deviation
for Bell-Shaped Curves:
The Empirical Rule
For any bell-shaped curve, approximately
• 68% of the values fall within 1 standard
deviation of the mean in either direction
• 95% of the values fall within 2 standard
deviations of the mean in either direction
• 99.7% of the values fall within 3 standard
deviations of the mean in either direction
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The Empirical Rule, the Standard
Deviation, and the Range
• Empirical Rule => the range from the
minimum to the maximum data values equals
about 4 to 6 standard deviations for data with
an approximate bell shape.
• You can get a rough idea of the value of the
standard deviation by dividing the range by 6.
Range
s
6
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Example 2.11 Women’s Heights (cont)
Mean height for the 199 British women is 1602 mm
and standard deviation is 62.4 mm.
• 68% of the 199 heights would fall in the range
1602  62.4, or 1539.6 to 1664.4 mm
• 95% of the heights would fall in the interval
1602  2(62.4), or 1477.2 to 1726.8 mm
• 99.7% of the heights would fall in the interval
1602  3(62.4), or 1414.8 to 1789.2 mm
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Example 2.11 Women’s Heights (cont)
Summary of the actual results:
Note: The minimum height = 1410 mm and the maximum
height = 1760 mm, for a range of 1760 – 1410 = 350 mm.
So an estimate of the standard deviation is:
Range 350
s

 58.3 mm
6
6
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Standardized z-Scores
Standardized score or z-score:
Observed value  Mean
z
Standard deviation
Example: Mean resting pulse rate for adult men is 70
beats per minute (bpm), standard deviation is 8 bpm.
The standardized score for a resting pulse rate of 80:
80  70
z
 1.25
8
A pulse rate of 80 is 1.25 standard deviations
above the mean pulse rate for adult men.
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The Empirical Rule Restated
For bell-shaped data,
• About 68% of the values have
z-scores between –1 and +1.
• About 95% of the values have
z-scores between –2 and +2.
• About 99.7% of the values have
z-scores between –3 and +3.
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