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159

Critical point. For example: x 05 leaves 5% probability in the tail.

3 x2 Critical Points

d.f.

X225

X205

Xi)25

X2oio

X2005

001

1.32

2.71

3.84

5.02

6.63

7.88

10.8

2.77

4.61

5.99

7.38

9.21

10.6

13.8

4.11

6.25

7.81

9.35

11.3

12.8

16.3

5.39

7.78

9.49

11.1

13.3

14.9

18.5

6.63

9.24

11.1

12.8

15.1

16.7

20.5

7.84

10.6

12.6

14.4

16.8

18.5

22.5

9.04

12.0

14.1

16.0

18.5

20.3

24.3

10.2

13.4

15.5

17.5

20.1

22.0

26.1

11.4

14.7

16.9

19.0

21.7

23.6

27.9

12.5

16.0

18.3

20.5

23.2

25.2

29.6

13.7

17.3

19.7

21.9

24.7

26.8

31.3

14.8

18.5

21.0

23.3

26.2

28.3

32.9

16.0

19.8

22.4

24.7

27.7

29.8

34.5

17.1

21.1

23.7

26.1

29.1

31.3

36.1

18.2

22.3

25.0

27.5

30.6

32.8

37.7

19.4

23.5

26.3

28.8

32.0

34.3

39.3

20.5

24.8

27.6

30.2

33.4

35.7

40.8

21.6

26.0

28.9

31.5

34.8

37.2

42.3

22.7

27.2

30.1

32.9

36.2

38.6

32.8

23.8

28.4

31.4

34.2

37.6

40.0

45.3

24.9

29.6

32.7

35.5

38.9

41.4

46.8

26.0

30.8

33.9

36.8

40.3

42.8

48.3

27.1

32.0

35.2

38.1

41.6

44.2

49.7

28.2

33.2

36.4

39.4

32.0

45.6

51.2

29.3

34.4

37.7

40.6

44.3

46.9

52.6

30.4

35.6

38.9

41.9

45.6

48.3

54.1

31.5

36.7

40.1

43.2

47.0

49.6

55.5

32.6

37.9

41.3

44.5

48.3

51.0

56.9

33.7

39.1

42.6

45.7

49.6

52.3

58.3

34.8

40.3

43.8

47.0

50.9

53.7

59.7

45.6

51.8

55.8

59.3

63.7

66.8

73.4

56.3

63.2

67.5

71.4

76.2

79.5

86.7

67.0

74.4

79.1

83.3

88.4

92.0

99.6

77.6

85.5

90.5

95.0

88.1

96.6

98.6



Critical point. For example:

F.05 leaves 5% probability in the tail.

4 F Critical Points

DEGREES OF FREEDOM FOR NUMERATOR

10 20 40

1 F25 5.83 7.50 8.20 8.58 8.82 8.98 9.19 9.32 9.58 9.71 9.85 F,o 39.9 49.5 53.6 55.8 57.2 58.2 59.4 60.2 61.7 62.5 63.3 Fm 161 200 216 225 230 234 239 242 248 251 254

2 F,5 2.57 3.00 3.15 3.23 3.28 3.31 3.35 3.38 3.43 3.45 3.48 F10 8.53 9.00 9.16 9.24 9.29 9.33 9.37 9.39 9.44 9.47 9.49 Fos 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.5 19.5 Fm 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.5 99.5

(- -Tool

Fmi 998 999 999 999 999 999 999 999 999 999 999

< 3 F,5 2.02 2.28 2.36 2.39 2.41 2.42 2.44 2.44 2.46 2.47 2.47

Fl0 5.54 5.46 5.39 5.34 5.31 5.28 5.25 5.23 5.18 5.16 5.13

2 F05 10.1 9.55 9.28 9.12 9.10 8.94 8.85 8.79 8.66 8.59 8.53

Fm 34.1 30.8 29.5 28.7 28.2 27.9 27.5 27.2 26.7 26.4 26.1

Fm 167 149 141 137 135 133 131 129 126 125 124 Q

S Q

u. W W <&

w Q

F.25

1.81

2.00

2.05

2.06

2.07

2.08

2.08

2.08

2.08

2.08

2.08

4.54

4.32

4.19

4.11

4.05

4.01

3.95

3.92

3.84

3.80

3.76

F.05

7.71

6.94

6.59

6.39

6.26

6.16

6.04

5.96

5.80

5.72

5.63

21.2

18.0

16.7

16.0

15.5

15.2

14.8

14.5

14.0

13.7

13.5

F.oo\

74.1

61.3

56.2

53.4

51.7

50.5

49.0

48.1

46.1

45.1

44.1

F.25

1.69

1.85

1.88

1.89

1.89

1.89

1.89

1.89

1.88

1.88

1.87

4.06

3.78

3.62

3.52

3.45

3.40

3.34

3.30

3.21

3.16

3.10

F.05

6.61

5.79

5.41

5.19

5.05

4.95

4.82

4.74

4.56

4.46

4.36

F.oi

16.3

13.3

12.1

11.4

11.0

10.7

10.3

10.1

9.55

9.29

9.02

F 001

47.2

37.1

33.2

31.1

29.8

28.8

27.6

26.9

25.4

24.6

23.8

F.25

1.62

1.76

1.78

1.79

1.79

1.78

1.77

1.77

1.76

1.75

1.74

3.78

3.46

3.29

3.18

3.11

3.05

2.98

2.94

2.84

2.78

2.72

F.05

5.99

5.14

4.76

4.53

4.39

4.28

4.15

4.06

3.87

3.77

3.67

13.7

10.9

9.78

9.15

8.75

8.47

8.10

7.87

7.40

7.14

6.88

35.5

27.0

23.7

21.9

20.8

20.0

19.0

18.4

17.1

16.4

15.8

F.25

1.57

1.70

1.72

1.72

1.71

1.71

1.70

1.69

1.67

1.66

1.65

F.io

3.59

3.26

3.07

2.96

2.88

2.83

2.75

2.70

2.59

2.54

2.47

F.05

5.59

4.74

4.35

4.12

3.97

3.87

3.73

3.64

3.44

3.34

3.23

12.2

9.55

8.45

7.85

7.46

7.19

6.84

6.62

6.16

5.91

5.65

F.oo]

29.3

21.7

18.8

17.2

16.2

15.5

14.6

14.1

12.9

12.3

11.7

1.54

1.66

1.69

1.66

1.66

1.65

1.64

1.63

1.61

1.59

1.58

3.46

3.11

2.92

2.81

2.73

2.67

2.59

2.54

2.42

2.36

2.29

F.05

5.32

4.46

4.07

3.84

3.69

3.58

3.44

3.35

3.15

3.04

2.93

11.3

8.65

7.59

7.01

6.63

6.37

6.03

5.81

5.36

5.12

4.86

F.ooi

25.4

18.5

15.8

14.4

13.5

12.9

12.0

11.5

10.5

9.92

9.33

F.25

1.51

1.62

1.63

1.63

1.62

1.61

1.60

1.59

1.56

1.55

1.53

3.36

3.01

2.81

2.69

2.61

2.55

2.47

2.42

2.30

2.23

2.16



DEGREES OF FREEDOM FOR NUMERATOR

10 20

1.01

-001

10 F25 F,o F.o5 F.oi F.ooi

12 F25 F.w FM FM Fn

Pi H

$ F.ooi

2 16 F.25

§

£>

U. 01

S 20 F25

Q f10

F.05

Fm F.ooi

.001 14 F.25

Q uu uu a:

u! 30 F25 F

J .05

Fm F.ooi

W 40 F25

D f10

F.05 F.01

F.ooi 60 f,5 F,o F0> F0, Fooi 120 F,< F.m F05 F.oi F.ooi

00 F.25

F.05

F.oi F.ooi

5.12 4.26 3.86 3.63 3.48

10.6 8.02 6.99 6.42 6.06

22.9 16.4 13.9 12.6 11.7

1.49 1.60 1.60 1.59 1.59

3.28 2.92 2.73 2.61 2.52

4.96 4.10 3.71 3.48 3.33 10.0 7.56 6.55 5.99 5.64

21.0 14.9 12.6 11.3 10.5 1.56 1.56 1.56 1.55 1.54 3.18 2.81 2.61 2.48 2.39 4.75 3.89 3.49 3.26 3.11

9.33 6.93 5.95 5.41 5.06 18.6 13.0 10.8 9.63 8.89

1.44 1.53 1.53 1.52 1.51

3.10 2.73 2.52 2.39 2.31

4.60 3.74 3.34 3.11 2.96

8.86 5.51 5.56 5.04 4.69

17.1 11.8 9.73 8.62 7.92 1.42 1.51 1.51 1.50 1.48 3.05 2.67 2.46 2.33 2.24 4.49 3.63 3.24 3.01 2.85 8.53 6.23 5.29 4.77 4.44

16.1 11.0 9.00 7.94 7.27

1.40 1.49 1.48 1.46 1.45

2.97 2.59 2.38 2.25 2.16

4.35 3.49 3.10 2.87 2.71 8.10 5.85 4.94 4.43 4.10

14.8 9.95 8.10 7.10 6.46

1.38 1.45 1.44 1.42 1.41

2.88 2.49 2.28 2.14 2.05

4.17 3.32 2.92 2.69 2.53

7.56 5.39 4.51 4.02 3.70

13.3 8.77 7.05 6.12 5.53

1.36 1.44 1.42 1.40 1.39

2.84 2.44 2.23 2.09 2.00 4.08 3.23 2.84 2.61 2.45

7.31 5.18 4.31 3.83 3.51 12.6 8.25 6.60 5.70 5.13

1.35 1.42 1.41 1.38 1.37

2.79 2.39 2.18 2.04 1.95

4.00 3.15 2.76 2.53 2.37

7.08 4.98 4.13 3.65 3.34

12.0 7.76 6.17 5.31 4.76

1.34 1.40 1.39 1.37 1.35 2.75 2.35 2.13 1.99 1.90 3.92 3.07 2.68 2.45 2.29

6.85 4.79 3.95 3.48 3.17

11.4 7.32 5.79 4.95 4.42

1.32 1.39 1.37 1.35 1.33 2.71 2.30 2.08 1.94 1.85 3.84 3.00 2.60 2.37 2.21 6.63 4.61 3.78 3.32 3.02

10.8 6.91 5.42 4.62 4.10

3.37 3.23 3.14 2.94 2.83 2.71

5.80 5.47 5.26 4.81 4.57 4.31 11.1 10.4 9.89 8.90 8.37 7.81

1.58 1.56 1.55 1.52 1.51 1.48

2.46 2.38 2.32 2.20 2.13 2.06

3.22 3.07 2.98 2.77 2.66 2.54

5.39 5.06 4.85 4.41 4.17 3.91

9.92 9.20 8.75 7.80 7.30 6.76 1.53 1.51 1.50 1.47 1.45 1.42

2.33 2.24 2.19 2.06 1.99 1.90 3.00 2.85 2.75 2.54 2.43 2.30 4.82 4.50 4.30 3.86 3.62 3.36

8.38 7.71 7.29 6.40 5.93 5.42 1.50 1.48 1.46 1.43 1.41 1.38

2.24 2.15 2.10 1.96 1.89 1.80 2.85 2.70 2.60 2.39 2.27 2.13

4.46 4.14 3.94 3.51 3.27 3.00

7.43 6.80 6.40 5.56 5.10 4.60 1.48 1.46 1.45 1.40 1.37 1.34 2.18 2.09 2.03 1.89 1.81 1.72 2.74 2.59 2.49 2.28 2.15 2.01 4.20 3.89 3.69 3.26 3.02 2.75

6.81 6.19 5.81 4.99 4.54 4.06

1.44 1.42 1.40 1.36 1.33 1.29

2.09 2.00 1.94 1.79 1.71 1.61 2.60 2.45 2.35 2.12 1.99 1.84 3.87 3.56 3.37 2.94 2.69 2.42 6.02 5.44 5.08 4.29 3.86 3.38

1.39 1.37 1.35 1.30 1.27 1.23 1.98 1.88 1.82 1.67 1.57 1.46 2.42 2.27 2.16 1.93 1.79 1.62

3.47 3.17 2.98 2.55 2.30 2.01 5.12 4.58 4.24 3.49 3.07 2.59 1.37 1.35 1.33 1.28 1.24 1.19

1.93 1.83 1.76 1.61 1.51 1.38

2.34 2.18 2.08 1.84 1.69 1.51 3.29 2.99 2.80 2.37 2.11 1.80

4.73 4.21 3.87 3.15 2.73 2.23

1.35 1.32 1.30 1.25 1.21 1.15 1.87 1.77 1.71 1.54 1.44 1.29

2.25 2.10 1.99 1.75 1.59 1.39 3.12 2.82 2.63 2.20 1.94 1.60 4.37 3.87 3.54 2.83 2.41 1.89 1.33 1.30 1.28 1.22 1.18 1.10

1.82 1.72 1.65 1.48 1.37 1.19 2.17 2.02 1.91 1.66 1.50 1.25 2.96 2.66 2.47 2.03 1.76 1.38 4.04 3.55 3.24 2.53 2.11 1.54 1.31 1.28 1.25 1.19 1.14 1.00 1.77 1.67 1.60 1.42 1.30 1.00

2.10 1.94 1.83 1.57 1.39 1.00 2.80 2.51 2.32 1.88 1.59 1.00

3.74 3.27 2.96 2.27 1.84 1.00



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