Tables

 

Areas in Normal Distribution

 

z

Area

-  

.000

-3.00

.001

-2.33

.01

-2.00

.02

-1.64

.05

-1.28

.10

-1.00

.16

-.84

.20

-.52

.30

-.25

.40

.00

.50

.25

.60

.52

.70

.84

.80

1.00

.84

1.28

.90

1.64

.95

2.00

.98

2.33

.99

3.00

.999

+  

1.00

 

Areas corresponding to selected z scores in normal distribution.

The t
Distribution

 

The critical values for t and z one-tailed tests at the .05 significance level, for different degrees of freedom, are shown in the table below.

 

 

 

t

t2

1

6.31

39.87

2

2.92

8.53

3

2.35

5.54

4

2.13

4.54

5

2.02

4.06

6

1.94

3.78

7

1.90

3.59

8

1.86

3.46

9

1.83

3.36

10

1.81

3.28

15

1.75

3.07

20

1.72

2.97

30

1.70

2.88

40

1.68

2.84

60

1.67

2.79

120

1.66

2.75

 

1.64

2.71

 

Values of the t and t2 corresponding to the five percent area of the t-distribution for selected degrees of freedom (one-tailed test). The degrees of freedom equal to n - 2. The t2 equals F for 1 degree of freedom. For infinitely large degrees of freedom, t equals z.

 

Using the t distribution for estimation of probability associated with the strength of a relationship in lieu of the normal distribution increases the threshold of the significance criterion and thus makes results less likely to be significant when a small number of subjects is used for analysis. For groups of subjects larger than 60, the z-test and t-tests can be used interchangeably.

 

 

The F Distribution

 

 

1

2

3

 

1

39.87

49.5

53.6

63.3

2

8.53

9.00

9.16

9.49

3

5.54

5.46

5.39

5.13

4

4.54

4.32

4.19

3.76

5

4.06

3.78

3.62

3.11

6

3.78

3.46

3.29

2.72

7

3.59

3.26

3.07

2.47

8

3.46

3.11

2.92

2.29

9

3.36

3.01

2.81

2.16

10

3.28

2.92

2.73

2.06

15

3.07

2.70

2.49

1.76

20

2.97

2.59

2.38

1.61

30

2.88

2.49

2.28

1.49

40

2.84

2.44

2.23

1.38

60

2.79

2.39

2.18

1.29

120

2.75

2.35

2.13

1.19

 

2.71

2.30

2.08

1.00

 

Values of F for selected degrees of freedom at the five percent level of significance (one-tailed test). For one degree of freedom, F equals t2. F(1, ¥) equals z2. Critical values in the F distribution and their equivalence with the other distributions are summarized below.

 

 

 

 

 

 

The Chi Square Distribution

 

 

 

 

1

2.71

3

6.25

5

9.24

10

15.99

20

28.41

30

40.26

40

51.81

50

63.17

100

118.5

 

 

 

 

For one degree of freedom, chi-square equals z-square. For infinite number of degrees of freedom, chi-square, divided by the degrees of freedom, equals F with both of its degrees of freedom equal to infinity.

 

Perspective on the gamma distributions

Within the statistical computer programs, the probabilities associated with the z, t F, and Chi Square ratios may be calculated by a single subroutine. This subroutine normalizes the F distribution as

 

                                                

 

After the normalization, this subroutine uses the polynomial approximations to find areas under the normal distribution, corresponding to standard z scores, as

 

                                              

 

where c1 = .196854, c2 = .115194, c3 = .000344, and c4 = .019527. The infinity is represented by a large number, usually equal to 1,000.

 

Since  F equals z-Square with (1, infinity) degrees of freedom, the probability associated with the z-Square ratio can be obtained as

 

p = fSig (1, 1000, z-Square).

 

Since F equals t-Square with (1, df) degrees of freedom, the probability associated with the t-Square ratio can be obtained as

 

p = fSig (1, df, t-Square).

 

This probability is obtained by calling the fSig subroutine as

 

p = fSig(df1,df2,F).

 

Since F equals Chi Square with (df, infinity) degrees of freedom, the probability associated with the chi square ratio can be obtained as

 

p = fsig(df, 1000, Chi-Square / df).