Chapter 5
Exploring Data: Distributions
February 9, 2010
Brandon Groeger
Outline
1. What is Statistics?
2. Data
3. Distributions
4. Histograms
5. Stemplots
6. Mean, Median, and Quartiles
7. Standard Deviation and Variance
8. Normal Distribution
9. Extensions and Applications
10. Discussion
What is Statistics?
• “Statistics is the science of collecting, organizing
and interpreting data”
• Statistical inference is drawing conclusions from
data.
Data
• Data is information
about an individual
or a group of
individuals (a
population).
• “A variable is any
characteristic of a
individual”
Name
Height
Weight
John
71 in.
160 lbs
Bob
67 in.
150 lbs
Jane
64 in.
130 lbs
Fred
78 in.
180 lbs
Distribution
• “The distribution of a variable tells us what
values the variable takes and how often it takes
these values.”
• Graphical representations of data make seeing
patterns easier.
Histograms
3
25
20
2
130-149
150-169
170-189
1
1
2
3
4
5
6
15
10
5
0
0
Weight (lbs)
100 Die Rolls
Making a Histogram
1.
2.
3.
Step 1: Define a set of
equally sized classes
Step 2: Determine the
number of individuals in
each class.
Step 3: Draw the histogram
Name
Height
Weight
John
71 in.
160 lbs
Bob
67 in.
150 lbs
Jane
64 in.
130 lbs
Fred
78 in.
180 lbs
2
60-64
65-69
70-74
75-79
1
0
Height (in.)
Interpreting Histograms
• Look for patterns, shape, the
center, and spread.
• Distributions can be
symmetric or skewed.
• An outlier is “an individual
value that falls outside the
overall pattern.”
Height (in.)
9
8
7
6
5
4
3
2
1
0
49 53 57 61 65 69 73 77 81 85 89
Stemplots
• 30 Test Scores(41, 52, 58, 63,
64, 65, 68, 70, 71, 71, 72, 75, 79,
82, 82, 83, 84, 85, 88, 89, 89,
90, 91, 92, 94, 98, 99, 100, 100,
100)
• In this stemplot the left
column(the stem) represents the
“tens place” of each test score
and the right column(the leaf)
represents the “ones place”.
• Stemplots can be easier to read
and more detailed than
Histograms for small amounts
of data.
Test Scores
Stem
Leaf
0
1
2
3
4 1
5 28
6 3458
7 011259
8 223546899
9 102489
10 00
Describing the Center: Mean
• The mean of a set of data is the sum of the data divided by the
number of data points.
• Mean =
x1  x2  ...  xn
x
n
• Example: Heights (64, 67, 71, 78)
• Mean = (64 + 67 + 71 + 78)/4 = 280/4 = 70
Describing the Center: Median
• “The median is the midpoint of a distribution, the number such that
half of the observations are smaller and the other half are larger.”
• Finding the median:
1. Arrange the data in order from smallest to largest
2. If the number of data points (n) is odd: median = the entry (n+1)/2
3. If n is even: median = the average of entry (n/2) and (n+1)/2
• Example: 30 Test Scores(41, 52, 58, 63, 64, 65, 68, 70, 71, 71, 72, 75,
79, 82, 82, 83, 84, 85, 88, 89, 89, 90, 91, 92, 94, 98, 99, 100, 100, 100)
• Median = Average(82,83) = 82.5
Describing Spread: Quartiles
• Quartiles divide a data set into four pieces, where each quartile has
one quarter of the data points.
• Finding the quartiles of a data set:
1. Find the median of the set this is the half way point (1/2) which
is the 2nd quartile (2/4).
2. Take all of the data points smaller than the median and find
their median this is the 1st quartile.
3. Take all of the data points larger than the median and find
their median this is the 3rd quartile .
Five Number Summary
• The five number summary of a distribution is the minimum, the 3
quartiles, and the maximum written in order.
• Example: 30 Test Scores(41, 52, 58, 63, 64, 65, 68, 70, 71, 71, 72, 75,
79, 82, 82, 83, 84, 85, 88, 89, 89, 90, 91, 92, 94, 98, 99, 100, 100,
100)
• Minimum = 41, 1st Quartile = 70, Median = 2nd Quartile= 82.5,3rd
Quartile = 91, Maximum = 100
Boxplots
• “A boxplot is a graph of the five number
summary”
100
90
80
70
60
50
40
30
20
10
0
Maximum
3rd Quartile
Median
1st Quartile
Minimum
Test Scores
Practice
• Make a boxplot for the
following set of monthly
S&P500 returns (-3.5%, -0.6%
4.8%, 1.1%, -8.6%, -1.0%, 1.2%,
-9.1%, -16.9%, -7.5%, 0.8%, 8.6%, -11.0%, 8.5%, 9.4%,
5.3%, 0.0%, 7.4%, 3.4%, 3.6%,
-2.0%, 5.7%, 1.8%)
15.0%
10.0%
5.0%
0.0%
-5.0%
•
•
•
•
•
Minimum: -16.9%
1st Quartile: -5.5%
Median: 0.8%
3rd Quartile: 3.4%
Maximum: 9.4%
-10.0%
-15.0%
-20.0%
Describing Spread:
Standard Deviation & Variance
• “The variance (s2) of a set of observations is an average of the
squares of the deviations of the observations from their mean.”
• “The standard deviation (s) is the square root of the variance.”
( x1  x) 2  ( x2  x) 2  ...  ( x2  x) 2
s 
n 1
2
• Note: Standard deviation is often calculated using n as the
denominator instead of n-1. This is called Bessel’s correction, which
corrects for bias.
Standard Deviation Example
• Weights in lbs: (130, 150, 160, 180)
• Mean = 155 lbs
• Variance = s2 = ((130-155) 2 + (150-155) 2 + (160155) 2 + (180-155) 2 ) / (4-1) = 433.33
• Standard deviation = s = (433.33)1/2 = 20.82 lbs
Normal Distributions
• A normal curve is the graph of
a normal distribution, which is
one of many types of
distributions.
• Many data sets including the
height of humans roughly
follow a normal distribution.
• 68-95-99.7 rule
A Normal Curve
Extensions
• Other distributions
▫ Uniform, Exponential, Gamma
• Regression analysis and fitting a trend line
• Other Statistics
▫ Geometric mean, Mode, Kurtosis
Applications
•
•
•
•
•
•
•
Manufacturing
Insurance
Investment/Banking
Marketing
Biology
Business Management
The Census
Trivia
• Abraham Wald (1902-1950): Where should extra
armor be added to WWII combat aircraft?
• 1999 Mars Climate Orbiter Crash
• 22% of American high school students reported
they smoke, but only 9.7% said that they smoked
20 out of the past 30 days.
Discussion
• Questions?
• Can you think of other extensions or
applications?
• How can you use statistics in everyday life?
• Homework: (7th edition) #9, 30a-b