MATH 204: PRINCIPLES OF STATISTICS 2USING THE FISHER-F TABLES

Tables VIII to XI in McClave and Sincich (pages 899 to 906) contain information on the 1− α probabilitypoints for the Fisher-F distribution for α = 0.1, 0.05, 0.025 and 0.01 respectively, and for different valuesof the degrees of freedom parameters. The values in the body of the table are the numbers x which solvethe equation

Pr[F > x] = α

when the statistic F has a Fisher-F distribution with ν1 and ν2 degrees of freedom, written

F ∼ Fisher-F(ν1, ν2)

where ν1 and ν2 are whole numbers greater than zero.

The table on the reverse of this sheet is the Fisher-F table for α = 0.05, equivalent to the table on pages901-902 of McClave and Sincich. We read ν1 from the column and ν2 from the row. For example,

• if ν1 = 10 and ν2 = 4, we know from the table that

Pr[ F > 5.96 ] = 0.05

• if ν1 = 6 and ν2 = 18, we know from the table that

Pr[ F > 2.66 ] = 0.05

• if ν1 = 20 and ν2 = 20, we know from the table that

Pr[ F > 2.12 ] = 0.05

The Fisher-F distribution is a non-symmetric probability distribution with a specific property that allowsthe tables in McClave and Sincich to tabulate only the right-hand tail of the distribution. If we need tolook up the left-hand tail, we can use the fact that if F ∼ Fisher-F(ν1, ν2), and 0 < p < 1

Pr[ F > x ] = p =⇒ Pr[ 1/F ≤ 1/x ] = p

so thatPr[ 1/F > 1/x ] = 1− p.

But it transpires that

F ∼ Fisher-F(ν1, ν2) =⇒ 1F∼ Fisher-F(ν2, ν1).

Therefore to look up the left-tail α probability point for the F ∼ Fisher-F(ν1, ν2) distribution, we look upthe right-tail 1−α probability point for the Fisher-F(ν2, ν1) distribution, and then take the reciprocal. Forexample,

• if F ∼ Fisher-F(10, 4), we use tables to discover that as

F0.05(4, 10) = 3.48

it follows thatPr[ F ≤ 1/3.48 ] = Pr[ F ≤ 0.29 ] = 0.05

giving the α = 0.05 probability point of the Fisher-F(10, 4) distribution as 0.29.

1

Tabl

eof

the

Fish

er-F

dist

ribu

tion

Entr

ies

inta

ble

are

the

α=

0.05

tail

quan

tile

ofFi

sher

-F(ν

1,ν

2)

dist

ribu

tion

ν 1gi

ven

inco

lum

ns,ν

2gi

ven

inro

ws.

ν 2\ν

11

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

116

1.45

199.

5021

5.71

224.

5823

0.16

233.

9923

6.77

238.

8824

0.54

241.

8824

2.98

243.

9124

4.69

245.

3624

5.95

246.

4624

6.92

247.

3224

7.69

248.

0124

8.31

248.

5824

8.83

249.

0524

9.26

218

.51

19.0

019

.16

19.2

519

.30

19.3

319

.35

19.3

719

.38

19.4

019

.40

19.4

119

.42

19.4

219

.43

19.4

319

.44

19.4

419

.44

19.4

519

.45

19.4

519

.45

19.4

519

.46

310

.13

9.55

9.28

9.12

9.01

8.94

8.89

8.85

8.81

8.79

8.76

8.74

8.73

8.71

8.70

8.69

8.68

8.67

8.67

8.66

8.65

8.65

8.64

8.64

8.63

47.

716.

946.

596.

396.

266.

166.

096.

046.

005.

965.

945.

915.

895.

875.

865.

845.

835.

825.

815.

805.

795.

795.

785.

775.

775

6.61

5.79

5.41

5.19

5.05

4.95

4.88

4.82

4.77

4.74

4.70

4.68

4.66

4.64

4.62

4.60

4.59

4.58

4.57

4.56

4.55

4.54

4.53

4.53

4.52

65.

995.

144.

764.

534.

394.

284.

214.

154.

104.

064.

034.

003.

983.

963.

943.

923.

913.

903.

883.

873.

863.

863.

853.

843.

837

5.59

4.74

4.35

4.12

3.97

3.87

3.79

3.73

3.68

3.64

3.60

3.57

3.55

3.53

3.51

3.49

3.48

3.47

3.46

3.44

3.43

3.43

3.42

3.41

3.40

85.

324.

464.

073.

843.

693.

583.

503.

443.

393.

353.

313.

283.

263.

243.

223.

203.

193.

173.

163.

153.

143.

133.

123.

123.

119

5.12

4.26

3.86

3.63

3.48

3.37

3.29

3.23

3.18

3.14

3.10

3.07

3.05

3.03

3.01

2.99

2.97

2.96

2.95

2.94

2.93

2.92

2.91

2.90

2.89

104.

964.

103.

713.

483.

333.

223.

143.

073.

022.

982.

942.

912.

892.

862.

852.

832.

812.

802.

792.

772.

762.

752.

752.

742.

7311

4.84

3.98

3.59

3.36

3.20

3.09

3.01

2.95

2.90

2.85

2.82

2.79

2.76

2.74

2.72

2.70

2.69

2.67

2.66

2.65

2.64

2.63

2.62

2.61

2.60

124.

753.

893.

493.

263.

113.

002.

912.

852.

802.

752.

722.

692.

662.

642.

622.

602.

582.

572.

562.

542.

532.

522.

512.

512.

5013

4.67

3.81

3.41

3.18

3.03

2.92

2.83

2.77

2.71

2.67

2.63

2.60

2.58

2.55

2.53

2.51

2.50

2.48

2.47

2.46

2.45

2.44

2.43

2.42

2.41

144.

603.

743.

343.

112.

962.

852.

762.

702.

652.

602.

572.

532.

512.

482.

462.

442.

432.

412.

402.

392.

382.

372.

362.

352.

3415

4.54

3.68

3.29

3.06

2.90

2.79

2.71

2.64

2.59

2.54

2.51

2.48

2.45

2.42

2.40

2.38

2.37

2.35

2.34

2.33

2.32

2.31

2.30

2.29

2.28

164.

493.

633.

243.

012.

852.

742.

662.

592.

542.

492.

462.

422.

402.

372.

352.

332.

322.

302.

292.

282.

262.

252.

242.

242.

2317

4.45

3.59

3.20

2.96

2.81

2.70

2.61

2.55

2.49

2.45

2.41

2.38

2.35

2.33

2.31

2.29

2.27

2.26

2.24

2.23

2.22

2.21

2.20

2.19

2.18

184.

413.

553.

162.

932.

772.

662.

582.

512.

462.

412.

372.

342.

312.

292.

272.

252.

232.

222.

202.

192.

182.

172.

162.

152.

1419

4.38

3.52

3.13

2.90

2.74

2.63

2.54

2.48

2.42

2.38

2.34

2.31

2.28

2.26

2.23

2.21

2.20

2.18

2.17

2.16

2.14

2.13

2.12

2.11

2.11

204.

353.

493.

102.

872.

712.

602.

512.

452.

392.

352.

312.

282.

252.

222.

202.

182.

172.

152.

142.

122.

112.

102.

092.

082.

0721

4.32

3.47

3.07

2.84

2.68

2.57

2.49

2.42

2.37

2.32

2.28

2.25

2.22

2.20

2.18

2.16

2.14

2.12

2.11

2.10

2.08

2.07

2.06

2.05

2.05

224.

303.

443.

052.

822.

662.

552.

462.

402.

342.

302.

262.

232.

202.

172.

152.

132.

112.

102.

082.

072.

062.

052.

042.

032.

0223

4.28

3.42

3.03

2.80

2.64

2.53

2.44

2.37

2.32

2.27

2.24

2.20

2.18

2.15

2.13

2.11

2.09

2.08

2.06

2.05

2.04

2.02

2.01

2.01

2.00

244.

263.

403.

012.

782.

622.

512.

422.

362.

302.

252.

222.

182.

152.

132.

112.

092.

072.

052.

042.

032.

012.

001.

991.

981.

9725

4.24

3.39

2.99

2.76

2.60

2.49

2.40

2.34

2.28

2.24

2.20

2.16

2.14

2.11

2.09

2.07

2.05

2.04

2.02

2.01

2.00

1.98

1.97

1.96

1.96

264.

233.

372.

982.

742.

592.

472.

392.

322.

272.

222.

182.

152.

122.

092.

072.

052.

032.

022.

001.

991.

981.

971.

961.

951.

9427

4.21

3.35

2.96

2.73

2.57

2.46

2.37

2.31

2.25

2.20

2.17

2.13

2.10

2.08

2.06

2.04

2.02

2.00

1.99

1.97

1.96

1.95

1.94

1.93

1.92

284.

203.

342.

952.

712.

562.

452.

362.

292.

242.

192.

152.

122.

092.

062.

042.

022.

001.

991.

971.

961.

951.

931.

921.

911.

9129

4.18

3.33

2.93

2.70

2.55

2.43

2.35

2.28

2.22

2.18

2.14

2.10

2.08

2.05

2.03

2.01

1.99

1.97

1.96

1.94

1.93

1.92

1.91

1.90

1.89

304.

173.

322.

922.

692.

532.

422.

332.

272.

212.

162.

132.

092.

062.

042.

011.

991.

981.

961.

951.

931.

921.

911.

901.

891.

8831

4.16

3.30

2.91

2.68

2.52

2.41

2.32

2.25

2.20

2.15

2.11

2.08

2.05

2.03

2.00

1.98

1.96

1.95

1.93

1.92

1.91

1.90

1.88

1.88

1.87

324.

153.

292.

902.

672.

512.

402.

312.

242.

192.

142.

102.

072.

042.

011.

991.

971.

951.

941.

921.

911.

901.

881.

871.

861.

85