Exploratory Data Analysis

711 Pages·1977·18.22 MB·English

by  John W. Tukey

62 exhibit 2/3: Easy re-expression
exhibit 2 of chapter 3: reference table
Break table for two-decimal logs
A) MAIN BREAK TABLE
Break log Break log Break log Break log Break log
9886 1567 2483 3936 6237
1012 .00 1603 .20 2541 .40 4027 .60 6383 .80
1035 .01 1641 .21 2600 .41 4121 .61 6531 .81
1059 .02 1679 .22 2661 .42 4217 .62 6683 .82
1084 .03 1718 .23 2723 .43 4315 .63 6839 .83
1109 .04 1758 .24 2786 .44 4416 .64 6998 .84
1135 05 1799 25 2851 .45 4519 .65 7161 .85
1161 .06 1841 .26 2917 .46 4624 .66 7328 .86
1189 .07 1884 .27 2985 .47 4732 .67 7499 .87
1216 .08 1928 .28 3055 .48 4842 .68 7674 .88
1245 .09 1972 .29 3126 .49 4955 .69 7852 .89
1274 .10 2018 .30 3199 .50 5070 .70 8035 .90
1303 .11 2065 .31 3273 .51 5188 .71 8222 .91
1334 .12 2113 .32 3350 .52 5309 .72 8414 .92
1365 .13 2163 .33 3428 .53 5433 .73 8610 .93
1396 .14 2213 .34 3508 .54 5559 .74 8810 .94
1429 .15 2265 .35 3589 .55 5689 .75 9016 .95
1462 .16 2317 .36 3673 .56 5821 .76 9226 .96
1496 .17 2371 .37 3758 .57 5957 .77 9441 .97
1531 .18 2427 .38 3846 .58 6095 .78 9661 .98
1567 .19 2483 .39 3936 .59 6237 .79 9886 .99
When in doubt use an even answer; thus, 1462 gives .16 and 1496 gives .18.
B) SETTING DECIMAL POINTS
1 1
+0 -1
10 0.1
+1 -2
100 0.01
+2 -3
1000 0.001
+3 -4
10,000 0.0001
+4 -5
100,000 0.00001
+5 -6
1,000,000 0.000001
C) EXAMPLES
Number B . A log number
log 137.2 2+.14= 2.14
log 0.03694 -2 + .57 = -1.43
log 0.896 -1 + .95 = -0.05
log 174,321 +5 + .24 = 5.24
See text, page 63, for example of use.
3D: quick roots and quick reciprocals/exhibit 7 71
exhibit 7 of chapter 3: reference table
Break table for (square) roots
A) EXAMPLES
Start by dividing number into periods of two digits, decimal point falrrngb!3tween
periods. Thus 124.2 is 1 24 2, but 1242 is 12 42. Like wise 0.00654 is 0065 4 or
654.
Number Periods from B from C Number
124.2 1 24 2 abo 112 11.2
1242 12 42 abo 35 35 .
. 00654 00654 .Ox 80 .080
B) BREAK TABLES to SET DECIMAL POINT --enter between bold figures,
leave with light figures
1
a
.x .01
abo
.Ox .00 01
abc.
.00x . 00 00 01
abcd.
.000x . 00 00 00 01
C) MAIN BREAK TABLE--in and out as in panel B
I Break I I Root I I Break I I Root I I Break I I Root I I Break I I Root I I Breakl I Root I
98 01 24964 5 66 1560 3540
100 160 240 40 60
1 02 01 26244 5 86 1640 3740
102 164 246 41 62
1 06 09 2 75 56 6 20 1722 39 69
104 168 252 42 64
1 10 25 28900 650 1806 42 25
106 172 258 43 66
1 1449 3 02 76 6 81 1892 44 69
108 176 264 44 68
1 18 81 3 1684 7 13 19 80 47 61
110 180 270 45 70
1 23 21 331 24 7 45 20 70 50 41
112 184 276 46 72
1 27 69 3 45 96 7 78 21 62 53 29
114 188 282 47 74
1 32 25 3 61 00 8 12 22 56 56 25
116 192 288 48 76
1 36 89 3 76 36 8 47 23 52 59 79
118 196 294 49 78
1 41 61 39204 882 24 50 62 41
120 200 30 50 80
1 48 84 40804 9 30 25 50 65 61
124 204 31 51 82
1 58 76 42436 9 92 26 52 68 89
128 208 32 52 84
1 69 00 4 41 00 10 56 27 56 72 25
132 212 33 53 86
1 79 56 4 57 96 11 22 28 62 7569
136 216 34 54 88
1 90 44 47524 11 90 29 70 79 21
140 220 35 55 90
201 64 49284 12 60 30 80 82 81
144 224 36 56 92
2 13 16 5 10 76 13 32- 31 92 8649
148 228 37 57 94
2 25 00 5 29 00 1406 3306 90 25
152 232 38 58 96
2 37 16 5 47 56 1482 34 22 9409
156 236 39 59 98
24964 56644 1560 3540 98 01
~~ff '~'
See page 75, f~ample of use.
72 exhibit 8/3: Easy re-expression
exhibit 8 of chapter 3: reference table
Break table for (negative) reciprocals (using -1000/number)
A) BREAK TABLES for SETTING DECIMAL POINT
Break Start Start Break
1000 1000
.x a.
10,000 100
.Ox abo
100,000 10
.0 Ox abc .
1,000,000 1.
.0 00x abed .
10,000,000 O. 1
.0 000x abcde .
100,000,000 0.01
Examples
Number A B -1000/number
124.2 a . -80 -8.0
.0 4739 abcde. -212 -212**.
1242. .x -80 -.80
B) MAIN BREAK TABLE--digits of negative reciprocal
I Break I I Value I I Break I I Value I I Break I I Value I I Break I \Value\ \ Break \ \ Value \
990 1639 2469 4115 617
-100 -60 -40 -240 -160
1010 1681 2532 4202 633
-98 -59 -39 -236 -156
1030 1709 2597 4274 649
-96 -58 -38 -232 -152
1053 1739 2667 4347 666
-94 -57 -37 -228 -148
1075 1770 2740 4425 685
-92 -56 -36 -224 -144
1099 1802 2816 4504 704
-90 -55 -35 -220 -140
1124 1835 2899 4587 725
-88 -54 -34 -216 -136
1149 1869 2985 4672 746
-86 -53 -33 -212 -132
1176 1905 3077 4762 769
-84 -52 -32 -208 -128
1205 1942 3175 4854 793
-82 , -51 -31 -204 -124
1235 1980 3287 4950 820
-80 -50 -30 -200 -120
1266 2020 3367 505 840
-78 -49 -294 -196 -118
1299 2062 3448 515 855
-76 -48 -288 -192 -116
1333 2105 3509 526 870
-74 -47 -282 -188 -114
1370 2151 3584 538 885
-72 -46 -276 -184 -112
1408 2198 3663 549 901
-70 -45 -270 -180 -110
1449 2247 3745 562 917
-68 -44 -264 -176 -108
1493 2299 3831 575 935
-66 -43 -258 -172 -106
1538 2353 3922 588 952
-64 -42 -252 -168 -104
1587 2410 4016 602 9'11
-62 -41 -246 -164 -102
1639 2469 4115 617 990
See text, page 78, for example of use.
JOHN W. TUKEY
Princeton University and
Bell Telephone Laboratories
Exploratory
Data
Analysis
ADDISON-WESLEY PUBLISHING COMPANY
Rcadinf.{. Massachllsctts • Mcnlo Park. Cal(f'ornia
London· Amsterdam· DOll Mills. Ontario· S,vdncy
This book is in the
ADDISONcWESLEY SERIES IN
BEHAVIORAL SCIENCE: QUANTITATIVE METHODS
Consulting Editor:
FREDERICK MOSTELLER
The writing of this work. together with some of
the research reported herein. was supported in
part by the United States Government.
Copyright [) 1977 by Addison-Wesley Publishing Company.
Inc. Philippines copyright 1977 by Addison-Wesley Publish
ing Company. Inc.
All rights reserved. No part of this publication may be repro
duced. stored in a retrieval system. or transmitted. in any
form or by any means. electronic. mechanical. photocopying.
recording. or otherwise. without the prior written permission
of the publisher. Printed in the United States of America.
Published simultaneously in Canada. Library of Congress
Cptalog Card No. 76-50g0.
C;(SBN 0-201-07616-0
CDEFGHIJ-HA-798
Dedicated to the memory of
CHARLIE WINSOR, biometrician, and
EDGAR ANDERSON, botanist,
data analysts both,
from whom the author learned much that
could not have been learned elsewhere

v
Preface
This book is based on an important principle:
It is important to understand what you CAN DO before you learn to measure
how WELL you seem to have DONE it.
Learning first what you can do will help you to work more easily and effec
tively.
This book is about exploratory data analysis, about looking at data to see
what it seems to say. It concentrates on simple arithmetic and easy-to-draw
pictures. It regards whatever appearances we have recognized as partial de
scriptions, and tries to look beneath them for new insights. Its concern is with
appearance, not with confirmation.
Examples, NOT case histories
The book does not exist to make the case that exploratory data analysis is
useful. Rather it exists to expose its readers and users to a considerable variety
of techniques for looking more effectively at one's data. The examples are not
intended to be complete case histories. Rather they show isolated techniques in
action on real data. The emphasis is on general techniques, rather than specific
problems.
A basic problem about any body of data is to make it more easily and
effectively handleable by minds--our minds, her mind, his mind. To this
general end:
<> anything that makes a simpler description possible makes the description
more easily handleable.
<> anything that looks below the previously described surface makes the
description more effective.
So we shall always be glad (a) to simplify description and (b) to describe one
layer deeper.
In particular:
<> to be able to say that we looked one layer deeper, and found nothing, is a
definite step forward--though not as far as to be able to say that we looked
deeper and found thus-and-such.
<> to be able to say that "if we change our point of view in the following way
... things are simpler" is always a gain--though not quite as much as to be
able to say "if we don't bother to change our point of view (some other)
things are equally simple".
vi Preface
Thus, for example, we regard learning that log pressure is almost a straight
line in the negative reciprocal of absolute temperature as a real gain, as
compared to saying that pressure increases with temperature at an evergrowing
rate. Equally, we regard being able to say that a batch of values is roughly
symmetrically distributed on a log scale as much better than to say that the raw
values have a very skew distribution.
In rating ease of description, after almost any reasonable change of point
of view, as very important, we are essentially asserting a belief in quantitative
knowledge--a belief that most of the key questions in our world sooner or later
demand answers to "by how much?" rather than merely to "in which direc
tion?" .
Consistent with this view, we believe, is a clear demand that pictures
based on exploration of data should force their messages upon us. Pictures that
emphasize what we already know--" security blankets" to reassure us--are
frequently not worth the space they take. Pictures that have to be gone over
with a reading glass to see the main point are wasteful of time and inadequate of
effect. The greatest value of a picture is when it forces us to notice what we
never expected to see.
We shall not try to say why specific techniques are the ones to use. Besides
pressures of space and time, there are specific reasons for this. Many of the
techniques are less than ten years old in their present form--some will improve
noticeably. And where a technique is very good, it is not at all certain that we
yet know why it is.
We have tried to use consistent techniques wherever this seemed reason
able, and have not worried where it didn't. Apparent consistency speeds
learning and remembering, but ought not to be allowed to outweigh noticeable
differences in performance.
In summary, then, we:
<> leave most interpretations of results to those who are experts III the
subject-matter field involved.
<> present techniques, not case histories.
<> regard simple descriptions as good in themselves.
<> feel free to ask for changes in point of view in order to gain such simplicity.
<> demand impact from our pictures.
<> regard every description (always incomplete!) as something to be lifted off
and looked under (mainly by using residuals).
<> regard consistency from one technique to another as desirable, not essen
tial.
Confirmation
The principles and procedures of what we call confirmatory data analysis
are both widely used and one of the great intellectual products of our century.