Statistics

#1. Standard normal probabilities (cont.) statistics tables
z
P(Z < z)
you are here 4   659
TfohresPe(aZre<tzhe) probabilities
positive. where z is

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359

0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753

0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141

0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517

0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621

1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830

1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015

1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177

1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441

1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545

1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633

1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706

1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817

2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857

2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890

2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916

2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952

2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964

2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974

2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981

2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990

3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993

3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995

3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997

3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

t-distribution table

#2. t-distribution critical values

This table gives you the values of t where P(T > t) = p. T follows a t-distribution P(T > t)

with ν degrees of freedom. Look up the values of ν and p and look up t.

t Look up ν in the first column... ...look up p in the first row...

Tail probability p

ν .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005
636.6
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 31.60
12.92
2 .816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 8.610
6.869
3 .765 .978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 5.959
5.408
4 .741 .941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 5.041
4.781
5 .727 .920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 4.587
4.437
6 .718 .906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 4.318 off the value
4.221 the table.
7 .711 .896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 4.140 ..o.tfhtenfrreoamd
4.073
8 .706 .889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 4.015
3.965
9 .703 .883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 3.922
3.883
10 .700 .879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 3.850
3.819
11 .697 .876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 3.792
3.768
12 .695 .873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 3.745
3.725
13 .694 .870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 3.707
3.690
14 .692 .868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 3.674
3.659
15 .691 .866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 3.646
3.551
16 .690 .865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 3.496
3.460
17 .689 .863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.416
3.390
18 .688 .862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.611 3.300
3.291
19 .688 .861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 99.9%

20 .687 .860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552

21 .686 .859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527

22 .686 .858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505

23 .685 .858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485

24 .685 .857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467

25 .684 .856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450

26 .684 .856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435

27 .684 .855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421

28 .683 .855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408

29 .683 .854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396

30 .683 .854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385

40 .681 .851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307

50 .679 .849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261

60 .679 .848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232

80 .678 .846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195

100 .677 .845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174

1000 .675 .842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098

∞ .674 .841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091

50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8%

Confidence level C

660   Appendix ii

statistics tables

#3. X 2 critical values

This table gives you the value of x where P(X ≥ x) = α. X α
has a χ2 distribution with ν degrees of freedom. Look up
the values of ν and α, and read off x. x

Look ucpolutmhne..v.alue of ν in the ...look up the value of α in ...then read off
first the first row... the value of x
from the table.
Tail probability α .0025 .001
ν .25 .20 .15 .10 .05 .025 .02 .01 .005 9.14 10.83
11.98 13.82
1 1.32 1.64 2.07 2.71 3.84 5.02 5.41 6.63 7.88 14.32 16.27
16.42 18.47
2 2.77 3.22 3.79 4.61 5.99 7.38 7.82 9.21 10.60 18.39 20.51
20.25 22.46
3 4.11 4.64 5.32 6.25 7.81 9.35 9.84 11.34 12.84 22.04 24.32
23.77 26.12
4 5.39 5.99 6.74 7.78 9.49 11.14 11.67 13.28 14.86 25.46 27.88
27.11 29.59
5 6.63 7.29 8.12 9.24 11.07 12.83 13.39 15.09 16.75 28.73 31.26
30.32 32.91
6 7.84 8.56 9.45 10.64 12.59 14.45 15.03 16.81 18.55 31.88 34.53
33.43 36.12
7 9.04 9.80 10.75 12.02 14.07 16.01 16.62 18.48 20.28 34.95 37.70
36.46 39.25
8 10.22 11.03 12.03 13.36 15.51 17.53 18.17 20.09 21.95 37.95 40.79
39.42 42.31
9 11.39 12.24 13.29 14.68 16.92 19.02 19.68 21.67 23.59 40.88 43.82
42.34 45.31
10 12.55 13.44 14.53 15.99 18.31 20.48 21.16 23.21 25.19 43.78 46.80
45.20 48.27
11 13.70 14.63 15.77 17.28 19.68 21.92 22.62 24.72 26.76 46.62 49.73
48.03 51.18
12 14.85 15.81 16.99 18.55 21.03 23.34 24.05 26.22 28.30 49.44 52.62
50.83 54.05
13 15.98 16.98 18.20 19.81 22.36 24.74 25.47 27.69 29.82 52.22 55.48
53.59 56.89
14 17.12 18.15 19.41 21.06 23.68 26.12 26.87 29.14 31.32 54.97 58.30
56.33 59.70
15 18.25 19.31 20.60 22.31 25.00 27.49 28.26 30.58 32.80 69.70 73.40
82.66 86.66
16 19.37 20.47 21.79 23.54 26.30 28.85 29.63 32.00 34.27 95.34 99.61
120.1 124.8
17 20.49 21.61 22.98 24.77 27.59 30.19 31.00 33.41 35.72 144.3 149.4

18 21.60 22.76 24.16 25.99 28.87 31.53 32.35 34.81 37.16

19 22.72 23.90 25.33 27.20 30.14 32.85 33.69 36.19 38.58

20 23.83 25.04 26.50 28.41 31.41 34.17 35.02 37.57 40.00

21 24.93 26.17 27.66 29.62 32.67 35.48 36.34 38.93 41.40

22 26.04 27.30 28.82 30.81 33.92 36.78 37.66 40.29 42.80

23 27.14 28.43 29.98 32.01 35.17 38.08 38.97 41.64 44.18

24 28.24 29.55 31.13 33.20 36.42 39.36 40.27 42.98 45.56

25 29.34 30.68 32.28 34.38 37.65 40.65 41.57 44.31 46.93

26 30.43 31.79 33.43 35.56 38.89 41.92 42.86 45.64 48.29

27 31.53 32.91 34.57 36.74 40.11 43.19 44.14 46.96 49.64

28 32.62 34.03 35.71 37.92 41.34 44.46 45.42 48.28 50.99

29 33.71 35.14 36.85 39.09 42.56 45.72 46.69 49.59 52.34

30 34.80 36.25 37.99 40.26 43.77 46.98 47.96 50.89 53.67

40 45.62 47.27 49.24 51.81 55.76 59.34 60.44 63.69 66.77

50 56.33 58.16 60.35 63.17 67.50 71.42 72.61 76.15 79.49

60 66.98 68.97 71.34 74.40 79.08 83.30 84.58 88.38 91.95

80 88.13 90.41 93.11 96.58 101.9 106.6 108.1 112.3 116.3

100 109.1 111.7 114.7 118.5 124.3 129.6 131.1 135.8 140.2

you are here 4   661



Index

Symbols χ2 (chi square) probability tables 575
χ2 (chi square) test 571
| symbol (see conditional probabilities) x (x bar) 445–447, 472–476

intersection distribution of 476–486
finding 159
P(A B) versus P(A | B) 165 A
P(Black Even) 167
P(Even) 167 accurate linear correlation 630

1/p, expectation 281 alternate hypothesis 529–530, 543

when large 407 average 46–82
when small 407 mean (see mean)
median (see median)
distribution (see Poisson distribution) mode (see mode)
types of 71
μ (mu) 50, 445
confidence intervals 498 average distance 105
interquartile range 105
ν (nu) 573
degrees of freedom 574 B

(sigma) 49 bar charts 10–20, 23
mean 49 frequency scales 13
percentage scales 12
(sigma) 107 scales 23
segmented bar chart 14
χ2 (chi square) 576 split-category bar chart 14

χ2 (chi square) distribution 567–604 Bayes’ Theorem 173, 178–179
cheat sheet 584
contingency table 587 bias 423–426, 434, 438
defined 572 in sampling 424–426, 438
degrees of freedom 574, 576, 595 sources 425
calculating 591
generalizing 596–597 bimodal 73
expected frequencies 587–588
goodness of fit 573, 579, 584 binomial distribution 289, 324, 384, 392–393, 544
independence 573, 586 approximating 389, 398, 407
main uses 573 approximating with normal distribution 386
significance 575 approximating with Poisson distribution 316–317
ν (nu) 573 central limit theorem 482

χ2 (chi square) hypothesis testing steps 576 this is the index   663

the index kth percentile 102
linear regression 640
binomial distribution (continued) linear transforms 220, 224, 233
discrete 395 line of best fit 640
expectation and variance 298, 301 negative linear correlation 640
finding mean and variance 389 normal distribution 359
guide 302
versus normal distribution 393, 395 approximating 396
normal probabilities 359
Binomial Distribution Up Close 297 one-tailed tests 539
p-value 539
binomial probabilities 384 percentiles 102
point estimator 447
bivariate data 608, 616, 640 Poisson distribution 324, 412
visualizing 609 population 438
positive linear correlation 640
blinding 646 probability distributions 220, 224
quartiles 97
box and whisker diagrams 100–102 range 97
sample 438
box plot 100 sampling distribution of means 485
sampling distribution of proportions 466
Bullet Points scatter diagrams 640
bias 438 significance level 539
binomial distribution 324 simple random sampling 438
bivariate data 640 standard deviation 122, 220
box and whisker diagram 102
cluster sampling 438 224
continuity correction 396 standard error of proportion 466
continuous data 337 standard error of the mean 485
continuous probability distributions 337 standard scores 122
correlation coefficient 640 stratified sampling 438
critical region 539 sum of squared errors 640
cumulative frequency 42 systematic sampling 438
discrete data 337 test statistic 539
expectation and variance of X 485 two-tailed tests 539
expectation of random variable X 224 univariate data 640
expectations 220, 233 upper and lower bounds 97
frequency density 30 variance of random variable X 224
geometric distribution 324 variances 122, 220, 233
histograms 30 z-scores 122
hypothesis tests 539 χ2 distribution 598
Type I error 566
Type II error 566 goodness of fit test 598
independent observations 378 test for independence 598
independent observations of X 233
independent random variables 233
interpercentile range 102
interquartile range 97

664   Index

C the index

categorical data 18, 73 defined 572
mean 62 degrees of freedom 574, 576, 595
median 62
calculating 591
categories versus numbers 18–23 generalizing 596–597
expected frequencies 587–588
causation versus correlation 614 goodness of fit 573, 579, 584
independence 573, 586
census 418 main uses 573
significance 575
central limit theorem 481–482, 485 ν (nu) 573
binomial distribution 482 chi square (χ2) hypothesis testing steps 576
Poisson distribution 482
chi square (χ2) probability tables 575
central tendency 45–82
chi square (χ2) test 571
charts and graphs 4
bar charts 10–20, 23 clustered sampling 434
bar chart scales 23
choosing right one 39–40 cluster sampling 433–434, 436, 438
comparing 6
cumulative frequency 35, 42 coefficient of determination 649
failure 9
frequency 8–9, 23 combinations (see permutations and combinations)
frequency scales 13
histograms (see histograms) combined weight
horizontal bar charts 11, 23 continuous 365
line charts 41, 42 distributed 367
multiple sets of data 14, 23 distributed normally 365
numerical data 23
percentage sales 12 complementary event 136
pie charts 8–9, 9, 23
proportions 9 completely randomized design (experiments) 647
scales 12
segmented bar chart 14 conditional probabilities 157–160
software 6 Bayes’ Theorem 173
split-category bar chart 14 P(A B) versus P(A | B) 165
vertical bar charts 10–11, 23 P(Black | Even) 170
probability tree 158–161
Chebyshev’s inequality 645
confidence intervals 487–520, 539
chi square (χ2) 576 cheat sheet 504
confidence level changes 518
chi square (χ2) distribution 567–604 four steps for finding 491–502
cheat sheet 584 Step 1: Choose your population statistic 492, 508
contingency table 587 Step 2: Find its sampling distribution 492, 509
Step 3: Decide on the level of confidence 494, 512
Step 4: Find the confidence limits 496–501, 513
introducing 490
point estimators 493
selecting appropriate confidence level 495
size of sample changes 518

you are here 4   665

the index line of best fit 618, 624, 640
finding equation 622
confidence intervals (continued) finding slope 623–624
slope of regression line 651 sum of squared errors 620–621
summary 503
t-distributions 509–515 negative linear correlation 613, 631, 640
probability tables 513 no correlation 613, 631
shortcuts 515 no linear correlation 630
small sample 510 outliers 634
standard score 511 perfect negative linear correlation 631
versus confidence level 507 perfect positive linear correlation 631
positive linear correlation 613, 631, 640
confidence level versus confidence interval 507 regression line 626
response variable 608
confidence limits 496, 502, 513 scatter diagrams 609, 612, 616, 618, 640
scatter plots 609
confounding 646 sum of squared errors 640
univariate data 608, 640
contingency table 587
correlation coefficient 631–634, 640
continuity correction 395–398, 412 formula 632
least square regression 648
Continuity Corrections Up Close 397
critical region 531–534, 539, 548
continuous data 327, 337, 365
frequency 328 Critical Regions Up Close 534
probability distribution 329–333
range of values 333 critical value 532
versus discrete data 366
cumulative frequency 34–38, 42
continuous probabilities 333 graph 35

continuous probability distributions 337 D
E(X) and Var(X) 654–655
data
continuous random variables 331 categorical and numerical data 18
categorical data 18
continuous scale versus discrete probability grouped 19
distribution 395 multiple sets of data 14
numerical data 18
control group 646 qualitative data 18

controls 646 deciles 98

correlation and regression 605–642 degrees of freedom 574, 576, 595
accurate linear correlation 630 calculating 591
bivariate data 608, 616, 640 generalizing 596–597
visualizing 609 number of 510
correlation coefficient 630–634, 640
correlation versus causation 614 dependent events 181, 189–190
dependent variable 608
explanatory variable 608
independent variable 608
least squares regression 626
linear regression 626, 640

666   Index

the index

dependent variables (experiments) 608, 646 distribution of Ps 464–466
expectation of Ps 462
discrete data 329, 337, 370 formulas 451
versus continuous data 326–327, 366 point estimators 443–447, 452

discrete probability distributions 197–240 for population variance 457
expectation 204–208 sampling distributions 485
linear transforms 233 population mean 443, 446
expectations 219 population parameters 444
independent observations 224, 225–226 population proportion 454–457
linear relationship between E(X) and E(Y) 217–218 population variance 448–450
linear transforms 219, 225–226 probabilities for a sample 459
expectation and variance 233 proportions, sampling distribution of 460
linear transforms versus playing multiple games 221 sample mean 445, 446
observation 222–224 sample variance 449, 452
observation shortcuts 223 sampling distribution 466
Pool Puzzle 215–216 continuity correction 469
random variables of proportions 460
adding 230 sampling distribution of means 471–479
independent 233 distribution of x 480
subtracting 231 expectation for X 474–475
shortcut or formula 236 variance of X 476
variance 205–208, 219 standard error 485
linear transforms 233 of mean 479
versus continuous scale 395 of proportion 466
variance of Ps 463
discrete random variables 202 x bar 445
μ 445
distribution
anatomy 645 events 132
mean 56
of X + Y 370 complementary 136
dependent 181
dotplots 644 exclusive 147–154

double blinding 646 versus exhaustive 150
independent 182–184
drawing lots 431, 434
versus dependent 189–190
E intersecting 147–154
mutually exclusive 147, 150
E(X) and Var(X) for continuous probability distributions
654–655 exclusive events 147–154, 150

empirical rule for normal distribution 645 exhaustive 149

estimating populations and samples 441–486 exhaustive events 150
central limit theorem 481–482, 485
binomial distribution 482 expectations 204–208, 219, 220, 367
Poisson distribution 482
1/p 281
binomial distribution 298

you are here 4   667

the index

expectations (continued) G
geometric distribution 280–281
independent observations 378 Gaussian distribution 352
linear transforms 233
Poisson distribution 308 geometric distribution 277–287, 297, 301, 324
two games 222–224 guide 284
inequalities 279
experimental units 646 pattern of expectations 280–281
variance 281–284
experiments 646
designing 647 Geometric Distribution Up Close 278

explanatory variable 608 goodness of fit 573
test 579
F
graphs (see charts and graphs)
factorials 246, 248
grouped data 19
Fireside Chats, Dependent and Independent discuss their
differences 186–187 H

Five Minute Mystery height probabilities 338–341
Case of the Broken Cookies 315
Solved 318 histograms 19–28
Case of the High Sunscreen Sales 611 frequency 24–30, 25
Solved 615 intervals 20
Case of the Lost Coffee Sales 421 making 20
Solved 429 making area proportional to frequency 26–28
Case of the Missing Parameters 357 mean 56
Solved 358 unequal intervals 24–30
Case of the Moving Expectation 211 when not to use 33
Solved 220
The Case of the Ambiguous Average 51 horizontal bar charts 11, 23
Solved 81
The Case of the Two Classes 185 horse racing 243–246
Solved 188
hypothesis tests 521–566
formulas for arrangements 248 alternate hypothesis 529–530, 543
critical region 531–534, 539, 548
frequencies 8, 23, 67–68, 73 critical value 532
comparing 14 null hypothesis 528, 543
continuous data 328 one-tailed tests 534, 539
cumulative frequency 34–38, 42 p-value 539
highest frequency group of values 52 power of a hypothesis test 561
histograms 24–30 process 526–539
percentages with no frequencies 12 overview 527
Step 1: Decide on the hypothesis 528–529, 543
frequency density 27–32, 68 Step 2: Choose the test statistic 531, 544
Step 3: Determine the critical region 532, 548
Frequency Density Up Close 29

frequency scales 13

668   Index

Step 4: Find the p-value 535–536 the index
Step 5: Is the sample result in the critical region?
L
537
Step 6: Make your decision 537 Law of Total Probability 172, 178
significance level 533, 538, 539
statistically significant 551 least squares regression 626, 648
test statistic 531, 539, 544, 547
two-tailed tests 534, 539 Least Squares Regression Up Close 626
Type I error 555–560, 566
Type II error 555–560, 566 leaves 644

I left-skewed data 62, 64

incorrect sampling unit 425 letters, using to represent numbers 48–49

independence 573 linear correlations 613, 630–631

independent events 182–183, 189–190 Linear Correlations Up Close 613
versus mutually exclusive 183
linear regression 626, 640, 650
independent observations 224–226, 377, 472
expectation 378 linear relationship between E(X) and E(Y) 217–218
of X 233
variance 378 linear transforms 219, 220, 224–226
versus linear transforms 376–378 distribution 376
expectation and variance 233
independent random variables 230–233, 368 versus independent observations 376–378
versus playing multiple games 221
indpendent variables 608, 646
line charts 41, 42
information
versus data 5 Line Charts Up Close 41
visualizing (see visualizing information)
line of best fit 618, 622, 640
interpercentile range 98, 102 finding equation 622
finding slope 623–624
interquartile range 92–93, 97 minimizing errors 620–621
average distance 105 non-linear 650
versus the median 97 sum of squared errors 620–621

intersecting events 147–154 lower bounds 86, 97
basketball scores 88
intersection 149–154
lower quartile 92
K finding 94

kth percentile 99, 102 M

matched pairs design (experiments) 647

mean 47–60
basketball scores 88
binomial distribution 389
calculating 50
calculating when to use 78

you are here 4   669

the index N

mean (continued) n! 248
categorical data 62
distributions 56 negative linear correlation 613, 631, 640
frequencies 52
frequency density 68 no correlation 613, 631
histograms 56
of two middle numbers 61 No Dumb Questions
outliers 57–59
positive and negative distances 105 adding probabilities 143
problems with 65–72 alternate hypothesis 530
skewed data 62, 64 approximating binomial distribution 398
standard deviations from 121 arranging objects in circle 248
using letters to represent numbers 48–49 average distance
versus median 62
X + Y 368 interquartile range 105
μ (mu) 50 Bayes’ Theorem 179
(sigma) 49 bias 426, 434
binomial distribution 301, 412
measuring probability 132 bivariate data 616
median 61–70 box and whisker diagram 101
breaking data into more than four pieces 97
calculating when to use 78 central limit theorem 485
categorical data 62 charts 5
frequency density 68 clustered sampling 434
in three steps 62 confidence intervals 491, 518, 539
middle quartile 92 confidence interval versus confidence level 507
problems with 65–72 continuity corrections 398, 412
skewed data 64 continuous data 370
versus mean 62 continuous distributions 352
versus the interquartile range 97 correlation coefficient 634
middle quartile 92 cumulative frequency 36
modal class 73 degrees of freedom 576, 595
mode 73–80 discrete data 370
calculating when to use 78 discrete random variable 203
categorical data 73 distribution of X + Y 370
three steps for finding 74 drawing lots 434
mu (see μ (mu)) E(X1 + X2) and E(2X) 224
multiple sets of data 14, 23 expectation 208, 219
mutually exclusive events 147, 150 factorials 248
frequency density 30
670   Index Gaussian distribution 352
geometric distribution 277, 284, 301
histograms 23, 30
how data is spread out 97

hypothesis tests 530, 552 the index

independent events 184 probability trees 165, 179
proportion versus probability 456
independent observations 378 questionnaires 426
random variables 233
independent versus mutually exclusive 184 right- and left-skewed data 62
roulette wheel 184
information versus data 5 sample mean 446
sample variance 452
interquartile range 97 sampling bias 434
sampling distribution 466
limit on intersecting events 154 sampling frame 426
scatter diagrams 616
linear transforms 219, 378 set theory 139
shortcuts 370
line charts 42 significance level 539
significance tests 552
line of best fit 624 slot machines 208
standard deviation 113, 122, 208
mean or median with categorical data 62 standard error 485

mean with skewed data 62 of proportion 466
standard scores 122, 347, 352
median 352
versus mean 62 outliers 122
versus the interquartile range 97 statistical sampling

n! 248 bias 426
clustered sampling 434
normal distribution drawing lots 434
accuracy of 398 increasing sample size 434
approximating binomial or Poisson distribution simple random sampling 434
412 stratified sampling 434
stratified sampling 434
normal probability tables 352 systematic sampling 434
t-distributions 518
null hypothesis 530 target population 426
Type I error 560
outliers 634 Type II error 560
variance 122, 208
P(Black | Even) 179 variance equations 113
variances 219
permutations and combinations 263 Venn diagrams 139, 165, 184
arranging by type 257 χ2 (chi square) distribution 595
χ2 (chi square) tests 576
point estimators 446, 452
and sampling distributions 485 no linear correlation 630

Poisson distributions 311, 314, 412 non-linear relationships 650
approximating binomial distribution 317, 398
you are here 4   671
population mean 446

positive and negative distances 105

probabilities written as fractions, decimals, or
percentages 139

probability 139
best method 143

probability density function 334

probability distributions 203
letters p and q 284
quiz show 290

probability for standardized range 347

probability of range 352

probability tables 352, 370

the index standardizing normal variables 344
tables 349–352, 352, 658–659
normal approximation 394 nu (see ν (nu))

normal distribution 325–360, 361–414 null hypothesis 528, 530, 543
accuracy of 398
approximating continuity correction 396 numbers, using letters to represent 48–49
approximating binomial distribution 386
approximating binomial or Poisson distribution 412 numerical data 18, 23
approximating binomial probabilities 397
binomial distribution 384, 389, 392–393 O
approximating 398, 407
continuous 395 observations 222–224
continuous data 337, 365 independent 224
continuous distributions 352 shortcuts 223
continuous probability distributions 337
defined 339–340 one-tailed tests 534, 539
discrete data 337
discrete data versus continuous data 326–327 outliers 57–59, 89–91, 93, 634
empirical rule 645 interquartile range 93
finding ≤ probabilities 397 standard scores 122
finding ≥ probabilities 397
finding between probabilities 397 P
frequency and continuous data 328
Gaussian distribution 352 p-value 535–536, 539
height probabilities 338–341
in place of binomial distribution 389 percentage sales 12
median 352
normal probability tables 352 percentages with no frequencies 12
Poisson distribution 386, 406
Pool Puzzle 399–400 percentiles 98–99, 102
probability = area 331 kth percentile 99, 102
probability density function 330–337, 337
probability for standardized range 347 perfect negative linear correlation 631
probability of range 352
probability tables 349–352 perfect positive linear correlation 631
standard score 345–347, 352
table 411 permutations and combinations 241–268
transforming 345 arrangements 246
versus binomial distribution 393, 395 arranging by type 252–257
versus t-distributions 515 arranging duplicates 254
arranging objects in circle 247–248
Normal Distribution Exposed 404 combinations 260–263, 293
examining combinations 260–263
normal probabilities 359 examining permutations 258–259
calculating 341–352 factorial 246
determining distribution 343 formulas for arrangements 248
permutations versus combinations 261
672   Index three-horse race 243–246

pie charts 8–9, 9, 23

the index

placebo 646 how probability relates to roulette 132
intersection 149–154, 153
point estimators 443–447, 452, 493, 519 Law of Total Probability 172, 178
and sampling distributions 485 measuring 132
for population variance 457 of getting a black or even 145–146
problem with 489 proportion 455
range of values 329
Poisson distribution 306–319, 324, 386, 406, 407, 412 union 149–154, 153
approximating binomial distribution 398 Venn diagram 136, 154
approximating the binomial distribution 316–317 written as fractions, decimals, or percentages 139
central limit theorem 482
expectation and variance 308 probability density 334
guide 319
when is large 407 function 330–337
when is small 407 never equaling 0 341
X + Y 312–313
probability distributions 220, 224, 363
Poisson Distribution Up Close 307
4X 376
Poisson variables, combining 313 binomial (see binomial distribution)
continuous data 329–333
Pool Puzzle geometric (see geometric distribution)
binomial distribution 299–300 large number of possibilities 273, 277
confidence intervals 499–500 letters p and q 284
continuity correction 399–400 new price and payouts 212–214
discrete probability distributions 215–216 normal (see normal distribution)
of X + Y 372
population 418, 438 patterns 274–277
chart 419 Poisson (see Poisson distribution)
mean 446 random variable X 210
proportion 454–455, 457 standard deviation 207
variance 448–450
versus samples 418 Probability Distributions Up Close 202
(see also estimating populations and samples)
probability tables 349–352, 352, 370, 513, 657–661
positive and negative distances 105
standard normal probabilities 658–659
positive linear correlation 613, 631, 640 t-distribution critical values 660
χ2 (chi square) critical values 661
possibility space 135
Probability Tables Up Close 351
precision, problem with 489
probability trees 158–161, 165, 180
probability 127–196
= area 331 hints 161
adding 142, 143
Bayes’ Theorem 173, 178 proportions 9
best method 143
conditional 157–160 probability 455
probability tree 158–161
events (see events) sampling distribution of 460
for a sample 459
distribution of PP46ss 3446642–466
expectation of
variance of Ps

standard error of 463

you are here 4   673

the index P(Even) 169
possibility space 135
Q probabilities 135
probability of ball landing on 7 133–134
qualitative data 18 sample space 135

quartiles 92 S
interquartile range 92–93
lower 92, 94 samples 418, 438
middle 92 biased 424–426
upper 92, 94 designing 422–423
mean 445, 446
questionnaires, bias 426 space 135
survey 418, 438
R unbiased 424–426
unreliability 420
randomization 646 variance 449, 452
(see also estimating populations and samples)
randomized block design (experiments) 647
sampling (see statistical sampling)
random number generators 431
sampling distribution 466
random variables 202 difference between two means 652
adding 230 difference between two proportions 653
continuous 331
independent 233 sampling distribution of means 471–479
subtracting 231 distribution of x 480
variance of X 476
range 86–103, 97, 329, 333
basketball scores 88 sampling distribution of proportion 460
calculating 86 distribution of Ps 464–466
lower bound 86 expectation of Ps 462
outliers 89–91 variance of Ps 463
problems with 90
quartiles 92 Sampling Distribution of Proportions Up Close 469
upper bound 86
Sampling Distribution of the Means Up Close 479
regression (see correlation and regression)
sampling frame 423–428, 438
replication 646 bias 425

response variable 608 sampling units 422, 428
bias 425
right-skewed data 62, 64
sampling without replacement 430
roulette 129–196
black and even pockets 156 sampling with replacement 430
board 129–130
how probability relates to 132 scales 12
independent events 184
measuring probability 132
P(Black | Even) 167–171

674   Index

scatter diagrams 609, 612, 616, 618, 640 the index
line of best fit 618
finding equation 622 standardizing normal variables 344
finding slope 623–624
sum of squared errors 620–621 standard normal probabilities 658–659

scatter plots (see scatter diagrams) standard scores 118–122, 345–347, 352
calculating 119
segmented bar chart 14 interpreting 120

set theory 139 Standard Scores Up Close 121

shortcuts 370 statistical sampling 415–440
bias in sampling 423–426, 434, 438
sigma ( ) 49 sources 425
choosing samples 430
sigma ( ) 107 cluster sampling 433, 433–434, 436, 438
defined 418
significance level 533, 538, 539 designing samples 422
drawing lots 431, 434
significance tests 552 how it works 419
incorrect sampling unit 425
simple random sampling 430–431, 434, 436, 438 increasing sample size 434
drawing lots 431 population 418, 438
random number generators 431 population chart 419
populations versus samples 418
skewed data 58–59, 64 random number generators 431
mean 62 representative sample 420
samples 438
Skewed Data Up Close 59 unreliability 420
sample survey 418, 438
skewed to the left 59 sampling bias 434
sampling chart 419
skewed to the right 58–59 sampling frame 423–428, 438
sampling units 422, 428
slope of regression line sampling without replacement 430
confidence intervals 651 sampling with replacement 430
simple random sampling 430–438
slot machines 198 choosing 431
discrete random variables 202 strata 432
low versus high variance 208 stratified sampling 432, 434, 436, 438
probability distributions 201 systematic sampling 433–434, 438
variance 207 target population 422, 428, 438
unreliability 420
split-category bar chart 14
statistics
standard deviation 107–110, 113–117, 207, 220 defined 2
from the mean 121 why learn 3
variance equations 113
(sigma) 107, 224 you are here 4   675

Standard Deviation Exposed 108

standard error 485
of mean 479
of proportion 463, 466

the index upper bounds 86, 97
basketball scores 88
statistics tables 657–661
standard normal probabilities 658–659 upper quartile 92
t-distribution critical values 660 finding 94
χ2 (chi square) critical values 661
V
stemplots 644
stems 644 variability 104–124
strata 432 average distance 105
stratified sampling 432–438 positive and negative distances 105
stratified sampling 436 variance (see variance)
summation symbol ( ) 49
sum of squared errors 640 variables 368
symmetric data 59 probabilities involving the difference between two 369
systematic sampling 433–434, 438
variance 106–113, 122, 205–208, 219, 220, 367
T binomial distribution 298, 389
calculating 111–113
t-distributions 509–515 quicker way 113
probability tables 513 geometric distribution 281–284
shortcuts 515 independent observations 378
small sample 510 linear transforms 233
standard score 511 of X 476
table 660 Poisson distribution 308
versus normal distributions 515 slot machines 207
standard deviation 107–110
target population 422, 426, 428, 438 (sigma) 107
test statistic 531, 539, 544, 547 two games 222–224
three-horse race 243–246 X + Y 368
two-tailed tests 534, 539
Type I error 555–560, 566 Variance Up Close 450
Type II error 555–560, 566
Venn diagrams 136, 139, 154, 165
U conditional probability 157
independent events 184
unbiased sample 424–425
uniform distribution 655 vertical bar charts 10–11, 23
union 149–154
univariate data 608, 640 visualizing information 1–44, 19–28
categorical and numerical data 18–23
676   Index cumulative frequency 34–38
histograms 19–28
statistics 2
(see also charts and graphs)

Vital Statistics the index

A or B 153 W
approximating binomial distribution 389
approximating Poisson distribution 407 Watch it!
arranging by type 254 criteria of np > 10 and nq > 10 389
Bayes’ Theorem 178 cumulative frequencies 35
combinations 263 exclusive versus exhaustive 150
conditions 165 how large n needs to be 465
cumulative frequency 34 independent random variables 230–232
event 132 independent versus mutually exclusive 183
formulas for arrangements 248 linear regression 626
frequency 8 percentages with no frequencies 12
independence 184 quartiles 92
independent observations 224 samples equation 451
interquartile range 93 subtracting random variables 231
Law of Total Probability 178 X1 + X2 and 2X 223
linear transforms 220
mean 54 Who Wants To Win A Swivel Chair 289, 381–386
mode 76 expectation and variance 304
outlier 58 generalizing probability for three questions 293
percentile 99 generalizing the probability 296
permutations 263 probability of getting exactly three questions right 304
probability 143 probability of getting exactly two questions right 304
quartiles 92 probability of getting no questions right 304
range 86 probability of getting two or three questions right 304
significance level 533 should you play or walk away 291
skewed data 58
standard score 346 width of data 88
uniform distribution 655
variance 106, 113 X

X + Y Distribution Up Close 368

X - Y Distribution Up Close 369

Z

z-scores 118–122
calculating 119
interpreting 120

you are here 4   677



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