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Page 1
Key Formulas
From Larson/Farber Elementary Statistics: Picturing the World, Second Edition
© 2002 Prentice Hall
CHAPTER 2
Class Width = 1 round up to next convenient number 2
Maximum data entry - Minimum data entry
of
Number of classes
Midpoint =
1Lower class limit2 + 1Upper class limit2
Chebychev’s Theorem The portion of any data set lying
within k standard deviations 1k 7 12 of the mean is at
1
least 1 - 2 .
k
Standard Score: z =
x - m
value - mean
=
s
standard deviation
2
Class frequency
Relative Frequency =
Sample size
Population Mean: m =
=
CHAPTER 3
f
n
Classical (or Theoretical) Probability:
gx
N
P1E2 =
gx
n
Sample Mean: x =
Weighted Mean: x =
Number of outcomes in E
Total number of outcomes in sample space
Empirical (or Statistical) Probability:
g1x # w2
P1E2 =
gw
Mean for Grouped Data: x =
g1x # f2
Frequency of event E
Total frequency
=
f
n
Probability of a Complement: P1E¿2 = 1 - P1E2
n
Range = 1Maximum entry2 - 1Minimum entry2
Probability of occurrence of both events A and B:
P1A and B2 = P1A2 # P1B ƒ A2
Population Variance: s2 =
P1A and B2 = P1A2 # P1B2 if A and B are independent
g1x - m22
N
Probability of occurrence of either A or B or both:
Population Standard Deviation:
s = 2s2 =
g1x - m22
C
P1A or B2 = P1A2 + P1B2 - P1A and B2
N
2
Sample Variance: s =
P1A or B2 = P1A2 + P1B2 if events are mutually
exclusive.
g1x - x22
n - 1
Sample Standard Deviation: s = 2s2 =
g1x - x22
C
n - 1
Permutations of n objects taken r at a time:
nPr
=
Sample Standard Deviation for Grouped Data:
s =
g1x - x22 f
C
n - 1
n!
, where r … n.
1n - r2!
Distinguishable Permutations: n1 alike, n 2 alike, Á ,
nk alike:
Empirical Rule (or 68-95-99.7 Rule) For data with a
(symmetric) bell-shaped distribution
n!
n1! # n2! # n2! Á nk!
1. About 68% of the data lies between m - s and
m + s.
where n1 + n2 + n3 + Á + nk = n.
2. About 95% of the data lies between m - 2s and
m + 2s.
Combination of n objects taken r at a time:
3. About 99.7% of the data lies between m - 3s and
m + 3s.
nCr
=
n!
1n - r2!r!
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Page 2
Key Formulas
From Larson/Farber Elementary Statistics: Picturing the World, Second Edition
© 2002 Prentice Hall
CHAPTER 4
CHAPTER 6
Mean of a Discrete Random Variable: m = gxP1x2
s2 = g1x - m22P1x2
c- Confidence Interval for m : x - E 6 m 6 x + E
s
where E = zc
if s is known and the population is
1n
s
normal or n Ú 30, and E = tc
if the population is
1n
Standard Deviation: s = 2s2
normal, s is unknown, and n 6 30.
Variance of a Discrete Random Variable:
Expected Value: E1x2 = m = gxP1x2
Binomial Probability of x successes in n trials:
P1x2 = nCxpxq n - x =
n!
pxqn - x
1n - x2!x!
Population Parameters of a Binomial Distribution:
Mean: m = np
Variance: s2 = npq
Standard Deviation: s = 1npq
zcs 2
b
E
Point Estimate for p, the population proportion of
x
n =
successes: p
n
Minimum Sample Size to Estimate m : n = a
c- Confidence Interval for Population Proportion p
n - E 6 p 6 p
n + E,
(when np Ú 5 and nq Ú 52: p
where
E = zc
Geometric Distribution:
n qn
p
.
B n
zc 2
b
E
The probability that the first success will occur on trial
number x is P1x2 = p1q2x - 1, where q = 1 - p.
n qn a
Minimum Sample Size to Estimate p : n = p
Poisson Distribution:
c- Confidence Interval for Population Variance s2 :
The probability of exactly x occurrences in an interval is
mxe -m
P1x2 =
, where e L 2.71828 .
x!
1n - 12s2
Standard Score, or z- Score:
x - m
value - mean
=
z =
s
standard deviation
Transforming a z- Score to an x- Value: x = m + zs
sx =
z =
s
1n
x - mx
x - m
=
sx
s> 1n
1n - 12s2
xL2
c- Confidence Interval for Population Standard
Deviation s:
CHAPTER 5
mx = m
xR2
6 s2 6
Central Limit Theorem
(Mean of the Sample Means)
Central Limit Theorem
(Standard Error)
Central Limit Theorem
C
1n - 12s2
xR2
6 s 6
C
1n - 12s2
xL2
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Page 3
Key Formulas
From Larson/Farber Elementary Statistics: Picturing the World, Second Edition
© 2002 Prentice Hall
CHAPTER 7
t =
x - m
z- Test for a Mean m : z =
, for s known with a
s> 1n
normal population, or for n Ú 30 .
t- Test for a Mean m : t =
x - m
s> 1n
, for s unknown, x
normally distributed, and n 6 30 .
1d.f. = n - 12
Note: n1p, n1q , n2p, and n2q must be at least 5.
z =
x =
n 22 - 1p1 - p22
n1 - p
1p
B
1
1
+
b
n1
n2
p qa
, where p =
1n - 12s2
1d.f. = n - 12
s2
Correlation Coefficient:
r =
ngxy - 1gx21gy2
2ngx2 - 1gx22 2n gy2 - 1gy22
CHAPTER 8
t- Test for the Correlation Coefficient:
Two-Sample z- Test for the Difference Between Means:
t =
(Independent samples; n1 and n2 Ú 30 or normally
distributed populations)
z =
1x1 - x22 - 1m1 - m 22
sx1 - x2
, where sx1 - x2
s21
s22
=
+
B n1
n2
Two-Sample t- Test for the Difference Between Means:
(Independent samples from normally distributed
populations, n1 or n2 6 30 )
t=
1x1 - x22 - 1m1 - m 22
C
1n1 -
(d.f. = n - 2 )
1 - r2
Bn - 2
where m =
ngxy - 1gx21gy2
ngx2 - 1gx22
b = y - mx =
and
gy
gx
- m
n
n
Coefficient of Determination:
If population variances are equal, d.f. = n1 + n2 - 2
12s21
r
Equation of a Regression Line: yn = mx + b
sx1 - x2
and sx1 - x2 =
x1 + x2
.
n1 + n2
CHAPTER 9
Chi-Square Test for a Variance or Standard Deviation:
2
n1gd22 - 1gd22
gd
, sd =
,
n
C
n1n - 12
Two-Sample z- Test for the Difference Between
Proportions:
n - p
p
2pq>n
sd> 1n
, where d =
and d.f. = n - 1.
z- Test for a Proportion p (when np Ú 5 and nq Ú 52 :
z =
d - md
+ 1n2 -
12s22
#
n1 + n2 - 2
1
1
+
.
B n1
n2
r2 =
g1yni - y22
explained variation
total variation
=
g1yi - y22
Standard Error of Estimate: se =
C
g1yi - yni22
n - 2
If population variances are not equal, d.f. is the smaller
s21
s22
+
.
of n1 - 1 or n2 - 1 and sx1 - x2 =
B n1
n2
c- Prediction Interval for y is yn - E 6 y 6 yn + E,
where
t- Test for the Difference Between Means:
(Dependent samples)
E = tcse
C
1 +
n1x0 - x22
1
+
.
n
ngx2 - 1gx22
1d.f. = n - 22
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Page 4
Key Formulas
From Larson/Farber Elementary Statistics: Picturing the World, Second Edition
© 2002 Prentice Hall
CHAPTER 10
Chi-Square: x 2 = g
CHAPTER 11
1O - E22
Test Statistic for Sign Test: When n 7 25,
E
z =
Goodness-of-Fit Test: d.f. = k - 1
Test of Independence:
Two-Sample F- Test for Variances:
Ú
s222
F =
s21
s22
1d.f.N = n1 - 1, and d.f.D = n2 - 12
One-Way Analysis of Variance Test:
gx 2
gni a xi b
MSB
SSB
N
, where MSB =
F =
=
MSW
k - 1
k - 1
and MSW
g1ni - 12s2i
SSW
=
=
.
N - k
N - k
0.51n
, where x is the smaller number of
+ or - signs and n is the total number of + and signs.
d.f. = 1no. of rows - 121no. of columns - 12
1s21
1x + 0.52 - 0.5n
When n … 25, the test statistic is the smaller number of
+ or - signs.
R - mR
,
sR
where R = sum of the ranks for the smaller sample,
Test Statistic for Wilcoxon Rank Sum Test: z =
mR =
n11n1 + n2 + 12
2
, sR =
B
n1n21n1 + n2 + 12
12
, and
n1 … n2 .
Test Statistic for the Kruskal-Wallis Test:
Given three or more independent samples, the test
statistic for the Kruskal-Wallis test is
( d.f.N = k - 1, d.f.D = N - k )
H =
R2k
R21
R22
12
a
+
+ Á +
b - 31N + 12.
N1N + 12 n1
n2
nk
1d.f. = k - 12
The Spearman Rank Correlation Coefficient:
rs = 1 -
6 gd2
n1n2 - 12
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Page 5
Table 4 —Standard Normal Distribution
Area
z
z
z
–3.4
–3.3
–3.2
–3.1
–3.0
–2.9
–2.8
–2.7
–2.6
–2.5
–2.4
–2.3
–2.2
–2.1
–2.0
–1.9
–1.8
–1.7
–1.6
–1.5
–1.4
–1.3
–1.2
–1.1
–1.0
–0.9
–0.8
–0.7
–0.6
–0.5
–0.4
–0.3
–0.2
–0.1
–0.0
0
.09
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
.08
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.07
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.06
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0722
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.05
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
Critical Values
Level of Confidence c
0.80
0.90
0.95
0.99
.04
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.03
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
c
zc
1.28
1.645
1.96
2.575
−z c
z
z=0
zc
.02
.0003
.0005
.0006
.0009
.0013
.0017
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.01
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0352
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.00
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
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Page 6
Table 4 —Standard Normal Distribution (continued)
Area
z
0
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.00
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
z
.01
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.02
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.03
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.04
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.05
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.06
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9278
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.07
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.08
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.09
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
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Page 7
Table 5—t-Distribution
1
α
2
−t
α
t
c-confidence interval
d.f.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
q
Level of
confidence, c
One tail, Two tails, t
Left-tailed test
0.50
0.25
0.50
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.674
α
t
−t
t
0.80
0.10
0.20
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.282
0.90
0.05
0.10
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.645
Right-tailed test
0.95
0.025
0.05
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
1.960
0.98
0.01
0.02
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.326
0.99
0.005
0.01
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.576
t
1
α
2
−t
t
t
Two-tailed test
2264_INS.qxd
2/6/02
11:56 AM
Page 8
Table 6— Chi-Square Distribution
1
α
2
α
χ2
χ2
χ2
χ2
L
Right tail
Degrees of
freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
1
α
2
χ2
R
Two tails
0.995
—
0.010
0.072
0.207
0.412
0.676
0.989
1.344
1.735
2.156
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
8.034
8.643
9.262
9.886
10.520
11.160
11.808
12.461
13.121
13.787
20.707
27.991
35.534
43.275
51.172
59.196
67.328
0.99
—
0.020
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.196
10.856
11.524
12.198
12.879
13.565
14.257
14.954
22.164
29.707
37.485
45.442
53.540
61.754
70.065
0.975
0.001
0.051
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.283
10.982
11.689
12.401
13.120
13.844
14.573
15.308
16.047
16.791
24.433
32.357
40.482
48.758
57.153
65.647
74.222
0.95
0.004
0.103
0.352
0.711
1.145
1.635
2.167
2.733
3.325
3.940
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.117
10.851
11.591
12.338
13.091
13.848
14.611
15.379
16.151
16.928
17.708
18.493
26.509
34.764
43.188
51.739
60.391
69.126
77.929
0.90
0.10
0.05
0.025
0.01
0.005
0.016
2.706
3.841
5.024
6.635
7.879
0.211
4.605
5.991
7.378
9.210 10.597
0.584
6.251
7.815
9.348 11.345 12.838
1.064
7.779
9.488 11.143 13.277 14.860
1.610
9.236 11.071 12.833 15.086 16.750
2.204 10.645 12.592 14.449 16.812 18.548
2.833 12.017 14.067 16.013 18.475 20.278
3.490 13.362 15.507 17.535 20.090 21.955
4.168 14.684 16.919 19.023 21.666 23.589
4.865 15.987 18.307 20.483 23.209 25.188
5.578 17.275 19.675 21.920 24.725 26.757
6.304 18.549 21.026 23.337 26.217 28.299
7.042 19.812 22.362 24.736 27.688 29.819
7.790 21.064 23.685 26.119 29.141 31.319
8.547 22.307 24.996 27.488 30.578 32.801
9.312 23.542 26.296 28.845 32.000 34.267
10.085 24.769 27.587 30.191 33.409 35.718
10.865 25.989 28.869 31.526 34.805 37.156
11.651 27.204 30.144 32.852 36.191 38.582
12.443 28.412 31.410 34.170 37.566 39.997
13.240 29.615 32.671 35.479 38.932 41.401
14.042 30.813 33.924 36.781 40.289 42.796
14.848 32.007 35.172 38.076 41.638 44.181
15.659 33.196 36.415 39.364 42.980 45.559
16.473 34.382 37.652 40.646 44.314 46.928
17.292 35.563 38.885 41.923 45.642 48.290
18.114 36.741 40.113 43.194 46.963 49.645
18.939 37.916 41.337 44.461 48.278 50.993
19.768 39.087 42.557 45.722 49.588 52.336
20.599 40.256 43.773 46.979 50.892 53.672
29.051 51.805 55.758 59.342 63.691 66.766
37.689 63.167 67.505 71.420 76.154 79.490
46.459 74.397 79.082 83.298 88.379 91.952
55.329 85.527 90.531 95.023 100.425 104.215
64.278 96.578 101.879 106.629 112.329 116.321
73.291 107.565 113.145 118.136 124.116 128.299
82.358 118.498 124.342 129.561 135.807 140.169