1 Chapter 7 Looking at Distributions. 2 Modeling by A Distribution For a given data set we want to...
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Transcript of 1 Chapter 7 Looking at Distributions. 2 Modeling by A Distribution For a given data set we want to...
1
Chapter 7Looking at Distributions
2
Modeling by A Distribution
For a given data set we want to know which distribution can fit each variable. This is a modeling problem.
When we have a knowledge to use a specific type distribution (normal, exponential, Poisson distributions) to fit the data, a goodness-fit-test will be useful.
Various Q-Q plots are very useful methods to find a suitable distribution to fit the data.
3
Two data sets
The contents in this chapter are from Chapter 7 of the textbook.
Our textbook chooses the data set of marathon.sav to show us how to use SPSS for looking at distribution. The Chicago Marathon has been run yearly since 1977.
As we use the student version of SPSS that has some limitation on the number of rows/columns,
we use a similar data set of mar1500.sav to instead.
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Data set “mar1500.sav”
The data set involves the following variables:
“age”, “sex”, “hours”, “agecat8”, and “agecat6”.
Hours = “completion time in hours” Agecat8: 1=24 or less, 2=25-39, 3=40-44, 4=45-
49, 5=50-54, 6=55-59, 7=60-64, 8=65+ Agecat6: 1=44 or less, 2=45-49, 3=50-54, 4=55-
59, 5=60-64, 6=65+
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Histogram
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Impressions on the histogram
The mean falls in 4.3 - 4.4 The distribution is not symmetric about the
mean. The distribution has a tail toward larger times. Low marathon times are difficult to achieve. It
is hard to break the world record. Since the distribution has a tail toward larger
values, the median should be somewhat less than the mean.
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Basic statisticsDescriptives
4.3306 .020164.2911
4.3702
4.30734.2765
.609.78070
2.197.705.511.00.482 .063.323 .126
MeanLower BoundUpper Bound
95% ConfidenceInterval for Mean
5% Trimmed MeanMedianVarianceStd. DeviationMinimumMaximumRangeInterquartile RangeSkewnessKurtosis
completiontime in hours
Statistic Std. Error
8
Basic statistics
The 5% trimmed mean excludes the 5% largest and the 5% smallest values. It is based on the 90% of cases in the middle.
The trimmed mean provides an alternative to the median when you have some outliers.
In this data the 5% trimmed mean doesn’t differ much from the usual mean, because the distribution is not too far from being symmetric.
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Comparisons of completing time on Gender
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Comparisons of completing time on Gender
Percentiles
3.5399 3.02563.7233 3.23734.0964 3.5950 4.1006 3.59514.5433 4.0469 4.5433 4.04695.0790 4.6183 5.0783 4.61725.6372 5.14085.9658 5.5035
Percentiles5102550759095
F Msex
F Msex
completion time inhours
completion time inhours
WeightedAverage(Definition 1) Tukey's Hinges
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Comparisons of completing time on Gender
The difference in all of the percentile values of completing times between men and women is about 0.4882 hour.
The weighted percentiles and Tukey’s hinges are two different ways of calculating sample percentiles. More details refer to P.120.
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Histogram of completion times for women
8. 006. 004. 00
80
60
40
20
0
Freq
uenc
y
Mean =4. 6274Std. Dev. =0. 74743
N =589
Hi stogram
for sex= F
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Histogram of completion times for men
6. 005. 004. 003. 002. 00
100
80
60
40
20
0
Freq
uenc
y
Mean =4. 1387Std. Dev. =0. 74105
N =911
Hi stogram
for sex= M
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Age and Gender
Report
time HR:MIN:SEC
4:35:42 84 0:49:08 4:29:204:05:50 69 0:43:59 4:02:124:22:14 153 0:49:03 4:19:384:34:28 350 0:41:28 4:31:574:05:14 479 0:43:14 3:59:514:17:35 829 0:44:52 4:15:504:32:45 76 0:43:11 4:28:024:03:17 142 0:44:30 3:58:294:13:33 218 0:46:09 4:12:034:58:23 38 0:55:24 4:47:134:10:22 102 0:39:23 4:01:094:23:24 140 0:49:01 4:11:594:52:03 28 0:40:23 4:46:004:15:20 73 0:48:05 4:10:564:25:31 101 0:48:46 4:27:224:36:21 5 0:40:08 4:21:034:25:36 25 0:35:23 4:20:534:27:24 30 0:35:42 4:20:585:31:17 2 0:50:32 5:31:175:01:26 11 0:47:01 5:03:025:06:01 13 0:46:42 5:03:026:56:33 2 0:16:06 6:56:335:19:27 8 1:00:13 5:17:025:38:52 10 1:07:16 5:41:534:37:31 585 0:44:56 4:32:274:08:15 909 0:44:29 4:02:424:19:43 1494 0:46:53 4:16:29
sexFMTotalFMTotalFMTotalFMTotalFMTotalFMTotalFMTotalFMTotalFMTotal
age group24 or less
25-39
40-44
45-49
50-54
55-59
60-64
65+
Total
Mean NStd.
Deviation Median
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Age and Gender
65+60-6455-5950-5445-4944 or l ess
age group
6. 00
4. 00
2. 00
0. 00
Mean completion time in hours M
Fsex
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Boxplots of completing times by age and gender
65+60-6455-5950-5445-4944 or l ess
age group
8. 00
7. 00
6. 00
5. 00
4. 00
3. 00
2. 00
comp
leti
on t
ime
in h
ours
MF
sex
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Remarks
Average completion times for men and women of different ages are shown.
For every age group, the average time for men is less than the average time for women.
For men and women younger than 45, age does not seem to matter very much.
For both men and women the variability of completion times is very stable except the eldest age group.
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Detecting outliers
Cases with values between 1.5 and 3 box lengths from the upper or lower edge of the box are called outliers and are designated with an “o”.
Cases with values of more than 3 box lengths from the upper or lower edge of the box are called extreme values. They are designated with “*”.
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Extreme Values
1500 7:42:021499 7:07:571498 6:57:191497 6:55:081496 6:54:12
55 3:03:4562 3:04:5977 3:09:2279 3:09:5380 3:10:14
1489 6:36:271488 6:31:481483 6:24:161482 6:23:311481 6:22:40
1 De Haven(USA) 2:11:40
2 2:16:343 2:24:444 2:33:305 2:38:39
1234512345123451
2345
Highest
Lowest
Highest
Lowest
sexF
M
time HR:MIN:SECCase Number name Value
20
A stem-and-leaf plot is a display very much like a histogram, but it includes more information of the data.
In a stem-and-leaf plot, each row corresponds to a stem and each case is represented by a leaf.
Stem-and-leaf plots
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The following are price of 15 students eating lunch at a fast-food restaurant:
5.35, 4.75, 4.30, 5.47, 4.85, 6.62, 3.54, 4.87,
6.26, 5.48, 7.27, 8.45, 6.05, 4.76, 5.91
1 3 | 5 The first value of 5.35 is rounded to 5.4
5 4 | 83998 The second value of 4.75 is rounded to 4.8
4 5 | 4559 Their stems are 5 and 4, respectively
3 6 | 631 Their leafs are 4 and 8, respectively
1 7 | 3
1 8 | 5
Stem-and-leaf plots
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Stem-and-leaf plots
completion time in hours Stem-and-Leaf Plot foragecat6= 45-49
Frequency Stem & Leaf
2.00 2 . 99 13.00 3 . 0022223344444 40.00 3 .
5555566777777788888888899999999999999999 35.00 4 . 00000001111111122222233333333334444 21.00 4 . 555666666777778888899 12.00 5 . 000111111234 9.00 5 . 667778889 4.00 6 . 0011 4.00 Extremes (>=6.2)
Stem width: 1.00 Each leaf: 1 case(s)