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Transcript
Exploratory Data Analysis
The goal of data analysis is to gain information from the data.
Exploratory data analysis: set of methods to display and summarize the data.
Data on just one variable: the distribution of the observations is analyzed by
I.
Displaying the data in a graph that shows overall patterns and unusual
observations (bar chart, histogram, density curve)
II.
Computing descriptive statistics that summarize specific aspects of the
data (center and spread).
Review of Histograms





A histogram represents percent by area.
The height of each block represents frequencies/percentages of
the observations falling in the interval.
The total area under a histogram is ______ if height in
frequencies
The total area under a histogram is ______ if height in
percentages
There is no fixed choice for the number of classes in a
histogram:
•
•
•

If class intervals are too small, the histogram will have spikes;
If class intervals are too large, some information will be missed.
Use your judgment!
Typically statistical software will choose the class intervals for
you, but you can modify them.
Center and Spread
Frequency
Distribution of city fuel consumption
16
14
12
10
8
6
4
2
0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Mph/gallon
Measuring Centers
The most common measures are the mean (or average) and the median.
1.
The Mean or Average x
To calculate the average x of a set of observations, add their value and divide by the
number of observations:
x1  x2  x3  ...  xn
x
n
Data: Number of home runs hit by Babe Ruth as a Yankee
54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22
The mean number of home runs hit in a year is:
x
54  59  35  41  46  ...  41  34  22 659

 43.9
15
15
2.
The median
The median M is the midpoint of a distribution, the number such that half the
observations are smaller and the other half are larger.
To find the median:
1. Sort all the observations in order of size from smallest to largest
2. If the number of observations n is odd, the median M is the center
observation in the ordered list; I.e. M=(n+1)/2-th obs.
3. If the number of observations n is even, the median M is the mean of the two
center observations in the ordered list.
Example 1: Ordered list of home run hits by Babe Ruth:
22 25 34 35 41 41 46 46 46 47 49 54 54 59 60
N=15 Median = 46
8th
Example 2: Ordered list of home run hits by Roger Maris:
8 13 14 16 23 26 28 33 39 61
N=10 Median = (23+26)/2=24.5
Symmetric distribution
50%
Mean versus Median
1. The mean and median of a
symmetric distribution are close
together
Mean
Median
2. In skewed distributions, the mean is farther out in the long tail
than is the median. The mean is more sensitive to extreme
values.Right-skewed distribution
Left-skewed distribution
50%
Median
Mean
50%
Mean
Median
Mean or Median?


The mean is a good measure for the
center of a symmetric distribution
The median is a resistant measure and
should be used for skewed distributions.
Its value is only slightly affected by the
presence of extreme observations, no
matter how large these observations are.
City
The Mode
Frequency
Distribution of city fuel consumption
16
14
12
10
8
6
4
2
0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Mph/gallon
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Largest(5)
Smallest(5)
18.9
1.629717
18
17
8.926327
79.67931
17.87193
3.710471
53
8
61
567
30
22
13
On average, the cars under study drive 18.9 miles per gallon, and 50% of the
cars under study drive at least 18 miles per gallon.
The mode is the observation value with the highest frequency
Spread of a Distribution
Two measures of spread:
1. The Quartiles:
First quartile Q1 = is the value such
that 25% of the observations fall at or
below it,
(Q1 is often called 25th percentile).
The third quartile Q3 = the value such
that 75% of the observations fall at or
below it, (Q3 is often called 75th
percentile).
Typically used if the distribution of
the observations is skewed.
Q1
25%
M Q3
Frequency
Distribution of city fuel consumption
16
14
12
10
8
6
4
2
0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Mph/gallon
First quartile (Q1) = 16, third quartile (Q3) = 21
What does this mean in terms of the data?
Percentiles (also called Quantiles):
In general the nth percentile is a value such that n% of the observations
fall at or below or it;
n%
nth percentile
In the example before:
5th percentile = 10.35
95th percentile = 24.1
10th percentile = 11
90th percentile = 22
Hence about 80% of the cars get between 11 and 22 miles per gallon.
Descriptive measures for
skewed distributions
If the histogram of the data is skewed, use the following descriptive
statistics:
Min, Q1, Median, Q3, Max
To describe the distribution of the observed variable.
In our example,
Min=8, Q1=16, Median=18, Q3=21, Max=61
The Standard Deviation
If a distribution is symmetric:
Use the average to measure the center and
the Standard Deviation to measure the spread.
The standard deviation s (or SD ) measures how far the observations are from the
average.
Example: A person’s metabolic rate= rate at which the body consumes energy.
Rates of 7 men in a study on dieting: 1792, 1666, 1614, 1460, 1867, 1439, 1362.
The mean is x  1600 and the s.d. s =189.24
Deviation=1600 –1439=161

1300
 
1400

1500
Deviation=1867 – 1600=267
x
1600
Metabolic rate


1700
1800

1900
Formula for the SD
In symbols, the standard deviation s of n observations x1 , x2 ,..., xn is
( x1  x ) 2  ( x2  x ) 2  ...  ( xn  x ) 2
s
n 1
The variance of an observed variable is defined as the square of the standard
deviation.
Variance = s2
Properties of the SD

It measures the spread about the mean.

Only used in association with the mean. Good descriptive measure for
symmetric distributions

If s = 0, all the observations have the same value

It is a POSITIVE value, the larger s is, the more spread out the
observations are around the mean

It is NOT a resistant measure, a few extreme observations may affect
its value (make it very large).

The variance is the square of the s.d.
Interpreting the SD
For many lists of observations – especially if their histogram is bell-shaped
1.
2.
Roughly 68% of the observations in the list lie within 1 standard
deviation of the average
95% of the observations lie within 2 standard deviations of the average
Ave-2s.d.
Ave-s.d.
Average
68%
95%
Ave+s.d.
Ave+2s.d.
Example
In a large university, data were collected to study the academic achievements
of computer science majors. We’ll consider the SAT math scores of 224 first
year CS students.
The average SATM score is 595.28 with s.d. s= 86.40
Are the average and s.d. good
descriptions of the SATM scores
distribution?
Roughly 68% of the students have
scores between 510 and 680
Roughly 95% of the students have
scores between 422 and 768
Histogram of the SATM Scores
CS students example:
Descriptive statistics
Mean = 595.28 Std Deviation = 86.40 Max= 800 Min= 300
Q1 = 540
Median = 600.00 Q3= 650 IQR=110 1.5xIQR=165
5th percentile = 460
95th percentile = 750
Histogram of the SATM Scores
422
95% of scores
768
Analysis of the scores
for male and female students:
SATM scores for men
SATM scores for women
Exploratory Data Analysis:
1. Always plot your data
2. Look for overall patterns & striking deviations such as
outliers
3. Calculate a numerical summary to describe the center and
the spread
4. NEXT STEP: sometimes the overall pattern is so regular that
we can describe it through a smooth curve, called a density
curve
Computing descriptive statistics
in Excel
There are two ways:
1. Use the formula palette – click on
the fx button
OR
2.
Use the Data Analysis Toolpak &
select descriptive statistics
The descriptive statistics tool
Input range: sequence of
cells containing the
data
Label in First row
Output range: tell Excel
where to put the
output
Summary statistics: to
be checked
Formulas for 5-number summary
Five number summary
City
Min
Q1
Median
Q3
Max
8
16
18
20.75
61
Highway
Min
Q1
Median
Q3
Max
13
22.25
25.5
28
68
Select an empty cell, and type the function name you want to compute or
use the function palette for the list of available functions.
For instance to compute the min of the fuel consumption data in the city,
type
=min(b2:b31)
Normal distributions
Normal curves provide a simple, compact way to describe
symmetric, bell-shaped distributions.
Normal curve
SAT math scores for CS students
Money spent in a supermarket
Is the normal curve a good approximation?
SAT math scores for CS students
The area under the histogram, i.e. the percentages of the observations, can be
approximated by the corresponding area under the normal curve.
If the histogram is symmetric, we say that the data are approximately normal
(or normally distributed).
We need to know only the average and the standard deviation of the
observations!!