Download Slide 1

Plan for Today:
Chapter 11: Displaying Distributions with Graphs
Chapter 12: Describing Distributions with Numbers
Histograms
Pie chart and bar graph are the common graphs of the
distribution of a categorical variable.
Histogram is the most common graph of the distribution
of a quantitative variable.
Histograms
the obs from
85 to 95
implies number
number /percentage
at this range
Note: There is no space between bars.
Overall Pattern of a Distribution
The center and the spread.
See if the distribution has a simple shape that you can
describe in a few words.
Histograms: center and the spread
Histogram A
Histogram B
Histograms: shape
Symmetric: if the right and left sides of the histogram are
approximately mirror images of each other.
Histograms: shape
Skewed to the right: if the right side of the histogram extends
much farther out than left side.
Histograms: shape
Skewed to the left: if the left side of the histogram extends
much farther out than right side.
Stemplot
A stemplot (a.k.a. stem-and-leaf plot) is quicker to make and
presents more detailed information.
Stemplot
The max temperatures for the first 11 days this February at
West Lafayette (I faked the number 19).
56 49 55 42 48 36 36 35 33 38 19
Largest place value
Next place to the right
1
2
3
4
5
9
Keep this row even you
don’t have any 20s
35668
289
56
Duplicates have to be
labeled separately.
Boxplots:
The median M is the midpoint of a distribution. Half the
observation are smaller that M and the other half are larger.
How to find the median:
1) Arrange all 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.
3) If the number of observations n is even, the median M is the
average of the two center observations in the ordered list.
Boxplots:
The median divided the sequence into left/right subgroups.
The first quartile Q1 is the median of the left subgroup.
The third quartile Q3 is the median of the right.
Boxplots:
Q1 = 10.5
Q3 = 26
[ 7 9 10 11 14 17 ] 19[ 20 21 25 27 29 30 ]
median
Boxplots (without Outliers):
Maximum
25% of the data
Q3
25% of the data
median
25% of the data
Q1
25% of the data
Minimum
Without outliers
Outliers:
The interquartile range (IQR) is the distance between first
quartile Q1 and third quartile Q3.
IQR = Q3 – Q1
Any data observation which lies more than 1.5*IQR lower
than the first quartile or 1.5*IQR higher than the third quartile
is considered an outlier.
IQR
1.5*IQR
1.5* IQR
Median
Q1
Q3
Modified Boxplots (with Outliers)
Largest non-outlier
point
Minimum(since we don’t
have any outliers
With outliers
Center and Spread :
We often use two indexes to measure the central tendency:
1) Median
2) Mean/ average:
sample mean:
x1  x2 
X
n
xn
Center and Spread :
We often use two indexes to measure the variability or “spread” :
1) Interquartile range (IQR)
2) Standard deviation (std dev):
sample std dev:
sample variance:
s
1 N
2
( xi  x )

N  1 n1
N
1
2
2
s 
( xi  x )

N  1 n 1
Center and Spread :
Mean and standard deviation have better numerical properties.
The median, Q1, Q3 suffer less impact at the present of outliers.
Center and Spread :
The max temperatures for the first 10 days this February at
West Lafayette. The researcher made a typo when he recorded
the value 49.
Before: 56 49 55 42 48 36 36 35 33 38
After: 56 149 55 42 48 36 36 35 33 38
Before
After
Median
40
40
Q1
36
36
Q3
49
55
Before
After
Mean
42.8
52.8
Std Dev
8.57
34.8