Transcript Six Fundamental Issues of Statistics in Theoretical & Methodological Context 1. Anecdotal accounts versus systematic evaluation of data. 2.

Six Fundamental Issues of
Statistics in Theoretical &
Methodological Context
1. Anecdotal accounts versus
systematic evaluation of data.
2. Social construction of reality.
3. How data were collected.
4. Beware of lurking variables.
5. Variation is everywhere.
6. Conclusions are always uncertain.
 What you’re going to know by the
end of the fall semester:

Descriptive statistics: how to describe the
features of data you’ve collected
 Basics of graphs & tables
 Overview of sampling design
 Exploratory data analysis: describe the
features of individual variables and the
relationships between variables
 Inferential statistics: how to draw
representative samples; & how to
use hypothesis tests, confidence
intervals, and significance tests to
estimate unknown characteristics of
a population based on a sample.
 Basics of how to evaluate data &
research.
 What
you’re going to know by
the end of the spring semester:
Exploratory Analysis
 Exploring data to describe characteristics
of individual variables and relationships
between variables is a strength of statistical
analysis.
OLS Regression
 For every additional year of education,
monthly earnings increase by $76 on
average, holding other variables constant.
 Females earn $13 less than males per
month on average, holding other variables
constant.
 That is, how to estimate the mean value
of an outcome variable, based on the values
of explanatory variables & in the context of
hypothesis testing/inferential statistics.
Regression with a Categorical
Dependent Variable
 For every higher category of educational
attainment, the odds of voting Democrat
instead of Republican increase by a multiple
of 1.31 on average, holding other variables
constant.
 That is, using explanatory variables to
estimate the mean value of a categorical
outcome variable (such as Democrat vs.
Republican), in the context of hypothesis
testing & inferential statistics.
But to get to those stages we’ll have to
learn the basics of statistics:
 E.g., what are the types of variables? How are
variables defined & measured?
 How should we collect data?
 How do we compute means & standard deviations, &
what are their weaknesses?
 The five-number summary: how do we compute it, &
what are its advantages?
 What are correlation & regression coefficients? How
are they based on means & standard deviations? What
are their weaknesses?
 What is association? What is causation?
 What are the basic differences between
experimental & observational research?
 What are the basic issues in probability &
sampling?
 What are confidence intervals, & how do
we compute them?
 What is the difference between statistical
significance & practical significance?
 What is hypothesis testing, & what are its
pitfalls?
 What does it mean to say how one variable
relates to another, ‘adjusting for other
factors’?
 What does all of this have to do with the
social construction of reality & with social
research?
 What are the similarities & differences
between quantitative & qualitative research?
 The challenge is to use statistics to
describe & analyze social relations.
 If a statistical procedure doesn’t
contribute to describing &
analyzing social relations, then—
from the standpoint of the social
sciences—don’t use it.
More Preliminaries
 See this course’s slide presentation
on ‘Evaluating Data Sets &
Research.’
 Read Babbie, The Practice of Social
Research; & Ragin, Constructing
Social Research.
Let’s get started!!!
At the Heart of Statistical
Commonsense
 Utts & Heckard’s Statistical Ideas and
Methods lays out ‘seven statistical stories
with morals.’
 Let’s peruse the stories & review their
morals, because they underpin
commonsense that’s easy to lose sight of
when we start grappling with technical
details, software, and social science
perspectives.
1. Graph the data.
2. When discussing change in the rate/risk
of something occurring, present the base
rate or baseline risk.
3. A properly conducted random sample
even of relatively small size can provide
reasonably accurate information about a
much larger population.
4. A representative sample is essential to
making sound conclusions about a
population’s characteristics.
5. Cause-effect conclusions can’t be made
from observational studies.
6. Cause-effect conclusions can be made from
properly conducted experiments, but the
results can’t necessarily be generalized to a
larger population.
7. There is a big difference between
‘statistical’ significance & ‘practical’
significance.
 Let’s keep these principles in mind.
 What’s the difference between
‘descriptive’ & ‘inferential’ statistics?
 Descriptive statistics: summarizing the
features of data.
 Inferential statistics: using principles of
probability to make inferences from a random
sample to a population.
 Moore/McCabe chaps. 1-2 emphasize
descriptive statistics; chaps. 3-12 lead to &
eventually emphasize inferential statistics.
Data, Statistics, Variables
 Data (a plural word): observations
that are put into informational
context.
 Statistics: a set of procedures for
collecting, organizing, & interpreting
numerical data.
 Statistics turns objects into individuals
(i.e. subjects or observations) for analysis.
 It analyzes the individuals (i.e. subjects or
observations) as variables.
 Variable: any characteristic that can take
different values for different individuals (i.e.
different subjects or observations).
Two basic kinds of variables:
 Categorical (or qualitative)
 Quantitative (or measurement)


Categorical variable: assigns an
individual (i.e. subject or observation) to
a category or group.
Two types of categorical variables:
(1) Nominal: simply names—Black,
Hispanic, Asian, or White.
(2) Ordinal: ranks, but not according to
any mathematical interval—low,
middle, or high ses.


Quantitative variable: takes numerical values
for which arithmetic operations such as adding
or averaging make sense.
Two types of quantitative variables:
(1) Discrete: values only differ by fixed amounts (e.g.,
household size: 1, 2, … members)
(2) Continuous: differences between two observations
can be arbitrarily small (e.g., age: years>months>
days>hours>minutes>seconds>milliseconds>…).
 Two types of quantitative continuous
variables:
(1) Interval: equal intervals but no true
zero point—e.g., Fahrenheit or Centigrade
temperature.
(2) Ratio: equal intervals with true zero
point—years of age, weight, earnings
(though debt may mean that ‘income’ has
no true zero point).
Why does type of variable matter?
 Because, as we’ll see during the
year, different types of statistical
procedures call for different
types of variables (& vice versa).
A Variable’s Frequency
Distribution
 A variable’s frequency distribution
(categorical or quantitative): tells us
what values the variable takes & how
often it takes these values.
 As we’ll see this year, specific
statistical procedures are
appropriate only for specific kinds
of distributions.
 Recall, in addition, that specific
kinds of variables call for specific
kinds of statistical procedures (&
vice versa).
 So, in terms of applied
statistics, we need to know (1)
the type of variable and (2) the
characteristics of its frequency
distribution.
 From the standpoint of social &
policy sciences, moreover, a
variable’s frequency distribution is
a point of departure for examining
social relations.
 E.g., what comments and questions
might the distribution of income
provoke?
Exploratory data analysis: how to
describe frequency distributions of
variables and relationships between
these distributions.
 Examine each variable by itself
(univariate analysis). Then examine
the relationships between the variables
(bivariate or multivariate analysis).

Whether the analysis is univariate or
bivariate/multivariate, begin by using
graphs to describe patterns. Why?
 We’ll begin by focusing on univariate
exploratory analysis.
 In univariate analysis:
 Begin by examining each variable with
graphs.
 If the overall graphic pattern is
sufficiently regular, then use numerical
summaries to describe it.
 We’ll start with univariate analysis of
categorical variables. Then we’ll do univariate
analysis of quantitative variables.
The ‘values’ of a categorical
variable are simply labels (i.e.
groupings):
 Black, Hispanic, Asian, White,
Other
 Female, Male
 Protestant, Catholic, Jewish,
Other
 What are the most basic
questions we have to ask about
such groupings?
How to describe a categorical
variable:
 Frequency table (which is roughly
equivalent to a graphic description)
 Bar graph
 Pie chart
 Frequency table
Education: % of Persons Aged 25-34
Education
%
Less than high school 12.00
High school
30.00
Some college
28.00
Bachelor’s
22.00
Advanced
8.00
___________________________
Total
100.00
N=723
 A frequency table, bar graph, or
pie chart lists a categorical variable’s
categories (‘values’) & gives either
the count or percent of individuals
(i.e subjects or observations) that
fall in each category.
 Frequency table
Education: % of Persons Aged 25-34
Education
%
Less than high school
12.00
High school
30.00
Some college
28.00
Bachelor’s
22.00
Advanced
8.00
___________________________
Total
N=723
100.00
How to do it in Stata
. use hsb2, clear
[to display the hsb2 data set]
. tabulate race
Freq.
Percent
Cum.
hispanic
24
12.00
12.00
african-
20
10.00
22.00
asian
11
5.50
27.50
white
145
72.50
100.00
Total
200
100.00
. tab race, plot
hispanic,
black,
asian,
white
Freq.
hispanic
24
*********
african-
20
*******
asian
11
****
white
145
**********************************
Total
200
.8
.2
.4
.6
.725
.12
.1
0
.055
0
1
2
3
hispanic, black, asian, white
4
. histogram race, discrete fraction gap(10) addlabel
. tab race, generate(raced)
. graph pie raced*
race==hispanic
race==asian
race==africanrace==w hite
Graphic options to describe a
quantitative variable:

stemplot: for small data sets (N<200 or so)
 dotplot
 boxplot
 histogram
 kdensity
 normal quantile plot
How to do it in Stata
. stem math
. dotplot math
. gr box math
. hist math, norm
. kdensity math, norm
. qnorm math, grid
Stem-and-leaf plot for math
3t 3
3f 5
3s 7
3. 88999999
4* 00000000001111111
4t 22222223333333
4f 444455555555
4s 66666666777
4. 888889999999999
5* 000000011111111
5t 2222223333333
5f 444444444455555
5s 66666667777777777777
5. 88888899
6* 000001111111
6t 222233333
6f 44444555
6s 666677
6. 899
7* 01111
7t 2223
7f 55
80
70
60
50
40
30
0
5
10
Frequency
. dotplot math
15
80
70
60
50
40
30
. gr box math
.04
.03
.02
0
.01
30
40
50
60
math score
. hist math, norm
70
80
.04
.03
.02
0
.01
30
40
50
60
math score
Kernel density estimate
Normal density
. kdensity, math norm
70
80
52.645
68.05473
30
39
40
50
52
60
70
70.5
80
37.23527
30
40
50
60
Inverse Normal
Grid lines are 5, 10, 25, 50, 75, 90, and 95 percentiles
. qnorm math
70
80

It takes more than one kind of graph
to explore a quantitative variable’s
distribution.
 E.g., a histogram or a kdensity graph
displays overall contour but may not
clearly reveal potential outliers.
 E.g., a boxplot identifies outliers and key
points of a distribution but does not
capture its fine grain.
 Another step toward bivariate
analysis: how are the values of a
quantitative variable distributed
according to a categorical variable?
 E.g., the distribution of math test
scores by female or male.
 In Stata, here’s how to examine quantitative
distributions by a categorical variable:
. bys female: summarize math
. bys female: summarize math, detail
. bys female: stem math
. graph box math, over(female, total) marker(1,
mlabel(id))
. dotplot math, by(female)
. tabulate female, su(math)
 Boxplots are generally the best way to
examine quantitative frequency distributions
by a categorical variable.
30
40
50
60
70
80
. gr box math, over(female, total)
male
female
(total)
How graphically to describe a
quantitative variable’s
frequency distribution:
 Look for overall pattern & striking
deviations.
 Describe the overall pattern by its
shape, center & spread.
 An outlier is an important kind of
deviation.
Shape:
 Does the distribution have one or
more major peaks, called modes?
 Is it approximately symmetric, or is it
skewed to the right (right or positive
skew) or skewed to the left (left or
negative skew)?
 Graphic descriptions lay the
foundation for sound data analysis.
 But they don’t crisply summarize a
distribution’s tendencies of shape,
center, & spread.
 For this we need numerical
summaries.
 For a quantitative variable, if the
graphic descriptions show that the
distribution is more or less consistent
in form (i.e., without pronounced
skewness & outliers), then use
appropriate numerical summaries to
describe its shape, center, & spread.
Center:
 What values converge at the center?
 The mean: one way of summarizing
the values that converge at the center
for a quantitative variable.
 xi
 
n
 The principal problem with the mean: it’s
highly sensitive to a few (or more) extreme
observations.
 That is, the mean is pulled strongly by
skewness & outliers.
 Hence, the mean is not a resistant
measure of a distribution’s center.
 The principal alternative to the mean as
a way of summarizing a quantitative
variable’s distribution’s center is the
median.
 The median: the midpoint of the
distribution (50th percentile), such that
half the observations are smaller & the
other half are larger.
 How to locate the median: list the observations
from lowest to highest value; find the location of
(n+1)/2. If the number of observations is even, the
median is the mean of the two center values.
 Alternative method: list the observations from
lowest to highest value; successively remove the
lowest & highest value observations until you are left
with a center value (or with two center values to
average).
 Advantage of the median: it’s resistant to extreme
observations, i.e. to skewness & outliers.
Examples: Median
 Locate the median for each of the
following.
(i) 1, 5, 7
(ii) 1, 2, 5, 7
(iii) 1, 2, 2, 7, 8
(iv) 8, -3, 5, 0, 1, 4, -1
For students registered at U.S.
universities, which is larger: mean age
or median age? Why?

 In the following histograms,
estimate the locations of the mean &
the median. How close to each other
or far apart are they, & why?
 The mean, median, & mode (the
value with the most frequent
observations) are three kinds of
averages (i.e. three measures of
central tendency).
 Returning to mean and median,
what does neither of them convey
about a quantitative variable’s
distribution?
Spread: dispersion around the center.
 Four basic measures of spread:
 Standard deviation: more on this later.
 Quartiles: 25th & 75th percentiles; more
on these later.
 Minimum & maximum (i.e. range)
 Outliers: striking deviations from the
overall pattern.
 Measures of spread: recall that univariate
graphs reveal not only the center but also
the shape and spread of a frequency
distribution.
 Spread, then, refers to dispersion
around the center.
 Spread includes outliers, but a
distribution’s minimum & maximum
values are not necessarily outliers.
 Recall that graphs such as histograms
& kdensity are not the best at detecting
outliers.
 Stemplots (for small numbers of
observations) and boxplots are
particularly good for detecting outliers.
Definition of Outlier
"Outliers are sample values that cause
surprise in relation to the majority of the
sample” (W.N. Venables and B.D. Ripley,
Modern Applied Statistics with S, 119).
Questions to ask about outliers:
1. Do they represent data errors? If so, correct or
delete them.
2. Do they represent some group, phenomenon, or
entity that is distinctive from the one you’re
trying to examine? If so, perhaps delete them
(or perhaps do another study to examine them).
3. Do they represent some divergent, insightful
aspect of the relationship you’re studying? If so,
keep them (but perhaps temper their influence).
 Always examine the data both with &
without the outliers.
 Consider reporting the results both with &
without the outliers.
 Remember: if they don’t represent errors,
outliers can yield important insights into
the group, phenomenon, or entity that
you’re studying.
Shape, center & spread: the fivenumber summary is the most
resistant numerical summary of a
variable’s distribution (i.e. most
resistant to influence of skewness &
outliers).
 Median: 50th percentile
 Quartiles: low (25th percentile) & high
(75th percentile)
 Range: minimum, maximum
 Inter-quartile range (IQR) is the 5-number
summary’s substitute for standard deviation
as a measure of spread around the center.
 IQR = upper quartile – lower quartile
 Five-number summary includes a procedure
for identifying outliers:
Outliers = IQR x 1.5
The boxplot—which graphs the 5-number
summary—is ideal for graphically displaying
outliers as defined by IQR x 1.5.
 The minimum & maximum are not
necessarily outliers.
 They, as well as other observations
that fall near them, are outliers only if
they represent striking deviations
from the overall pattern (defined by
the five-number summary as IQR x
1.5).
How to do it in Stata
. summarize math, detail
. summarize math, detail
math score
Percentiles
Smallest
1%
36
33
5%
39
35
10%
40
37
Obs
200
25%
45
38
Sum of Wgt.
200
50%
52
Mean
52.645
Largest
Std. Dev.
9.368448
87.76781
75%
59
72
90%
65.5
73
Variance
95%
70.5
75
Skewness
99%
74
75
Kurtosis
.2844115
2.337319
 The downloadable ‘univar’ is a useful
alternative way of summarizing a
quantitative variable:
. univar math
-------------- Quantiles -------------Variable
n
Mean
S.D.
Min
.25
Mdn
.75
Max
--------------------------------------------------------------------------------------------math
200
52.65
9.37
33.00
45.00
52.00
59.00
75.00
‘tabstat’ is another useful
alternative:

. tabstat math, stats(min p25 median p75 max)
variable |
min
p25
p50
p75
max
---------+------------------------------------------------math |
33
45
52
59
75
 ‘q’, instead of p25 p50 p75, also displays
quartiles.
 ‘iqr’ displays inter-quartile range.
Boxplot:
 Graphic description of fivenumber summary
 Useful in general to compare
distributions of two or more
quantitative variables
 Does not display the fine-grain
features of a distribution, but is
ideal for displaying outliers as
defined by IQR x 1.5.
30
40
50
60
70
80
. graph box math, marker(1, mlabel(id))
30
40
50
60
70
80
. gr box read write math, marker(1, mlab(id))
reading score
math score
writing score
How to graph boxplots by
a categorical variable
. gr box math, over(female, total)
marker(1, mlab(id))
 The five-number summary is the
most resistant numerical description
of a quantitative variable’s
distribution.
 Why, then, is it not the most
commonly used numerical description?
 Because the mean & its associated measure of
spread—the standard deviation—describe the shape,
center & spread of an important type of distribution:
the normal distribution.
 Another, related reason: they form the basis for
computing two important statistics: correlation
coefficients & regression coefficients.
 Let’s consider the standard deviation. Then we’ll
consider linear transformations & the normal
distribution.
Standard deviation: measures the
average spread of the observations
around a variable’s mean.
1. Compute the quantitative variable’s mean.
2. Subtract each observation’s value from the
mean & square the result.
3. Add the results & divide by n – 1 (the figure
obtained being the ‘variance’).*
4. Take the square root of the variance to obtain
the standard deviation.
* n – 1: ‘degrees of freedom,’ a measure of the
available amount of statistical information.
The more, the better.
 Variance:
( xi  x )2
2
n 1
 Standard deviation:

(
x
i  x )2
 
n 1
 Remember: there’s more to
spread than just the range.
 There’s also standard
deviation or inter-quartile
range: measures of dispersion
around a distribution’s center.
Two distributions have the same
mean, but one has a larger standard
deviation. How do their shapes differ?

 Can the standard deviation ever be
zero?
 Can the standard deviation ever be
negative?
 For a set of positive numbers, can
the standard deviation ever be larger
than the mean?
Answers
 The standard deviation cannot be
negative, but it can be zero.
 If the standard deviation is larger
than the mean, there are extreme
values & possibly skewness.
 Ideally the standard deviation is
about half the value of the mean.

See Freedman et al., Statistics.

Variance:
( xi  x )2
2
n 1
 Standard
deviation:

(
x
i  x )2
 
n 1
Why do we use the standard deviation
rather than the variance?
 Squaring the deviations makes them more tightly
linked to the mean (i.e. the center).
 Squaring the deviations makes them correspond
to the spread of the normal distribution.
 Squaring the deviations measures dispersion
around the mean in the variable’s original scale
(e.g., dollars, kilograms, SAT scores).
 Problem: the standard deviation, like the mean,
is highly vulnerable to outliers, and its value is
misleading if there’s notable skewness.
 Use the mean & standard deviation only
when a quantitative variable’s frequency
distribution is relatively symmetric & free of
outliers.
 Otherwise it may be possible to apply remedial
actions (e.g., transforming or removing outliers).
For Future Reference:
Coefficient of Variation
 This is not covered in Moore/McCabe/
Craig but is used when comparing the
distributions of two or more sets of
observations with differing measurement
scales.
 It measures variation without reference
to a data set’s measurement scale.
CV = standard deviation/mean
 How to do it in Stata
. use hsb2, clear
. tabstat math, stats(cv)
variable |
cv
-------------------------math
| .1779551
--------------------------
 ‘tabstat’ is a flexible command for summarizing
statistics—e.g., to summarize stats by a categorical
variable:
. tabstat math, stats(mean median sd min max)
by(female)
female
mean
p50
sd
min
max
male
52.94505
52
9.664784
35
75
female
52.3945
53
9.151015
33
72
Total
52.645
52
9.368448
33
75
Summary
1. Graphically describe a frequency
distribution’s overall pattern & striking
deviations: its shape, center & spread
(including outliers).
2. Among numerical descriptions, the fivenumber summary—median, quartiles &
minimum/maximum—is the most
resistant to extreme observations (i.e. to
skewness & outliers).
3. Use the mean & standard deviation only if
the distribution is relatively symmetric &
free of outliers (or possibly take remedial
action to make it so).
4. If they don’t represent errors, outliers
may yield important insights.
5. What are the implications of a variable’s
distribution for understanding social
relations & public policies?
 Next let’s consider linear
transformations.
Linear Transformations
 The governor proposes to give all state
employees a flat raise of $100 a month. What
would this do to the average monthly salary of
state employees, & to the standard deviation?
 What would a 5% increase in the salaries, across
the board, do to the average monthly salary, & to
the standard deviation?
 Let’s figure out how to answer these questions.
 Linear transformations: change
the original variable x into a new
variable.
 The value of the transformed
variable is given by the equation
xnew=a + bx.
a: constant
b: slope
 Adding the constant a (positive or
negative) shifts all values of x upward or
downward by the same amount but does not
change measures of spread.
 That is, if you add the same number to
every observation, that number just gets
added to the average; the standard
deviation, IQR, & range do not change.
 Multiplying each by the positive a changes
the size of all units of x’s measurement.
 That is, if you multiply every entry on a list
by the same positive number, the average &
the standard deviation get multiplied by that
number.
How to do linear transformations in
Stata? Let’s find out & look at an
example of the principals we just
introduced:

. u hsb2, clear
. su math
. gen mathp10=math + 10
. gen mathX10=math*10
. dotplot math mathp10 mathX10
. su math mathp10 mathX10
800
600
400
0
200
math score
mathp10
mathX10
. su math mathp10 mathX10
Variable |
Obs
Mean
Std. Dev.
Min
Max
-----------------------------------------------------------------------math
|
200
52.645
9.368448
33
75
mathp10 |
200
62.645
9.368448
43
85
mathX10 |
200
526.45
93.68448
330
750
Let’s next turn our attention to
a tool whose purpose, like that of
graphs & numerical summaries,
is to help us grasp the main
features of a distribution.

Density curve: an idealized,
smooth curve that is always on
or above the horizontal axis &
has area exactly 1 underneath it.

 Density curves may be symmetrical
or asymmetrical, and in each case
there are countless possible shapes.
 Density curves are idealized, smooth
curves that help us look beyond the
ragged details of a distribution to
grasp its overall pattern.
 kdensity read, norm
.04
.03
0
Density
.02
.01
20
40
60
reading score
Kernel density estimate
Normal density
80
A symmetrical, unimodal, &
bell-shaped density curve is
called a normal curve.

 This, however, is just one
kind of density curve.
Here are the main features of density
curves:
 The curve is always on or above the
horizontal axis & has area exactly 1 underneath
it.
 Median: divides the area under the density
curve in half; it is the midpoint.
 Mean: the balance point of the density curve.
 The median & the mean are the same for a
symmetric density curve, but not for a skewed
density curve.
 Any area under a density curve
represents a proportion of a variable’s total
number of observations.

 The Greek letter refers to the mean &
the Greek letter
refers to the standard
deviation of a density curve.

Before we inspect some examples:
 Recall that, in a density curve, the median
divides the area under the density curve in
half while the mean is the ‘balance point.’
 Recall, too, that the median & mean are
located in the same place in a symmetrical
density curve, but not in an asymmetrical
density curve.
 Let’s see how these features work
in skewed density curves.
 In each of the following skewed
density curves, which line
represents the mean, which line
represents the median, & which
line is irrelevant? Why?
Answers
(a) Mean is C; median is B
(b) Mean & median are A
(c) Mean is A; median is B
 Next, let’s look at a normal
distribution: a symmetrical,
unimodal, bell-shaped density
curve.
 Where is the mean & the median,
& why?
 Normal distributions: symmetrical,
unimodal, & bell-shaped density
curves, which, although their details
may vary, always have the same
overall shape.
 Is any symmetrical density curve a
normal distribution? Draw three
symmetrical distributions that aren’t
normal curves.
Normal distributions—which are
idealized forms—are accurately
described by their mean & their
standard deviation.

 The previous two curves have
different shapes. Why?
 Why, nonetheless, are they
both normal curves?
 Remember that any given area
under a density curve represents a
proportion of all a variable’s
observations.
 Such an area of a density curve is
relative to its total area, which equals
1.
Let’s look at such areas in
a normal density curve.

 In reality, no variable’s frequency
distribution will be nearly as smooth
as a density curve.
 That’s because a density curve is a
conceptual tool: like graphs &
numerical summaries, a density curve
is a way of summarizing the overall
pattern of a distribution.
 Whether ‘normal’ or not,
density curves stand with graphs
& numerical summaries as ways
of distinguishing a variable’s
forest from its trees.
 This brings us to the 68 – 95 –
99.7 Rule.
The 68 – 95 – 99.7 Rule
 About 68% of the observations of a
normal curve fall within one standard
deviation of the mean.
 About 95% of the observations of a
normal curve fall within two standard
deviations of the mean.
 About 99.7% of the observations of a
normal curve fall within three standard
deviations of the mean.
 How does the 68 – 95 - 99.7
Rule pertain to the features of
a density curve?
 And to what kind of density
curve in particular does the
rule pertain?
 The Rule pertains to areas under a
density curve, whose total area
equals 1. Each area represents a
proportion of the total number of
observations.
 The Rule pertains specifically to
one kind of density curve, the
normal distribution.
 For real-world data, the 68 – 95 –
99.7 Rule never truly characterizes a
distribution.
 For real-world data it’s an
approximation, but as we’ll see in
future weeks, with enough
observations the Rule commonly
comes close enough to be used.
Here is a commonly utilized way of using
proportions of the area under the normal
distribution.
 E.g., you take your six-month old child to the
pediatrician for a routine check-up. You are told that
the child’s weight ranks at the 42nd percentile of the
weights of the population of children her age.
 E.g., you take the GRE. The results indicate that
you scored at the 87th percentile of the population of
GRE-takers.
 What do such percentiles mean?
 What do they have to do with
density curves—in particular, with the
normal distribution?
Answer: they are pegged to
the standard normal
distribution.

 Here we’ll see in fact how
approximations to the normal
distribution are commonly used.
 We always need to verify empirically,
though, that a real-world frequency
distribution indeed approximates a
normal curve.
 Later on today we’ll see how one
kind of graph—the normal quantile
plot—is used to assess this
approximation.
 The standard normal distribution (or
curve): it is the normal distribution with
mean 0 & standard deviation 1. That is:
N(0,1). The standard normal distribution is
symmetric around the mean 0.
 Standardizing a variable: transforming
the variable’s original unit of measurement
(say, the baby’s weight, or your GRE score)
to the mean & standard deviation of the
standard normal distribution N(0,1).
 Another way of putting it: standardizing a
value means transforming the original xvalue (again, the baby’s weight, or your
GRE score) to standard units to show how
many standard deviations the original xvalue is above or below the mean of the
standard normal distribution N(0,1).
 That is, a standard score tells us how
many standard deviation the original
observation falls away from the mean of
the standard normal distribution, & in what
direction.
 How many standard deviations away from
the mean of the standard normal
distribution, & in what direction, does the
baby’s weight fall? What about your GRE
score?
 Standardized scores enable us to
compare the values of two or
more variables measured in
differing units (e.g., pounds
versus inches) relative to their
respective means.
E.g., how does your baby’s
weight compare to her height
among the population of babies of
that age?

 E.g., an SAT score of 543 versus
an ACT score of 22?
 The standardized values for any
distribution always have mean 0 &
standard deviation 1: that is, N(0,1).
 Observations larger than the mean
are positive when standardized, &
observations smaller than the mean
are negative.
 A standardized value is called a zscore. Transform an x-value (e.g., the
baby’s weight or your GRE score) into
a z-score as follows:
z
xi - 

Then either consult the Standard
Normal Table (Table A)—the old way of
doing things—or, better, do a software
computation to locate the z-score on
the standard normal curve.

The table entry or software output for
each z-score gives the area under the
standard normal curve to the left of
the z-score.
 In other words, your baby’s weight is
converted into a z-score—a point on the
horizontal axis relative to the mean of the
standard normal distribution.
 The Standard Normal Table, or software
computation, indicates that your baby
weighs more than 42% of the population of
babies of the same age: the area to the left
of the z-score represents the 42% of the
population of babies that weigh less than
your baby.
 Your GRE score is converted into a zscore—a point on the horizontal axis
relative to the mean of the standard normal
distribution.
 The Standard Normal table, or software
computation, indicates that you scored
higher than 87% of the population that
takes the GRE: i.e. the area to the left of
your z-score represents the 87% of the
population that scored below you on the
GRE.
 Going in the opposite direction: how to
transform a z-score back into the original
x-value?
xi
=
+ (z-score* )
 That is, how to convert the baby’s zscore back into her weight, & your z-score
back into your GRE score?
Examples: Standardization

You give an exam to the large group of students
in your course ‘The Metaphysics of Flathead
Worms.’ The mean score is 50 & the standard
deviation is 10.
(i) Convert each of the following test scores to
standard units: 60, 45, 75.
(ii) Find the original-scale test scores that in
standard units are: 0, 1.5, -2.8.

Fill in the blanks:
(a) The area between +/- ______ under the normal
curve equals 68% of the total.
(b) The area between +/- ______ under the normal
curve equals 95% of the total.
(c) The area between +/- ______ under the normal
curve equals 99.7% of the total.
How to do it with Stata’s
calculator
  =50,

 =10
xi = 60.
What is the z-score?
. display (60 - 50)/10
1

So, z-score = 1.
 What percentage of students on the
standard normal distribution have zscores lower than 1?
. display norm(1)
.84134475

The answer = .8413 (i.e. 84.13% of the
students have scores lower than z-score =
1)
What if you already know the score’s
relative location on the standard
normal distribution but you want to
work backward to find the z-score?

. di invnorm(.84134475)
1
(i.e. z-score = 1)
A Standard-Normal Way
to Assess Outliers
 For quantitative variables, here’s an
alternative to the 5-number summary way of
assessing outliers:
(1) Standardize the distribution’s values.
(2) If n<=80, then a standard value of
+-2.5 might be considered an outlier. If
n>80, then +-3.0 might be considered
an outlier.
How to do it in Stata
. egen zmath=std(math)
. su math zmath
. gr box zmath, marker(1, mlab(id))
. list zmath if abs(zmath)>=3.0
Some Questions about Distributions
True or false?

All density curves are symmetric.
 All symmetrical density curves are normal
curves.
 All normal curves are standard normal
distributions.
 All distributions of real data can appropriately be
expressed as standard normal distributions.
Concerning Distributions in General
 Analyzing a frequency distribution’s shape,
center & spread (including outliers)
contextualizes each individual observation as
part of an overall pattern (such as greater or
lesser inequality, specific features of
inequality).
 What are implications of such
contextualization for understanding social
relations & public policies?
Always graph a distribution
before computing any statistic
that is sensitive to skewness &
outliers.

 E.g., always use graph a
distribution to check for skewness
& outliers before computing a
variable’s mean or standard
deviation.
 A good practice is routinely to use
more than one kind of graph (e.g.,
histogram & boxplot).
 One kind of graph may do better at
displaying certain features of a
distribution’s shape, center, and spread
(e.g., histogram or kdensity) while
another may do better at displaying
potential outliers (e.g., boxplot).
How to do it in Stata
. qnorm read, grid
52.23
69.09458
20
40
60
80
35.36542
20
40
60
Inv erse N ormal
Gri d li nes are 5, 10, 25, 50, 75, 90, and 95 perc enti les
80
.04
.03
.02
0
.01
Density
. kdensity read, norm
20
40
60
reading score
Kernel density estimate
Normal density
80
.04
.03
.02
0
.01
Density
. histogram read, norm
30
40
50
60
reading score
70
80
30
40
50
60
70
80
. dotplot read
0
10
20
Frequency
30
. stem read
2. | 8
3* | 1
3t |
3f | 4444445
3s | 66677
3. | 99999999
4* | 11
4t | 222222222222233
4f | 444444444444455
4s | 6777777777777777777777777777
4. | 8
5* | 000000000000000000
5t | 222222222222223
5f | 45555555555555
5s | 77777777777777
5. |
6* | 0000000001
6t | 3333333333333333
6f | 555555555
6s | 6
6. | 88888888888
7* | 11
7t | 33333
7f |
7s | 66
 Stemplots are useful for relatively small amounts of data.
30
40
50
60
70
80
. gr read box
Summary
1. How do we examine a frequency
distribution for a quantitative variable?
2. What’s a density curve? What’s its
purpose?
3. What’s a normal curve?
4. Are all symmetrical curves normal?
5. What’s the relation of the mean &
median in a normal curve?
6. What’s the 68-95-99.7 Rule for a normal
distribution?
7. What’s a standard normal distribution?
8. What does it mean to standardize a
value or a variable, & how do we do it?
9. What does standardization enable us to
do for two or more variables that are
measured in differing units?
10. What does it mean to unstandardize a
value, & how do we do it?
11. Why should we first graph a
distribution, & why should we commonly
use more than one kind of graph?
12. How should we treat outliers?
13. What are the implications of a variable’s
distribution for (a) applied statistics and
(b) understanding social relations &
public policies?
Some Stata tips relevant to the
Moore/McCabe homework problems
 To list the observed values of a dataset: list
 To list the observed values of a variable: list math
 To list the observed values of a variable in order
from low to high:
sort math
list math
 To list the observed values from high to low:
gsort -math
 To see a data set in the spreadsheet:
browse
 To see a variable in the spreadsheet:
br math
 To graph, e.g., writing score by math score:
scatter write math || qfit write math
 To graph write & math by a categorical variable:
scatter write math, (by female) || qfit write
math
 If a categorical variable won’t display
graphically (because it’s coded as a ‘string’
[i.e. text] variable), do the following to create
a new, ‘destringed’ variable to use in the
calculations:
encode gender, generate(gend)
tabulate gend
 How to create a standard normal distribution
of random data from scratch & then
summarize, list, & graph the data:
. clear
. set obs 100
[# desired observations]
. set seed 123
[to replicate the data]
. generate newvar = uniform()
. summarize newvar
. list newvar
. stem newvar
. gr box newvar
. dotplot newvar
. kdensity newvar, norm
. histogram newvar, norm
. qnorm newvar, grid
 Line graph:
. twoway line pasadena year
 To set a graph’s y & x scales:
twoway line pasadena year,
ylabel(0(25)100) xlabel(1950(25)2000)
Remember
 First graphically then—if the distribution is
sufficiently regular—numerically describe a
quantitative variable’s frequency distribution.
 If the shape is quite skewed, use the fivenumber summary as the numerical
description.
 Describe the distribution in terms of shape,
center, & spread, including outliers.
 There’s more to spread than minimum
& maximum.
 The minimum & maximum aren’t
necessarily outliers.