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Chapter 4 Displaying and Summarizing
Quantitative Data
Math2200
Example: Tsunamis and Earthquake
- The most disastrous tsunami
- Dec 26, 2004, in Sumatra
- Earthquake: magnitude 9.0
- Killed 225,000 people
Question
• Was the earthquake that caused it truly unusually big?
• US National Geophysical Data Center
• Data on the magnitude of underlying earthquakes for 1240 historical tsunamis
• How do we learn the data
Year Country Magnitude
-1300 GREECE 6
-479 GREECE 6.7
-426 GREECE 7.1
-373 GREECE 7.3
-330 GREECE 7
-227 GREECE 7.2
-57 ALBANIA 6.6
-26 CYPRUS ISLAND 7.3
2003 NEW ZEALAND 7.5
2003 JAPAN 8.1
2003 JAPAN 6.8
2003 USA 7.2
2004 INDONESIA 6.5
2004 JAPAN 7.2
2004 JAPAN 7.4
2004 INDONESIA 9
Histogram
• Display a quantitative variable by discretizing it into equal-width bins
• Counts for the bins give the distribution of the quantitative variable
• Make a bar chart based on these counts and align the bar according to the bin values, we get a histogram– Do not leave gaps between bars
Historgram of earthquake magnitude
Magnitude
Fre
qu
en
cy
3 4 5 6 7 8 9
05
01
00
15
02
00
25
03
00
35
0
With different number of bins
Magnitude
Fre
quen
cy
3 4 5 6 7 8 9
050
100
150
Magnitude
Fre
quen
cy
3 4 5 6 7 8 9
050
100
150
Magnitude
Fre
quen
cy
3 4 5 6 7 8 9
050
150
250
350
Magnitude
Fre
quen
cy
3 4 5 6 7 8 9
010
020
030
040
050
0
Summarize the histogram
• Magnitudes are typically around 7
• Most are between 5.5 and 8.5
• Minimum is around 3
• Maximum is around 9
• Why there is a sharp peak in the middle?
Stem-and-Leaf Plot• John W. Tukey
• Useful for small data sets
• Similar to histogram, but the bars give numerical values more than counts
5 | 6 6 | 0444 6 | 8888 7 | 2222 7 | 6666 8 | 000044 8 | 8Pulse-rates of 24 woman(8|8 means 88 beats/min)
Handwriting
• Handwriting may not give the same space for different digits. That violates the area principle
• When you make a stem-and-leaf plot, be sure to give each digit the same width.
Dotplot
• Replace digits in stem-and-leaf plot by dots
How to summarize the distribution of a quantitative variable?
• shape
mode, symmetry, outlier
• center
mean, median
• Spread
sd, IQR
Shape• Peak / Mode
– Is there a peak? If so, how many peaks?– For quantitative variables, the mode is where
the peak is at.– No peak: uniform– One peak: unimodal– Two peaks: bimodal– More than two peaks: multimodal
uniform
x
Fre
quen
cy
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
12
unimodal
x
Fre
quen
cy
-2 -1 0 1 2
05
1015
20
bimodal
x
Fre
quen
cy
-2 0 2 4 6
020
4060
8010
0
multimodal
x
Fre
quen
cy
-5 0 5
020
4060
8010
0
Shape• Symmetry
– Tail: thinner ends of a distribution– Skewed: If one tail stretches out farther than the
other, we say the histogram is skewed to the side of the longer tail
skewed to the right
x1
Fre
quen
cy
0.0 0.5 1.0 1.5 2.0 2.5 3.0
020
4060
8010
0
symmetric
x
Fre
quen
cy
-3 -2 -1 0 1 2 3
020
4060
8010
0skewed to the left
x2
Fre
quen
cy
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
020
4060
8010
0
Shape
• Outliers– Those that stand
away from the body of the distribution
– The judgment is vague sometimes
Fre
qu
en
cy
-6 -4 -2 0 2
05
10
15
Center
• When a histogram is symmetric and unimodal, the center is obvious– The corresponding numerical value can be
taken as the sample average, or say the sample mean
– The sample mean is actually where the histogram balances
Center
• For skewed distribution– The sample mean is dragged to the side of
the longer tail– Usually, much more than 50% values will be
less or larger than the sample mean– Median is more appropriate
• Median is the value that splits the data in half
Finding the median
• Suppose that we have n numbers
• Order them first– If n is odd, the median is middle value. That
is, the value in the (n+1)/2 position– If n is even, we take median as the average of
the values in positionsn/2 and n/2+1
Mean versus median
• Extreme values / outliers:– Median only considers the order of the values,
so it is resistant to extreme values– Mean is very sensitive
• Skewed distribution– Median is preferred than mean
• Unimodal and symmetric distribution– Mean is preferred because it uses more
information from the data
Spread
To quantify the variation
• Range
• Interquartile range (IQR)
• Standard deviation
Range
• Range = max – min
• Very sensitive to extreme values
Interquartile Range
• Quartiles– Q1 (lower quartile or the 25th percentile): one quarter
of the data lies below Q1– Q2 (median or the 50th percentile)– Q3 (upper quartile or the 75th percentile): one quarter
of the data lies above Q3• IQR = Q3-Q1
– Not sensitive to extreme values• How to find Q1 and Q3?
– Split the order values into two halves using the median
– Q1 is the median of the first half– Q3 is the median of the second half
Standard deviation
• Sample variance = average of squared deviations
• Standard deviation (sd)– Sensitive to extreme values
σX in TI-83
How to obtain these numbers using TI-83?
• Press STAT• Move the cursor to CALC• Press 1• The screen shows 1-Var Stats• Put the list you want the statistics for. For examp
le, L1.• Press ENTER, then you will see
– Sample mean, sample sum, sample sum squares, sample standard deviation (Sx),σx (the same as except divided by n instead of n-1), sample size n, minimum, Q1, median, Q3, maximum
Summary
• Make a picture– Histogram, stem-and-leaf plot, dot plot
• Shape– How many modes?– Symmetric?– Outliers?
• If there are outliers, summarize once with the outliers and another time without the outliers
• Center and spread– Skewed distribution: median and IQR– Symmetric and unimodal distribution: mean and sd
What can go wrong?• Do not use what we learned in chapter 4 for a categorica
l variable– Do not make histogram of a categorical variable– Do not look for shape and center and spread of a bar chart– Do not use mean, sd, IQR, etc. for a categorical variable
• Graph with bars are not always histograms or bar charts• Choose a bin width appropriate to the data• Check the summary numbers. Do they make sense?• Do not worry about small differences when using differen
t methods– No need to use too many digits for the summary numbers– Using one or two more digits than data is enough
• Do not round in the middle of a calculation• Multiple modes, outliers (make a picture)