Download 01 - DePaul University

Looking at Data - Distributions
Displaying Distributions with Graphs
IPS Chapter 1.1
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 1.1)
Displaying distributions with graphs

Variables

Types of variables

Graphs for categorical variables


Bar graphs

Pie charts
Graphs for quantitative variables

Histograms

Stemplots

Stemplots versus histograms

Interpreting histograms

Time plots
Key things to focus on:

Categorical v.s. Quantitative variables

Types of graphs used for categorical as opposed to quantitative
variables.

Histograms

How they are created / choosing appropriate bin size

Commonly encountered histogram shapes

How to interpret them
Variables
In a study, we collect information—data—from individuals. Individuals
can be people, animals, plants, or any object of interest.
A variable is any characteristic of an individual. A variable varies among
individuals.
Example: age, height, blood pressure, ethnicity, leaf length, first language
* The distribution of a variable tells us what values the variable takes
and how often it takes these values.
Two types of variables

Variables can be either quantitative…

Something that takes numerical values for which arithmetic operations,
such as adding and averaging, make sense.

Example: How tall you are, your age, your blood cholesterol level, the
number of credit cards you own.

… or categorical.

Something that falls into one of several categories. What can be counted
is the count or proportion of individuals in each category.

Example: Your blood type (A, B, AB, O), your hair color, your ethnicity,
whether you paid income tax last tax year or not.
How do you know if a variable is categorical or quantitative?
Ask:
 What are the n individuals/units in the sample (of size “n”)?
 What is being recorded about those n individuals/units?
 Is that a number ( quantitative) or a statement ( categorical)?
Categorical
Quantitative
Each individual is
assigned to one of
several categories.
Each individual is
attributed a
numerical value.
Individuals
in sample
DIAGNOSIS
AGE AT DEATH
Patient A
Heart disease
56
Patient B
Stroke
70
Patient C
Stroke
75
Patient D
Lung cancer
60
Patient E
Heart disease
80
Patient F
Accident
73
Patient G
Diabetes
69
Charting – It’s all about the data…



The purpose of a graph or chart is to try and provide a ‘visual’
description of the data.
The type of chart chosen, the scales used, the labels, the
description, summary lines, etc are all tools whose ultimate
purpose is to give the most accurate understanding of the
data.
By the same token, it is possible to create graphs that give a
thoroughly misleading picture of the data!


This is disappointingly common.
So, whether you choose a pie chart, bar chart, histogram,
scatterplot, etc, the question you must always be asking
yourself is: Does this chart give the most accurate
understanding of the data?
Ways to chart categorical data
Because the variable is categorical, the data in the graph can be
ordered any way we want (alphabetical, by increasing value, by year,
by personal preference, etc.)

Bar graphs
Each category is
represented by
a bar.

Pie charts
The slices must
represent the parts of one whole.

Perhaps the greatest benefit of the pie chart is the ability to emphasize
each category’s relation to the whole.
Example: Top 10 causes of death in the United States 2001
Rank Causes of death
Counts
% of top
10s
% of total
deaths
1 Heart disease
700,142
37%
29%
2 Cancer
553,768
29%
23%
3 Cerebrovascular
163,538
9%
7%
4 Chronic respiratory
123,013
6%
5%
5 Accidents
101,537
5%
4%
6 Diabetes mellitus
71,372
4%
3%
7 Flu and pneumonia
62,034
3%
3%
8 Alzheimer’s disease
53,852
3%
2%
9 Kidney disorders
39,480
2%
2%
32,238
2%
1%
10 Septicemia
All other causes
629,967
26%
For each individual who died in the United States in 2001, we record what was
the cause of death. The table above is a summary of that information.
Example: Top 10 causes of death in the United States 2001
Rank
Causes of death
Counts
% of top 10s
% of total deaths
1
Heart disease
700,142
37%
29%
2
Cancer
553,768
29%
23%
3
Cerebrovascular
163,538
9%
7%
4
Chronic respiratory
123,013
6%
5%
5
Accidents
101,537
5%
4%
6
Diabetes mellitus
71,372
4%
3%
7
Flu and pneumonia
62,034
3%
3%
8
Alzheimer’s disease
53,852
3%
2%
9
Kidney disorders
39,480
2%
2%
Septicemia
32,238
2%
1%
10
All other causes
629,967
26%
• Pie chart or Bar chart?
• How would you order the data?
• alphabetically?
• size?
• ickiness of the disease?
• REMEMBER: The purpose of a chart/graph is to provide the best
picture of the data!
ise
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Counts (x1000)
800
700
600
500
400
300
200
100
0
Sorted alphabetically
Bar graph sorted by rank
 Much easier to analyze
Bar graphs
Top 10 causes of deaths in the United States 2001
The number of individuals
who died of an accident in
2001 is approximately
100,000.
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800
700
600
500
400
300
200
100
0
He
ar
td
Counts (x1000)
Each category is represented by one bar. The bar’s height shows the count (or
sometimes the percentage) for that particular category.
Pie charts
Each slice represents a piece of one whole. The size of a slice depends on what
percent of the whole this category represents.
Percent of people dying from
top 10 causes of death in the United States in 2000
Pie charts
• A pie chart makes it easy to analyze in terms of the relative proportions of each
category.
• Notice here that we have changed from the numbers of people dying to the
percent of people dying
• To make a pie chart, typically use percentages, and they have to add up to
100% or you won’t have the whole pie.
• That is why pie charts typically have an “other”.
• This chart doesn’t have an ‘other’ since we’re talking only about the top
10 diseases, not all diseases.
Make sure your
labels match
the data.
Make sure
all percents
add up to 100.
Percent of deaths from top 10 causes
Percent of
deaths from
all causes
• The top pie chart is the one we
have just been looking at.
• In the bottom one I have
included deaths from all other
causes in addition to the top 10.
• Adding this additional category
changes the percentages on the
original 10.
• Heart disease was 37% of total
before, now is a smaller percent,
29%, because we are looking at
all deaths.
Child poverty before and after government
intervention—UNICEF
What does this chart tell you?
•The United States has the highest rate of child
poverty among developed nations (22% of under 18).
•Its government does the least—through taxes and
subsidies—to remedy the problem (size of orange
bars and percent difference between orange/blue
bars).
Could you transform this bar graph to fit in 1 pie
chart? In two pie charts? Why?
The poverty line is defined as 50% of national median income.
Quick Review:

Categorical data: The variables do not have a quantity to them.
Rather, each variable is placed in a group, and we are comparing
the size of the different groups.
 Eg: Top causes of death in the USA: We have several categories
(diseases) and we compare these diseases.
 Pie charts and bar charts are frequently good choices for charting
categorical data.

Quantitative data: Variables that have a numerical value to them,
such as the percentage of Hispanic people per capita in each state,
or the average corn-crop yield by county in Indiana.
 We typically use different kinds of graphs such as histograms,
scatterplots, stem-and-leaf charts to graph quantitative data.
Ways to chart quantitative (as opposed to
categorical) data

Histograms and stemplots

These are summary graphs for a single variable. They are very useful to
understand the pattern of variability in the data.


Single variable as opposed to comparing two variables. More on this later.

Important topic: We will discuss variability over the next 1-2 lectures.
Line graphs: time plots

Use when there is a meaningful sequence, like time. Including a line
connecting the points sometimes helps emphasize any change over time.

Remember: Every decision you make such as whether or not to
include a summary line affects the “picture” told by your graph.
** Histograms
The range of values that a
variable can take is divided
into equal size intervals.
The histogram shows the
number of individual data
points that fall in each
interval.
The first column represents all states with a Hispanic percent in their
population between 0% and 4.99%. The height of the column shows how
many states (27) have a percent in this range.
The last column represents all states with a Hispanic percent in their
population between 40% and 44.99%. There is only one such state: New
Mexico, at 42.1% Hispanics.
Stem plots


Another way of graphing quantitative data.
Below is a list of people’s ages taken from a single row at a
play. We will graph these ages using a ‘stem plot’
9, 9, 22, 32, 33, 39, 39, 42, 49, 52, 58, 70
Stem plots
9, 9, 22, 32, 33, 39, 39, 42, 49, 52, 58, 70
How to make a stemplot:
1) Separate each observation into a stem, consisting of
all but the final (rightmost) digit, and a leaf, which is
that remaining final digit. Stems may have as many
digits as needed, but each leaf contains only a single
digit.
2) Write the stems in a vertical column with the smallest
value at the top, and draw a vertical line at the right
of this column.
3) Write each leaf in the row to the right of its stem, in
increasing order out from the stem.
STEM
LEAVES
State
Percent
Alabama
Alaska
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
Florida
Georgia
Hawaii
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Maryland
Massachusetts
Michigan
Minnesota
Mississippi
Missouri
Montana
Nebraska
Nevada
NewHampshire
NewJersey
NewMexico
NewYork
NorthCarolina
NorthDakota
Ohio
Oklahoma
Oregon
Pennsylvania
RhodeIsland
SouthCarolina
SouthDakota
Tennessee
Texas
Utah
Vermont
Virginia
W ashington
W estVirginia
W isconsin
W yoming
1.5
4.1
25.3
2.8
32.4
17.1
9.4
4.8
16.8
5.3
7.2
7.9
10.7
3.5
2.8
7
1.5
2.4
0.7
4.3
6.8
3.3
2.9
1.3
2.1
2
5.5
19.7
1.7
13.3
42.1
15.1
4.7
1.2
1.9
5.2
8
3.2
8.7
2.4
1.4
2
32
9
0.9
4.7
7.2
0.7
3.6
6.4
Step 1:
Sort the
data
State
Percent
Maine
W estVirginia
Vermont
NorthDakota
Mississippi
SouthDakota
Alabama
Kentucky
NewHampshire
Ohio
Montana
Tennessee
Missouri
Louisiana
SouthCarolina
Arkansas
Iowa
Minnesota
Pennsylvania
Michigan
Indiana
W isconsin
Alaska
Maryland
NorthCarolina
Virginia
Delaware
Oklahoma
Georgia
Nebraska
W yoming
Massachusetts
Kansas
Hawaii
W ashington
Idaho
Oregon
RhodeIsland
Utah
Connecticut
Illinois
NewJersey
NewYork
Florida
Colorado
Nevada
Arizona
Texas
California
NewMexico
0.7
0.7
0.9
1.2
1.3
1.4
1.5
1.5
1.7
1.9
2
2
2.1
2.4
2.4
2.8
2.8
2.9
3.2
3.3
3.5
3.6
4.1
4.3
4.7
4.7
4.8
5.2
5.3
5.5
6.4
6.8
7
7.2
7.2
7.9
8
8.7
9
9.4
10.7
13.3
15.1
16.8
17.1
19.7
25.3
32
32.4
42.1
Percent of Hispanic residents
in each of the 50 states
Step 2:
Assign the
values to
stems and
leaves
Stem Plot

To compare two related distributions, a back-to-back stem plot with
common stems is useful.

Stem plots do not work well for large datasets(tedious / impractical).

When the observed values have too many digits, trim the numbers
before making a stem plot.

When plotting a moderate number of observations, you can split
each stem.
Stemplots versus histograms
Stemplots are quick and dirty histograms that can easily be done by
hand, and therefore are very convenient for back of the envelope
calculations. However, they are rarely found in scientific or layperson
publications.
Interpreting histograms
When describing the distribution of a quantitative variable, we look for the
overall pattern and for striking deviations from that pattern. We can describe
the overall pattern of a histogram by its shape, center, and spread.
Histogram with a line connecting
each column  too detailed
Histogram with a smoothed curve
highlighting the overall pattern of
the distribution
** Distributions:
Most common shapes

Symmetric
distribution
A distribution is symmetric if the right and left
sides of the histogram are approximately mirror
images of each other.

A distribution is skewed to the right if the right
side of the histogram (side with larger values)
extends much farther out than the left side. It is
skewed to the left if the left side of the histogram
Skewed
distribution
extends much farther out than the right side.
Complex,
multimodal
distribution

Not all distributions have a simple overall shape,
especially when there are few observations.
** Outliers
An important kind of deviation is an outlier. Outliers are observations
that lie outside the overall pattern of a distribution. Always look for
outliers and try to explain them.
The overall pattern is fairly
symmetrical except for 2
states that clearly do not
belong to the main trend.
Alaska and Florida have
unusual representation of
the elderly in their
population.
A large gap in the
distribution is typically a
sign of an outlier.
Alaska
Florida
How to create a histogram
Deciding on the appropriate bin size can be an iterative process – you
may need to try a few times to decide on an optimal size.
Ideally:

Not too many bins with either 0 or 1 counts

Not overly summarized that you lose all the information

Not so detailed that it is no longer summary
 rule of thumb: start with 5 to 10 bins
Look at the distribution and refine your bins
(There isn’t a unique or “perfect” solution)
Same data set
Not
summarized
enough
Too summarized
Line graphs: time plots
In a time plot, time always goes on the horizontal, x axis.
We describe time series by looking for an overall pattern and for striking
deviations from that pattern. In a time series:
A trend is a rise or fall that
persists over time, despite
small irregularities.
A pattern that repeats itself
at regular intervals of time is
called seasonal variation.
Retail price of fresh oranges
over time
Time is on the horizontal, x axis.
The variable of interest—here
“retail price of fresh oranges”—
goes on the vertical, y axis.
This time plot shows a regular pattern of yearly variations. These are seasonal
variations in fresh orange pricing most likely due to similar seasonal variations in
the production of fresh oranges.
There is also an overall upward trend in pricing over time. It could simply be
reflecting inflation trends or a more fundamental change in this industry.
A time plot can be used to compare two or more
data sets covering the same time period.
1918 influenza epidemic
# Cases # Deaths
Date
1918 influenza epidemic
800 700
700 600
600 500
500 400
400 300
300 200
200 100
100
0
0
wek
ee1
we k 1
wek
ee3
we k 3
wek
ee5
we k 5
wek
ee7
we k
7
wek
we ee9k
ewk 9
ee11
we k 1
ewk 1
e1
we ek3
ewk 13
e1
we ek5
ewk 15
ee17
k
17
0
# deaths reported
800
10000
9000
10000
8000
9000
7000
8000
6000
7000
5000
6000
4000
5000
3000
4000
3000
2000
2000
1000
1000 0
we
0
0
130
552
738
414
198
90
56
50
71
137
178
194
290
310
149
Incidence
36
531
4233
8682
7164
2229
600
164
57
722
1517
1828
1539
2416
3148
3465
1440
# cases diagnosed
week 1
week 2
week 3
week 4
week 5
week 6
week 7
week 8
week 9
week 10
week 11
week 12
week 13
week 14
week 15
week 16
week 17
1918 influenza epidemic
# Cases
# Cases
# Deaths
# Deaths
The pattern over time for the number of flu diagnoses closely resembles that for the
number of deaths from the flu, indicating that about 8% to 10% of the people
diagnosed that year died shortly afterward, from complications of the flu.
** Scales matter
Death rates from cancer (US, 1945-95)
Death rates from cancer (US, 1945-95)
Death rate (per
thousand)
250
200
150
100
250
Death rate (per thousand)
How you stretch the axes and choose your
scales can give a different impression.
200
150
100
50
50
0
1940
1950
1960
1970
1980
1990
0
1940
2000
1960
1980
2000
Years
Years
Death rates from cancer (US, 1945-95)
250
Death rates from cancer (US, 1945-95)
220
Death rate (per thousand)
Death rate (per thousand)
200
150
100
50
0
1940
1960
Years
1980
2000
A picture is worth a
thousand words,
200
BUT
180
160
There is nothing like
hard numbers.
 Look at the scales.
140
120
1940
1960
1980
Years
2000
Why does it matter?
What's wrong with
these graphs?
Cornell’s tuition over time
Careful reading
reveals that:
1. The ranking graph covers an 11-year period, the
tuition graph 35 years, yet they are shown
comparatively on the cover and without a
horizontal time scale.
2. Ranking and tuition have very different units, yet
both graphs are placed on the same page without
a vertical axis to show the units.
Cornell’s ranking over time
3. The impression of a recent sharp “drop” in the
ranking graph actually shows that Cornell’s rank
has IMPROVED from 15th to 6th ...
Look at your axis labels carefully!!




Pay close attention to the labels on each axis
Pay close attention to the units of those labels (e.g.
kilograms, cost in 100s of dollars, average per
million people,etc, etc)
Sometimes you have to think about them for a few
minutes before you can really understand them, or
spot potential flaws.
This is one of the most important points of this
lecture.
Looking at Data - Distributions
Describing distributions with numbers
IPS Chapter 1.2
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 1.2)
Describing distributions with numbers
Measures of center: mean, median


Mean versus median

Measures of spread: quartiles, standard deviation

Five-number summary and boxplot

Choosing among summary statistics

Changing the unit of measurement
Measure of center: the mean
The mean or arithmetic average
To calculate the average, or mean, add
all values, then divide by the number of
individuals. It is the “center of mass.”
Sum of heights is 1598.3
divided by 25 women = 63.9 inches
Heights of 25 women in inches
58 .2
59 .5
60 .7
60 .9
61 .9
61 .9
62 .2
62 .2
62 .4
62 .9
63 .9
63 .1
63 .9
64 .0
64 .5
64 .1
64 .8
65 .2
65 .7
66 .2
66 .7
67 .1
67 .8
68 .9
69 .6
woman
(i)
height
(x)
woman
(i)
height
(x)
i=1
x1= 58.2
i = 14
x14= 64.0
i=2
x2= 59.5
i = 15
x15= 64.5
i=3
x3= 60.7
i = 16
x16= 64.1
i=4
x4= 60.9
i = 17
x17= 64.8
i=5
x5= 61.9
i = 18
x18= 65.2
i=6
x6= 61.9
i = 19
x19= 65.7
i=7
x7= 62.2
i = 20
x20= 66.2
i=8
x8= 62.2
i = 21
x21= 66.7
i=9
x9= 62.4
i = 22
x22= 67.1
i = 10
x10= 62.9
i = 23
x23= 67.8
i = 11
x11= 63.9
i = 24
x24= 68.9
i = 12
x12= 63.1
i = 25
x25= 69.6
i = 13
x13= 63.9
n=25
S=1598.3
Mathematical notation:
x1  x2  ...  xn
x
n
1 n
x   xi
n i 1
Don’t get
psyched out!
1598.3
x
 63.9
25
Learn right away how to get the mean using your calculators.
Another summation example
Consider the following dataset: 3, -11, 6, 0, 22
Find the value of x given the following formula:
( n * 2) ( xi  3)
Answer: Take each value in the dataset.
Call the first value x1. Call the second value x2. Call the last value (the fifth
number), x5.
‘n’ refers to the quantity of values in the dataset.
So in this case, as there are 5 numbers in the dataset, n = 5.
First do the summation: (3+3) + (-11+3) + (6+3) + (0+3) + (22+3)
This gives you 35.
Then multiply that by (n*2) which is 10.
This gives you an answer of 350.
** Another measure of center: the median
The median is the midpoint of a distribution—the number such
that half of the observations are smaller and half are larger.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
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10
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12
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2
3
4
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7
8
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10
11
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
25 12
6.1
1. Sort observations by size.
n = number of observations
______________________________
2.a. If n is odd, the median is
observation (n+1)/2 down the list
 n = 25
(n+1)/2 = 26/2 = 13
Median = 3.4
2.b. If n is even, the median is the
mean of the two middle observations.
Survival years for
Disease X
n = 24 
n/2 = 12
Median = (3.3+3.4) /2 = 3.35
1
2
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5
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0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
Comparing the mean and the median
•
The mean and the median are the same only if the distribution is symmetrical.
•
The median is a measure of center that is resistant to skew and outliers. The mean is not.
•
Outliers “pull” the mean towards themselves.
Mean and median for a
symmetric distribution
Mean
Median
Mean and median for
skewed distributions
Left skew
Mean
Median
Mean
Median
Right skew
One other measure: Mode





It is the category that occurs most
frequently.
In a histogram, it’s the highest bin.
A histogram that clusters around
one general peak (even if it’s more
than one single bar) is called
‘unimodal’.
If there are two peaks, ‘bimodal’
More than that, we just say
‘multimodal’.
In the dataset: 2, 3, 4, 4, 6  the mode would
be 4 since it occurs the most frequently.
However, the mode is only useful when you
have reasonably large datasets. Even so, we
will use it only to summarize a histogram as
discussed above.
Incidence of IBD in Denmark
Mean and median of a distribution with outliers
Percent of people dying
x  3.4
x  4.2
Without the outliers
With the outliers
The mean is pulled to the
The median, on the other hand,
right a lot by the outliers
is only slightly pulled to the right
(from 3.4 to 4.2).
by the outliers (from 3.4 to 3.6).
Effect of outliers
on the mean and
median
Percent of people dying
x  3.4
x  4.2
Without the outliers
With the outliers
Note the presence of outliers – those two fortunate people who managed to live several
years longer than the others. These two large values moved the mean up from 3.4 to 4.2
However, the median , the number of years it takes for half the people to die only went from
3.4 to 3.6.
Note that this says that the median is fairly resistant, but not 100% resistant. The median is
not sensitive to the size of the outlier, rather, iIt is sensitive to the number of outliers.
This is typical behavior for the mean and median. The mean is sensitive to outliers, because
when you add all the values up to get the mean the outliers are weighted disproportionately
by their large size.
However, when you get the median, they are just another two points to count –the actual
size of those values does not affect things.
Measures:

Describe these two graphs.

Which measure of center
would you use for the top
graph? For the bottom
graph? Why?

Answer: When you have a fairly
symmetric distribution with few
outliers or where the outliers are
not very significant in terms of
size, mean is often a good bet
(e.g. top graph). If you have a
few outliers and/oroutliers with
very large values, they will
probably throw off your mean. In
that case, you’d probably want
to go with median (bottom
graph).
*** Impact of skewed data
Symmetric distribution…
Disease X:
x  3.4
M  3.4
Mean and median are the same.
… and a right-skewed distribution
Multiple myeloma:
x  3.4
M  2.5
The mean is pulled toward
the skew.
** Measures of variation




Most people intuitively ‘get’ the benefit of knowing the center of a
distribution (e.g. the ‘average’ salary of first-year doctors). However, a piece
of data that is sadly neglected but is EVERY bit as important, is the variation
of the data.
Just as there are different ways of describing the center of a distribution
(e.g. mean, median, mode), there are different techniques for describing the
spread of a distribution.
As with the center, you must know which description of the spread is the
best of the most accurate tool for describing the spread.
Common techniques for describing the variation in a dataset:
 Range: the highest and lowest values in the dataset. Important, but
outliers can give people a highly inaccurate picture (imagine if you
looked at the range of salaries).
 Quartiles – dividing the range into four
 Standard Deviation / Variance: this is one of the most effective means
of describing the spread, and a tool that we will come back to constantly
throughout this course.
* Measure of spread: the quartiles
The first quartile, Q1, is the value in the
sample that has 25% of the observations
(data points) at or below it. (It is the
median of the lower half of the sorted
data, excluding M).
The third quartile, Q3, is the value in the
sample that has 75% of the data at or
below it. (It is the median of the upper
half of the sorted data, excluding M).
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25
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7
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5
1
2
3
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5
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7
1
2
3
4
5
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
6.1
Survival time (years)
n=25
Q1= first quartile = 2.2
M = median = 3.4
Q3= third quartile = 4.35
5-Number Summary and Box Plot


Once you have divided your dataset into quartiles, you now have
one technique for creating a neat little summary. It is called the ‘5
Number Summary’ and is made up of:
 Lowest number
 First (lower) quartile
 Median (not the mean!)
 Third (upper) quartile
 Highest number
Once you have this summary in hand, you can even ‘draw’ it using a
simple (but very convenient) plot known as a box plot.
** Five-number summary and boxplot
6
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2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
Largest = max = 6.1
BOXPLOT
7
Q3= third quartile
= 4.35
M = median = 3.4
6
Years until death
25
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20
19
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12
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10
9
8
7
6
5
4
3
2
1
5
4
3
2
1
Q1= first quartile
= 2.2
Smallest = min = 0.6
0
Disease X
Five-number summary:
min Q1 M Q3 max
Boxplots for skewed data
Years until death
Comparing box plots for a normal
and a right-skewed distribution
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Boxplots remain
true to the data and
depict clearly
symmetry or skew.
Disease X
Multiple Myeloma
* OUTLIERS – “Suspected Outliers”
Outliers are data points that require some thought. The first step is to decide whether a data point should
indeed be labeled as an outlier. Once you have decided that it is an outlier, the next question is what you
want to do with it.
There are two options for dealing with outliers – you can include them in your analysis, or you can leave
them out.

Exclude outliers: Suppose you have a datapoint that is extremely high – and you think it was
recorded in error. In this case, you would not want to include this value in your calculations since
values like mean and standard deviation would be thrown off by this bad datapoint.

However, if you choose to leave out a datapoint, you MUST include in your paper a discussion of
your reasons for doing so.

Include outliers: The other option, of course, is to include the outlier(s) in your calculations and
analysis.

Discussion question: Suppose we wanted to determine the average height of DePaul students
and we use our class as a sample. However, that particular day, we are being visited by an
incoming freshman who just hpapens to be the tallest person in the world. Would you include
him/her in your analysis?
* OUTLIERS – Identification of the Outlier
At what point do we typically label a datapoint as an outlier? Here is one
way:
1.
Determine the distance from the suspicious data point to the nearest
quartile (Q1 or Q3).
2.
Determine the distance between Q1 and Q3, called the interquartile
range, or IQR.
3.
We call an observation a suspected outlier if it falls more than 1.5
times the size of the interquartile range (IQR) below the first quartile
or above the third quartile.
This technique is called the “1.5 * IQR rule for outliers.”
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1
7.9
6.1
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
“Modified” Boxplot
8
7
Q3 = 4.35
6
Years until death
25
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14
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12
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10
9
8
7
6
5
4
3
2
1
Distance to Q3
7.9 − 4.35 = 3.55
5
Interquartile range
Q3 – Q 1
4.35 − 2.2 = 2.15
4
3
2
1
Q1 = 2.2
0
Disease X
Individual #25 has a value of 7.9 years, which is 3.55 years above
the third quartile. This is more than 3.225 years (i.e.1.5 * IQR).
Thus, we can say that individual #25 is a suspected outlier.
** The “Five-Number Summary”


A quick-and-dirty way to give a reasonable description of center and
spread.
It is made up of:
1.
2.
3.
4.
5.
Minimum
Q1
M
Q3
Maximm
For the previous slide, the five-number summary would be:
0.6 2.2 3.4 4.35 6.1 (If we exclude the outlier)
Quick Reminder:
• Boxplots are a very good way to visualize the spread around a median.
• Boxplots can be used no matter what the distribution, (e.g. normal,
skewed, random, etc).
• Boxplots are a good way to contrast variables having different
distributions with each other. Put one or more boxplots side by side.
Standard Deviation and Z-Scores
One of the most important concepts in the course…
Another way of describing variation



Recall that we usually describe a distribution in terms of its shape,
center, and variation.
One method of describing the variation has already been discussed:
List the quartiles / 5-number summary.
Another extremely important method of describing variation is the
‘variance’ or ‘standard deviation’.
Variation
In every set of values (“dataset”—know this term!) e.g. ages, heights, incomes, crop yields,
spitball-distances, etc there will always be variation. The question is, how do we describe just
how varied our data is? Are all the points closely clustered around some value or are they all over
the map?
Having a sense of how much the various datapoints are spread out around the center (mean or
median) is an important piece of information. Consider employee pay at DePaul University: if we
looked at every income from student workers to the Provost/President, we would see a
tremendous variation around the center as some people make a great deal more than the center,
and some make much less. However, if we focused only on student employees, we would find
that the variation around the center is considerably less.
Suppose we were looking at a widget used in the construction of a high-performance laser.
Suppose this widget is intended to be exactly 2.3 inches in diameter. A manufacturer must make
their parts extremely close to this diameter in order for them to fit properly. We would assume that
the average diameter of, say, 10000 parts would indeed be very close to 2.3. However, the
question from a manufacturing perspective would be: How much variation is there among the
parts? The manufacturer would hope that the variation would be extremely small, that is, nearly all
the parts are very close in size to the mean. If there is a large variation, it would mean that there
are several parts that are inappropriately sized, and many of our very expensive lasers are going
to be faulty.
The most common statistic we used for measuring variability (spread) is called the standard
deviation.
SD (‘s’) is one way of describing the variation of a dataset.
There are other ways. (e.g. Recall that there is more than one
way to describe the center of a dataset).
SD is a great choice because it allows us to easily draw lots of
conclusions and answer various questions about our data.
* However, SD is best used with data that has a symmetrical
distribution (ie doesn’t work well with skewed data). This is
because in order to calculate the SD, we need to use the mean
of that data. Recall that mean is only a good choice for the
center if your data is symmetric.
The Standard Deviation measures variation by looking at how
far the observations are from their mean.
Variance and Standard Deviation are
essentially the same thing

Variance and Standard Deviation are essentially the same thing.
The SD is simply the square root of the variance.




Eg: If Variance is 25, then SD = 5.
If SD = 7, then Variance = 49.
Some people prefer to use variance when discussing spread, others
prefer standard deviation. I hope it’s clear that either one can easily be
converted to the other.
For the time being, I’ll focus on standard deviation.
Measure of spread: the standard deviation
The standard deviation “s” is used to describe the variation around the
mean. Like the mean, it is not resistant to skew or outliers.
Remember that
“resistance” is an
important concept!
1. First calculate the variance s2.
n
1
2
s2 
(
x

x
)

i
n 1 1
2. Then take the square root to get
the standard deviation s.
x
Mean
± 1 s.d.
1 n
2
s
(
x

x
)
 i
n 1 1
(see PPT note)
1 n
2
s 
(
x

x
)

i
n 1 1
2
x
Mean
± 1 s.d.
Take difference of EACH
datapoint from the mean
and square it:
1. Calculate the mean
2. Do a summation of
(every datapoint –
mean)
3. Divide by by n-1
4. To get from variance to
s, simply take the square
root.
Calculating standard deviation ‘s’
Women height (inches)
i
xi
x
(xi-x)
(xi-x)2
1
59
63.4
-4.4
19.0
2
60
63.4
-3.4
11.3
3
61
63.4
-2.4
5.6
4
62
63.4
-1.4
1.8
5
62
63.4
-1.4
1.8
6
63
63.4
-0.4
0.1
7
63
63.4
-0.4
0.1
8
63
63.4
-0.4
0.1
9
64
63.4
0.6
0.4
10
64
63.4
0.6
0.4
11
65
63.4
1.6
2.7
Degrees freedom (df) = (n − 1) = 13
12
66
63.4
2.6
7.0
s2 = variance = 85.2/13 = 6.55 inches squared
13
67
63.4
3.6
13.3
14
68
63.4
4.6
21.6
Sum
0.0
Sum
85.2
n
1
2
s
(
x

x
)
 i
(n  1) 1
Mean = 63.4
Sum of squared deviations from mean = 85.2
s = standard deviation = √6.55 = 2.56 inches
Mean
63.4
Calculating these by hand can be tedious. We will do a couple of
examples, but will quickly switch over to doing them using a
computer.
Properties of Standard Deviation
s is a very useful tool for data analysis
A few properties of s:
 s measures spread about the mean and should be used only when
the mean is the measure of center.

s = 0 only when all observations have the same value and there is
no spread. Otherwise, s > 0.

s is not resistant to outliers.

s has the same units of measurement as the original observations
(e.g. gpa, inches-tall, miles per hour, etc)
Software output for summary statistics:
Excel - From Menu:
Tools/Data Analysis/
Descriptive Statistics
Give common
statistics of your
sample data.
Minitab
(Choosing among summary statistics)

Because the mean is not resistant to
outliers or skew, it should probably only
Height of 30 Women
be used to describe distributions that are
fairly symmetrical and that don’t have
68
 Boxplot for symmetric data: Plot the
67
mean and use the standard deviation for
66
Boxplot for non-symmetric data: If
there is skew or outliers are present, use
Height in Inches
outliers.
error bars.

69
65
64
63
62
61
the median in the five number summary
60
(which you could plot as a boxplot).
59
However, if there is a skew, or you can
58
not use the mean, then you shold not use
SD for your data analysis. So in this case
you would use the quartiles for your bars.
Box Plot
Boxplot
Mean ±
+/-SD
SD
Mean
(Changing the unit of measurement)
Variables can be recorded in different units of measurement. Most
commonly, one measurement unit is a “linear transformation” (later)
of another measurement unit: xnew = a + bx.
Temperatures can be expressed in degrees Fahrenheit or degrees Celsius.
TemperatureFahrenheit = 32 + (9/5)* TemperatureCelsius  a + bx.
Key Point: Linear transformations (changing the unit) do NOTchange
the basic shape of a distribution (skew, symmetry, multimodal).
However, they do change the values for center and spread:
Multiplying each observation by a positive number ‘b’ multiplies both
measures of center (mean, median) and spread (IQR, s) by ‘b’.

Adding some number ‘a’ (positive or negative) to each observation adds
‘a’ to measures of center and to quartiles but it does not change measures
of spread (IQR, s).

Looking at Data - Distributions
** Density Curves and Normal Distributions
IPS Chapter 1.3
© 2009 W.H. Freeman and Company
Objectives (IPS Chapter 1.3)
Density curves and Normal distributions

Density curves

Measuring center and spread for density curves

Normal distributions

The 68-95-99.7 rule

Standardizing observations

Using the standard Normal Table

Inverse Normal calculations

Normal quantile plots
* Density Curve



Density curve: When applied to a histogram, a density curve is
drawn as a smooth approximation to the irregular bars.
One of the reasons we like density curves, is that by estimating the
area under the curve, we can make various predictions and
calculations about the population.
For example, suppose we take our sample of 25 women’s heights,
plot them on a histogram, and then create a density curve. Using
this sample of 25 women, we could attempt to infer information
about all women in the population such as:
 What percentage of women in our population would be more than
6’ tall?
 What percentage of women are between 5’0 and 5’5?
 What is the likelihood of encountering women sorter than 4’6?
 What is the height of the tallest 90th percetile of women?
 etc
*** Area Under the Density Curve
A density curve is a mathematical model of a distribution.
The total area under the curve, by definition/convention, is equal to 1, that is,
100%.
The area under the curve for a range of values (e.g. scores between 6.0 and 8.0)
is the proportion of all observations for that range.
Histogram of a sample with the
smoothed, density curve
describing theoretically the
population.
Area under the curve contd
In the first graph, the areas under the shaded bars (showing students who score less than
6.0 on the exam) of the histogram is 30.3% of the total.
When the software draws its density curve, the area under the curve for students scoring
less than 6.0 is 29.3%. Note how close this is to the exact figure of 30.3%.
In other words, an accurately drawn density curve is a pretty effective tool for summarizing
the information the data.We could leave out the histogram and simply draw the curve and
we’d still have a pretty acurate picture of what was going on.
Density curves come in any
imaginable shape.
Some are well known
mathematically and others aren’t.
(Median and mean of a density curve)
The median of a density curve is the equal-areas point: the point that
divides the area under the curve in half. (Recall that area under the
curve is a proportion).
The mean of a density curve is the balance point, at which the curve
would balance if it were made of solid material.
Finding the mean by using a density curve

You can ballpark where the mean would be if you have a density
curve: Eyeball the curve and try to figure out where it would
‘balance’.
Median and mean of a density curve
The median and mean are the same for a symmetric density curve.
The mean of a skewed curve is pulled in the direction of the long tail.
A density curve is an idealized
description of the data


In this example, the DC is exactly symmetric, while the histogram is
only approximately symmetric.
It will become important to distinguish the mean/sd of the ‘idealized’
density curve from the mean/sd of the actual observations (see note).
(IMP: See ppt note)
* Normal Distribution


Definition: A continuous (solid line) distribution of probabilities that,
ideally, will give a good description of data that clusters about a
mean.
Examples of data that cluster around a mean:




people’s heights
SAT scores
Annual yields of corn crops in Indiana
However, do be aware that just because many things fall into a
Normal distribution, many other things do not.

For example, income distribution is not Normal. (It is typically rightskewed).
“Normal” Curves


Ie: The density curve that is drawn over the histogram of data that
follows a Normal distribution.
Normal curves have the following properties:






Symmetric
Unimodal
Bell-shaped
Curves like this are called ‘Normal curves’ and the data distributions
they describe are called ‘Normal distributions’
Note that it is possible to have a symmetric, bell-shaped curve that
is not Normal.
The idea of a Normal curve does not imply that other kinds of curves
are somehow abnormal!
(Normal distributions)
Normal – or Gaussian – distributions are a family of symmetrical, bellshaped density curves defined by a mean m (mu) and a standard
deviation s (sigma) : N(m,s).
1
f ( x) 
e
2
1  xm 
 

2 s 
2
x
e = 2.71828… The base of the natural logarithm
π = pi = 3.14159…
(See PPT note)
x
A family of density curves
Here, means are the same (m = 15)
while standard deviations are
different (s = 2, 4, and 6).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Here, means are different
(m = 10, 15, and 20) while standard
deviations are the same (s = 3)
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
** Why do we care about Normal
curves/distributions?


Much of the data we examine in the real world turns out to have a
normal distribution (exam scores, survival length for cancer patients,
crop yields, SAT results, people’s heights, birth rate of rabbits,
games of chance, etc, etc, etc)
A great deal of study and research has gone into the properties of
Normal distributions/curves. As a result there are many “tricks” /
inferences / shortcuts that we can make when working with ‘Normal’
data.
Determining the area under the curve
A fairly straight-forward two step process:
1. Convert the value you are interested in (e.g. a Grade equivalent score of 7.2) into a zscore.
2. Look up your z-score in a z-table.
•
The value you find is the area under the curve to the left of your ‘x’.
* A z-what?
• A z-score measures the number of standard
deviations that a data value x is from the mean m.
•
E.g. a z-score of 1 means your value lies exactly 1 sd above
the mean.
•
E.g. a z-score of -0.5 means your value lies exactly one-half sd
below the mean.
•
Know this definition!
** Calculating the z-score
A z-score measures the number of standard deviations that a data
value x is from the mean m.
z
(x  m )
s
• x
the value under consideration
• μ (mu)
the mean of the population
• σ (sigma)
the standard deviation of the population
Use the z-table



Also known as the “standard Normal table”
This table tells us the area under the curve to the left of the desired zscore.
So if your calculated z-score is -1.42, the value you find on the ztable tells you the area under the curve to the left of -1.42.


If your calculated z-score is 0.89, the value you find on the z-table
tells you the area under the curve to the left of 0.89.


The value is: 0.778 (or 77.8%)
The value is: 0.8133 (or 81.33%)
This is a widely used table.
 I guarantee your textbook will include a z-table
 There is a link to one at the top of the course web page
 Google z-table and you’ll find hundreds!
Using the standard Normal table (z-table)
Example: What is the area under the curve to the left of z = -2.40?
.0082 is the
area under
the curve
left of
z = 2.40
.0080 is the area
left of z = -2.41
0.0069 is the area
left of z = -2.46
(…)
The Worst Thing You Can Do:

Is to keep plugging numbers into formulas until you find an answer
that seems right.

When answering these types of questions, it is
VITAL that you make sure you understand the
question being asked in terms of how it looks on
the graph.

In fact, truly understanding the question being asked should be your
goal not just for Normal distribution / z-score type questions, but for
nearly all statistical questions.
Example

Women’s heights follow the N(64.5,2.5) distribution. What percent of
women are shorter than 67 inches tall (that’s 5’6”)?

Note: N(64.5, 2.5) is a way of saying that this dataset:




Follows a Normal distribution (“N”)
With a mean of 64.5
With a standard deviation of 2.5
Example: The original versions of the SAT had a distribution of N(500,100)

The distribution was Normal
Mean of 500

SD of 100

** Ex. Women heights
N(µ, s) =
N(64.5, 2.5)
Women’s heights follow the N(64.5,2.5)
distribution. What percent of women are
Area= ???
shorter than 67 inches tall (that’s 5’6”)?
mean µ = 64.5"
standard deviation s = 2.5"
x (height) = 67"
Area = ???
m = 64.5” x = 67”
z=0
z=1
Start by calculating the z-score.
z
(x  m)
s
(67  64.5) 2.5
, z

 1   1 SD' s above the mean
2.5
2.5
Use a z-table to find the area under the curve to the left of 1.
The table tells us that the area under the Normal curve to the left of +1 SDs is 084 (or 84%).
Conclusion:
84% of women are 67 inches tall or shorter.
Percent of women shorter than 67”
For z = 1.00, the value is
0.8413.
This tells us that the area under
the standard Normal curve to the
left of z is 0.8413.
N(µ, s) =
N(64.5”, 2.5”)
Area ≈ 0.84
Area ≈ 0.16
m = 64.5” x = 67”
z=1
Tips on using Table A
Because the Normal distribution is
symmetrical, if you need to
Area = 0.9901
determine the area to the right of
a z-value, there are 2 ways that
Area = 0.0099
you can go about it.
z = -2.33
area right of z = area left of -z
Review this
figure!!
area right of z =
1
-
area left of z
The National Collegiate Athletic Association (NCAA) requires Division I athletes to
score at least 820 on the combined math and verbal SAT exam to compete in their
first college year. The SAT scores of 2003 were approximately normal with mean
1026 and standard deviation 209.
What proportion of all students would be NCAA qualifiers (SAT ≥ 820)?
x  820
m  1026
s  209
(x  m)
z
s
(820  1026)
209
 206
z
 0.99
209
Table A : area under
z
the curve to the left of
z - .99 is 0.1611
or approx. 16%.
area right of 820
=
=
total area
1
-
area left of 820
0.1611
≈ 84%
Note: The actual data may contain students who scored
exactly 820 on the SAT. However, the proportion of scores
exactly equal to 820 is 0 for a normal distribution is a
consequence of the idealized smoothing of density curves.
(See PPT note)
The NCAA defines a “partial qualifier” eligible to practice and receive an athletic
scholarship, but not to compete, with a combined SAT score of at least 720.
What proportion of all students who take the SAT would be partial qualifiers?
That is, what proportion have scores between 720 and 820?
x  720
m  1026
s  209
(x  m)
z
s
(720  1026)
209
 306
z
 1.46
209
Table A : area under
z
area between
720 and 820
≈ 9%
=
=
area left of 820
0.1611
-
area left of 720
0.0721
N(0,1) to the left of
z - .99 is 0.0721
About 9% of all students who take the SAT have scores
or approx. 7%.
between 720 and 820.
The useful thing about working
with normally distributed data is
that we can manipulate it, and
then find answers to questions
that involve comparing seemingly
non-comparable distributions.
We do this by “standardizing” the
data. All this involves is changing
the scale so that the mean now = 0
and the standard deviation =1. If
you do this to different distributions
it makes them comparable.
z
(x  m )
s
N(0,1)
SAT vs ACT


Question: Suppose that student A scores 1140 on their SAT, and
student B scores 18.2 on their ACT. You are an admissions
counselor and you need to make a decision based exclusively on
their test score. Can you use this data to decide?
Answer: If you can convert these numbers to their corresponding zscores, then absolutely! To do so, you would, of course, need to
know the mean and standard deviation of the two exams. This
information is routinely provided by the testing services.
Ex. Gestation time in malnourished mothers
What is the effect of better maternal care on gestation time and preemies?
The goal is to obtain pregnancies 240 days (8 months) or longer.
What improvement did we get
by adding better food?
m 266
s15
m 250
s20
180
200
220
240
260
280
Gestation time (days)
Vitamins only
Vitamins and better food
300
320
Under each treatment, what percent of mothers failed to carry their babies at
least 240 days?
Vitamins Only
m=250, s=20,
x=240
x  240
m  250
s  20
z
(x  m)
s
(240  250)
20
170
 10
z
 0.5
20
(half a standard deviation)
Table A : area under N(0,1) to
z
the left of z - 0.5 is 0.3085.
190
210
230
250
270
290
Gestation time (days)
Vitamins only: 30.85% of women would be expected
to have gestation times shorter than 240 days.
(Or we could also say that about 70% of vitamin-only
women have gestations >240 days).
310
Vitamins and better food
m=266, s=15,
x=240
x  240
m  266
s  15
z
(x  m)
s
(240  266)
15
 26
206
z
 1.73
15
(almost 2 sd from mean)
Table A : area under N(0,1) to
z
the left of z - 1.73 is 0.0418.
221
236
251
266
281
296
311
Gestation time (days)
Vitamins and better food: 4.18% of women
would be expected to have gestation
times shorter than 240 days.
Compared to vitamin supplements alone, vitamins and better food resulted in a much
smaller percentage of women with pregnancy terms below 8 months (4% vs. 31%).
*** The 68-95-99.7% Rule
for Normal Distributions

This is essentially a “shortcut” for a
mental ballpark of the areas under the
normal curve. It is definitely worth
memorizing.

About 68% of all observations are
contained between 1 standard
deviation of the mean.

About 95% of all observations are
contained between 2SDs of the mean.

About 99.7% observations are
contained between 3 SDs of the mean.
*** The 68-95-99.7% Rule for Normal Distributions

About 68% of all women are
Inflection point
between 62 and 67 inches tall.

About 95% of all women are
between 59.5 and 69.5”.

Almost all (99.7%) women are
between 57 and 72”.
mean µ = 64.5
standard deviation s = 2.5
N(µ, s) = N(64.5, 2.5)
Reminder: µ (mu) is the mean of the idealized curve, while x¯ is the mean of a sample.
s (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.
(See PPT note)
* The 68-95-99.7% Rule for Normal Distributions

From the rule, we can also come
up with some additional numbers:

Inflection point
If 68% of all women are between
62 and 67 inches tall, this means
that 32% are outside of that range.
In other words, 16% are shorter
than 62 inches, and 16% are taller
than 67.

Similarly, if 95% of all women are
between 59.5 and 69.5”, then 5%
are outside of that range. In other
words, 2.5% are shorter than 59.5
and 2.5% are taller than 69.5”.
mean µ = 64.5
standard deviation s = 2.5
N(µ, s) = N(64.5, 2.5)
* Going in the other direction…
We may also want to find the observed range of values that correspond
to a given proportion/ area under the curve.
For that, we use Table A backward:

we first find the desired
area/ proportion in the
body of the table,

we then read the
corresponding z-value
from the left column and
top row.
For an area to the left of 1.25 % (0.0125),
the z-value is -2.24
Vitamins and better food
How long are the longest 75% of pregnancies when mothers with malnutrition are
given vitamins and better food?
m  266
s  15
upper area  75%
lower area  25%
x?
Table A : z value for the
lower area 25% under
N(0,1) is about - 0.67.
206
(x  m)
z
 x  m  ( z *s )
s
x  266  (0.67 *15)
x  255.95  256
m=266, s=15,
upper area 75%
upper 75%
221
236
?
251
266
281
296
Gestation time (days)
Remember that Table A gives the area to the
left of z. Thus, we need to search for the
lower 25% in Table A in order to get z.
 The 75% longest pregnancies in this group are about 256 days or longer.
311
Tips on using Table A
To calculate the area between two z- values, first get the area under each zvalue. Then simply subtract one from the other..
A common mistake made by
students is to subtract both z
values. But the Normal curve
is not uniform.
area between z1 and z2 =
area left of z1 – area left of z2
** Is the data Normal?




Deciding whether data does indeed show a Normal (or, close to
Normal) distribution is a very important question.
All the examples we’ve been discussing above involving z-scores
assume that the data is Normal. If the data was not Normal, all of
our answers and calculations would be flawed.
Recall that there are other types of distributions out there besides
the Normal distribution.
 Binomial
 Poisson
 Etc
Each type of distribution has its own characteristic formulas,
calculations, inference techniques, etc. Again, one of the more
common and useful distributions, and therefore the one we will
spend lots fo time discussing, is the Normal distribution.
A graphing technique for assessing if data is Normal:
Normal quantile plots
Simply eyeballing a curve to see if it is bell-shaped, symmetric, etc is not very
accurate. One mathematical way of determining whether a distribution is indeed
approximately normal is to plot the data on a normal quantile plot.
The data points are ranked and the percentile ranks are converted to z-scores
with Table A. The z-scores are then used for the x axis against which the data
are plotted on the y axis of the normal quantile plot.

If the distribution is indeed normal the plot will show a straight line, indicating a
good match between the data and a normal distribution.

Systematic deviations from a straight line indicate a non-normal distribution.
Outliers appear as points that are far away from the overall pattern of the plot.
Good fit to a straight line: the
distribution of rainwater pH
values is close to normal.
Curved pattern: the data are not
normally distributed. Instead, it shows
a right skew: a few individuals have
particularly long survival times.
Normal quantile plots are complex to do by hand, but they are standard
features in most statistical software.
(The “standard Normal distribution”)
Because all Normal distributions share the same properties, we can
standardize our data to transform any Normal curve N(m,s) into the
standard Normal curve N(0,1).
N(64.5, 2.5)
N(0,1)
=>
x
z
Standardized height (no units)
For each x we calculate a new value, z (called a z-score).