Transcript Chapter 4 - WordPress.com

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
extraordinary?
• US National Geophysical Data Center
• Data on the magnitude of underlying
earthquakes for 1240 historical tsunamis
• How do we learn the data?
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
200
150
100
50
0
Frequency
250
300
350
Historgram of earthquake magnitude
3
4
5
6
Magnitude
7
8
9
150
100
0
50
Frequency
100
50
0
Frequency
150
With different number of bins
3
4
5
6
7
8
9
3
4
5
8
9
7
8
9
500
400
300
100
200
Frequency
250
150
0
50
0
Frequency
7
Magnitude
350
Magnitude
6
3
4
5
6
Magnitude
7
8
9
3
4
5
6
Magnitude
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?
– Due to historical estimates given by experts
but not modern seismographs
How to make a histogram using TI-83?
• Input the data
– Suppose our data is 1.34, 2.21, 3.64, 6.2
– 2nd + ( + 1.34, 2.21, 3.64, 6.2 + 2nd + ) + STO + 2nd + 1
– That gives L1 = {1.34, 2.21, 3.64, 6.2}
• Make the histogram
– Press 2nd+Y= (STAT PLOT)
– You will see
STAT PLOTS
1: Plot 1 … Off
– Move the cursor to highlight 1:, then press ENTER
– Select the following:
• ON
• Type: histogram (3rd one)
• Xlist: L1
– Press ZOOM
– Press 9
• To adjust the graph window, press WINDOW and change the
options
Stem-and-Leaf Plot
• John W. Tukey
> stem(pulse,scale=2)
The decimal point is 1 digit(s) to the right of the |
5|6
6 | 0444
6 | 8888
7 | 2222
7 | 6666
8 | 000044
8|8
• Useful for small data
sets
• Similar to histogram,
but the bars give
numerical values more
than counts
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
• center
• spread
Shape
• Peak / Mode
– Is there a peak? If so, how many peaks?
– For categorical variables, the mode is the value that
appears most often
– 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
0
0
20
20
60
-2
80
100
100
0.2
0
0.4
0.6
2
4
x
0.8
6
60
80
0.0
40
Frequency
40
Frequency
0
0
2
5
6
8
10
Frequency
4
Frequency
15
10
20
12
uniform
unimodal
1.0
-2
-5
-1
0
x
x
bimodal
multimodal
0
x
1
2
5
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 left
80
60
80
0.0
0.5
1.0
1.5
x1
2.0
2.5
3.0
20
0
0
20
40
40
Frequency
60
Frequency
0
20
40
Frequency
60
80
100
100
symmetric
100
skewed to the right
-3
-2
-1
0
x
1
2
3
-3.0
-2.5
-2.0
-1.5
x2
-1.0
-0.5
0.0
Shape
10
5
0
Frequency
– Those that stand
away from the body
of the distribution
– The judgment is
vague sometimes
15
• Outliers
-6
-4
-2
0
2
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, e.g. income
– 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
example, 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 today for a categorical
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
different 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