Tools
Intervals & Tests
Hypothesis Test Of Sample Mean Example
Hypothesis Test Of Two Sample Variances Example
Hypothesis Test Of A Standard Deviation Compared To A Standard Value Example
Distributions
Area Under the Standard Normal Curve
Non-Normal Distributions in the Real World
Rayleigh Distribution for True Position
F.99 (n1,n2)
n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator
n2/n1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
1 |
4052 |
4999.5 |
5403 |
5625 |
5764 |
5859 |
5928 |
5982 |
6022 |
6056 |
2 |
98.50 |
99.00 |
99.17 |
99.25 |
99.30 |
99.33 |
99.36 |
99.37 |
99.39 |
99.40 |
3 |
34.12 |
30.82 |
29.46 |
28.71 |
28.24 |
27.91 |
27.67 |
27.49 |
27.35 |
27.23 |
4 |
21.20 |
18.00 |
16.69 |
15.98 |
15.52 |
15.21 |
14.98 |
14.80 |
14.66 |
14.55 |
5 |
16.26 |
13.27 |
12.06 |
11.39 |
10.97 |
10.67 |
10.46 |
10.29 |
10.16 |
10.05 |
6 |
13.75 |
10.92 |
9.78 |
9.15 |
8.75 |
8.47 |
8.26 |
8.10 |
7.98 |
7.87 |
7 |
12.25 |
9.55 |
8.45 |
7.85 |
7.46 |
7.19 |
6.99 |
6.84 |
6.72 |
6.62 |
8 |
11.26 |
8.65 |
7.59 |
7.01 |
6.63 |
6.37 |
6.18 |
6.03 |
5.91 |
5.81 |
9 |
10.56 |
8.02 |
6.99 |
6.42 |
6.06 |
5.80 |
5.61 |
5.47 |
5.35 |
5.26 |
10 |
10.04 |
7.56 |
6.55 |
5.99 |
5.64 |
5.39 |
5.20 |
5.06 |
4.94 |
4.85 |
11 |
9.65 |
7.21 |
6.22 |
5.67 |
5.32 |
5.07 |
4.89 |
4.74 |
4.63 |
4.54 |
12 |
9.33 |
6.93 |
5.95 |
5.41 |
5.06 |
4.82 |
4.64 |
4.50 |
4.39 |
4.30 |
13 |
9.07 |
6.70 |
5.74 |
5.21 |
4.86 |
4.62 |
4.44 |
4.30 |
4.19 |
4.10 |
14 |
8.86 |
6.51 |
5.56 |
5.04 |
4.69 |
4.46 |
4.28 |
4.14 |
4.03 |
3.94 |
15 |
8.68 |
6.36 |
5.42 |
4.89 |
4.56 |
4.32 |
4.14 |
4.00 |
3.89 |
3.80 |
16 |
8.53 |
6.23 |
5.29 |
4.77 |
4.44 |
4.20 |
4.03 |
3.89 |
3.78 |
3.69 |
17 |
8.40 |
6.11 |
5.18 |
4.67 |
4.34 |
4.10 |
3.93 |
3.79 |
3.68 |
3.59 |
18 |
8.29 |
6.01 |
5.09 |
4.58 |
4.25 |
4.01 |
3.84 |
3.71 |
3.60 |
3.51 |
19 |
8.18 |
5.93 |
5.01 |
4.50 |
4.17 |
3.94 |
3.77 |
3.63 |
3.52 |
3.43 |
20 |
8.10 |
5.85 |
4.94 |
4.43 |
4.10 |
3.87 |
3.70 |
3.56 |
3.46 |
3.37 |
21 |
8.02 |
5.78 |
4.87 |
4.37 |
4.04 |
3.81 |
3.64 |
3.51 |
3.40 |
3.31 |
22 |
7.95 |
5.72 |
4.82 |
4.31 |
3.99 |
3.76 |
3.59 |
3.45 |
3.35 |
3.26 |
23 |
7.88 |
5.66 |
4.76 |
4.26 |
3.94 |
3.71 |
3.54 |
3.41 |
3.30 |
3.21 |
24 |
7.82 |
5.61 |
4.72 |
4.22 |
3.90 |
3.67 |
3.50 |
3.36 |
3.26 |
3.17 |
25 |
7.77 |
5.57 |
4.68 |
4.18 |
3.85 |
3.63 |
3.46 |
3.32 |
3.22 |
3.13 |
26 |
7.72 |
5.53 |
4.64 |
4.14 |
3.82 |
3.59 |
3.42 |
3.29 |
3.18 |
3.09 |
27 |
7.68 |
5.49 |
4.60 |
4.11 |
3.78 |
3.56 |
3.39 |
3.26 |
3.15 |
3.06 |
28 |
7.64 |
5.45 |
4.57 |
4.07 |
3.75 |
3.53 |
3.36 |
3.23 |
3.12 |
3.03 |
29 |
7.60 |
5.42 |
4.54 |
4.04 |
3.73 |
3.50 |
3.33 |
3.20 |
3.09 |
3.00 |
30 |
7.56 |
5.39 |
4.51 |
4.02 |
3.70 |
3.47 |
3.30 |
3.17 |
3.07 |
2.98 |
40 |
7.31 |
5.18 |
4.31 |
3.83 |
3.51 |
3.29 |
3.12 |
2.99 |
2.89 |
2.80 |
60 |
7.08 |
4.98 |
4.13 |
3.65 |
3.34 |
3.12 |
2.95 |
2.82 |
2.72 |
2.63 |
120 |
6.85 |
4.79 |
3.95 |
3.48 |
3.17 |
2.96 |
2.79 |
2.66 |
2.56 |
2.47 |
∞ |
6.63 |
4.61 |
3.78 |
3.32 |
3.02 |
2.80 |
2.64 |
2.51 |
2.41 |
2.32 |
F.99(n1, n2)
n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator
n1
n2/n1 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
∞ |
1 |
6106 |
6157 |
6209 |
6235 |
6261 |
6287 |
6313 |
6339 |
6366 |
2 |
99.42 |
99.43 |
99.45 |
99.46 |
99.47 |
99.47 |
99.48 |
99.49 |
99.50 |
3 |
27.05 |
26.87 |
26.69 |
26.60 |
26.50 |
26.41 |
26.32 |
26.22 |
26.13 |
4 |
14.37 |
14.20 |
14.02 |
13.93 |
13.84 |
13.75 |
13.65 |
13.56 |
13.46 |
5 |
9.89 |
9.72 |
9.55 |
9.47 |
9.38 |
9.29 |
9.20 |
9.11 |
9.02 |
6 |
7.72 |
7.56 |
7.40 |
7.31 |
7.23 |
7.14 |
7.06 |
6.97 |
6.88 |
7 |
6.47 |
6.31 |
6.16 |
6.07 |
5.99 |
5.91 |
5.82 |
5.74 |
5.65 |
8 |
5.67 |
5.52 |
5.36 |
5.28 |
5.20 |
5.12 |
5.03 |
4.95 |
4.86 |
9 |
5.11 |
4.96 |
4.81 |
4.73 |
4.65 |
4.57 |
4.48 |
4.40 |
4.31 |
10 |
4.71 |
4.56 |
4.41 |
4.33 |
4.25 |
4.17 |
4.08 |
4.00 |
3.91 |
11 |
4.40 |
4.25 |
4.10 |
4.02 |
3.94 |
3.86 |
3.78 |
3.69 |
3.60 |
12 |
4.16 |
4.01 |
3.86 |
3.78 |
3.70 |
3.62 |
3.54 |
3.45 |
3.36 |
13 |
3.96 |
3.82 |
3.66 |
3.59 |
3.51 |
3.43 |
3.34 |
3.25 |
3.17 |
14 |
3.80 |
3.66 |
3.51 |
3.43 |
3.35 |
3.27 |
3.18 |
3.09 |
3.00 |
15 |
3.67 |
3.52 |
3.37 |
3.29 |
3.21 |
3.13 |
3.05 |
2.96 |
2.87 |
16 |
3.55 |
3.41 |
3.26 |
3.18 |
3.10 |
3.02 |
2.93 |
2.84 |
2.75 |
17 |
3.46 |
3.31 |
3.16 |
3.08 |
3.00 |
2.92 |
2.83 |
2.75 |
2.65 |
18 |
3.37 |
3.23 |
3.08 |
3.00 |
2.92 |
2.84 |
2.75 |
2.66 |
2.57 |
19 |
3.30 |
3.15 |
3.00 |
2.92 |
2.84 |
2.76 |
2.67 |
2.58 |
2.49 |
20 |
3.23 |
3.09 |
2.94 |
2.86 |
2.78 |
2.69 |
2.61 |
2.52 |
2.42 |
21 |
3.17 |
3.03 |
2.88 |
2.80 |
2.72 |
2.64 |
2.55 |
2.46 |
2.36 |
22 |
3.12 |
2.98 |
2.83 |
2.75 |
2.67 |
2.58 |
2.50 |
2.40 |
2.31 |
23 |
3.07 |
2.93 |
2.78 |
2.70 |
2.62 |
2.54 |
2.45 |
2.35 |
2.26 |
24 |
3.03 |
2.89 |
2.74 |
2.66 |
2.58 |
2.49 |
2.40 |
2.31 |
2.21 |
25 |
2.99 |
2.85 |
2.70 |
2.62 |
2.54 |
2.45 |
2.36 |
2.27 |
2.17 |
26 |
2.96 |
2.81 |
2.66 |
2.58 |
2.50 |
2.42 |
2.33 |
2.23 |
2.13 |
27 |
2.93 |
2.78 |
2.63 |
2.55 |
2.47 |
2.38 |
2.29 |
2.20 |
2.10 |
28 |
2.90 |
2.75 |
2.60 |
2.52 |
2.44 |
2.35 |
2.26 |
2.17 |
2.06 |
29 |
2.87 |
2.73 |
2.57 |
2.49 |
2.41 |
2.33 |
2.23 |
2.14 |
2.03 |
30 |
2.84 |
2.70 |
2.55 |
2.47 |
2.39 |
2.30 |
2.21 |
2.11 |
2.01 |
40 |
2.66 |
2.52 |
2.37 |
2.29 |
2.20 |
2.11 |
2.02 |
1.92 |
1.80 |
60 |
2.50 |
2.35 |
2.20 |
2.12 |
2.03 |
1.94 |
1.84 |
1.73 |
1.60 |
120 |
2.34 |
2.19 |
2.03 |
1.95 |
1.86 |
1.76 |
1.66 |
1.53 |
1.38 |
∞ |
2.18 |
2.04 |
1.88 |
1.79 |
1.70 |
1.59 |
1.47 |
1.32 |
1.00 |
F Distribution (α = 5%)
F.95(n1, n2)
n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator
n2/n1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
1 |
161.4 |
199.5 |
215.7 |
224.6 |
230.2 |
234.0 |
236.8 |
238.9 |
240.5 |
241.9 |
2 |
18.51 |
19.00 |
19.16 |
19.25 |
19.30 |
19.33 |
19.35 |
19.37 |
19.38 |
19.40 |
3 |
10.13 |
9.55 |
9.28 |
9.12 |
9.01 |
8.94 |
8.89 |
8.85 |
8.81 |
8.79 |
4 |
7.71 |
6.94 |
6.59 |
6.39 |
6.26 |
6.16 |
6.09 |
6.04 |
6.00 |
5.96 |
5 |
6.61 |
5.79 |
5.41 |
5.19 |
5.05 |
4.95 |
4.88 |
4.82 |
4.77 |
4.74 |
6 |
5.99 |
5.14 |
4.76 |
4.53 |
4.39 |
4.28 |
4.21 |
4.15 |
4.10 |
4.06 |
7 |
5.59 |
4.47 |
4.35 |
4.12 |
3.97 |
3.87 |
3.79 |
3.73 |
3.68 |
3.64 |
8 |
5.32 |
4.46 |
4.07 |
3.84 |
3.69 |
3.58 |
3.50 |
3.44 |
3.39 |
3.35 |
9 |
5.12 |
4.26 |
3.86 |
3.63 |
3.48 |
3.37 |
3.29 |
3.23 |
3.18 |
3.14 |
10 |
4.96 |
4.10 |
3.71 |
3.48 |
3.33 |
3.22 |
3.14 |
3.07 |
3.02 |
2.98 |
11 |
4.84 |
3.98 |
3.59 |
3.36 |
3.20 |
3.09 |
3.01 |
2.95 |
2.90 |
2.85 |
12 |
4.75 |
3.89 |
3.49 |
3.26 |
3.11 |
3.00 |
2.91 |
2.85 |
2.80 |
2.75 |
13 |
4.67 |
3.81 |
3.41 |
3.18 |
3.03 |
2.92 |
2.83 |
2.77 |
2.71 |
2.67 |
14 |
4.60 |
3.74 |
3.34 |
3.11 |
2.96 |
2.85 |
2.76 |
2.70 |
2.65 |
2.60 |
15 |
4.54 |
3.68 |
3.29 |
3.06 |
2.90 |
2.79 |
2.71 |
2.64 |
2.59 |
2.54 |
16 |
4.49 |
3.63 |
3.24 |
3.01 |
2.85 |
2.74 |
2.66 |
2.59 |
2.54 |
2.49 |
17 |
4.45 |
3.59 |
3.20 |
2.96 |
2.81 |
2.70 |
2.61 |
2.55 |
2.49 |
2.45 |
18 |
4.41 |
3.55 |
3.16 |
2.93 |
2.77 |
2.66 |
2.58 |
2.51 |
2.46 |
2.41 |
19 |
4.38 |
3.52 |
3.13 |
2.90 |
2.74 |
2.63 |
2.54 |
2.48 |
2.42 |
2.38 |
20 |
4.35 |
3.49 |
3.10 |
2.87 |
2.71 |
2.60 |
2.51 |
2.45 |
2.39 |
2.35 |
21 |
4.32 |
3.47 |
3.07 |
2.84 |
2.68 |
2.57 |
2.49 |
2.42 |
2.37 |
2.32 |
22 |
4.30 |
3.44 |
3.05 |
2.82 |
2.66 |
2.55 |
2.46 |
2.40 |
2.34 |
2.30 |
23 |
4.28 |
3.42 |
3.03 |
2.80 |
2.64 |
2.53 |
2.44 |
2.37 |
2.32 |
2.27 |
24 |
4.26 |
3.40 |
3.01 |
2.78 |
2.62 |
2.51 |
2.42 |
2.36 |
2.30 |
2.25 |
25 |
4.24 |
3.39 |
2.99 |
2.76 |
2.60 |
2.49 |
2.40 |
2.34 |
2.28 |
2.24 |
26 |
4.23 |
3.37 |
2.98 |
2.74 |
2.59 |
2.47 |
2.39 |
2.32 |
2.27 |
2.22 |
27 |
4.21 |
3.35 |
2.96 |
2.73 |
2.57 |
2.46 |
2.37 |
2.31 |
2.25 |
2.20 |
28 |
4.20 |
3.34 |
2.95 |
2.71 |
2.56 |
2.45 |
2.36 |
2.29 |
2.24 |
2.19 |
29 |
4.18 |
3.33 |
2.93 |
2.70 |
2.55 |
2.43 |
2.35 |
2.28 |
2.22 |
2.18 |
30 |
4.17 |
3.32 |
2.92 |
2.69 |
2.53 |
2.42 |
2.33 |
2.27 |
2.21 |
2.16 |
40 |
4.08 |
3.23 |
2.84 |
2.61 |
2.45 |
2.34 |
2.25 |
2.18 |
2.12 |
2.08 |
60 |
4.00 |
3.15 |
2.76 |
2.53 |
2.37 |
2.25 |
2.17 |
2.10 |
2.04 |
1.99 |
120 |
3.92 |
3.07 |
2.68 |
2.45 |
2.29 |
2.17 |
2.09 |
2.02 |
1.96 |
1.91 |
∞ |
3.84 |
3.00 |
2.60 |
2.37 |
2.21 |
2.10 |
2.01 |
1.94 |
1.88 |
1.83 |
F.99 (n1, n2)
n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator
n2/n1 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
∞ |
1 |
243.9 |
245.9 |
248.0 |
249.1 |
250.1 |
251.1 |
252.2 |
253.2 |
254.3 |
2 |
19.41 |
19.43 |
19.45 |
19.45 |
19.46 |
19.47 |
19.48 |
19.49 |
19.50 |
3 |
8.74 |
8.70 |
8.66 |
8.64 |
8.62 |
8.59 |
8.57 |
8.55 |
8.53 |
4 |
5.91 |
5.86 |
5.80 |
5.77 |
5.75 |
5.72 |
5.69 |
5.66 |
5.63 |
5 |
4.68 |
4.62 |
4.56 |
4.53 |
4.50 |
4.46 |
4.43 |
4.40 |
4.36 |
6 |
4.00 |
3.94 |
3.87 |
3.84 |
3.81 |
3.77 |
3.74 |
3.70 |
3.67 |
7 |
3.57 |
3.51 |
3.44 |
3.41 |
3.38 |
3.34 |
3.30 |
3.27 |
3.23 |
8 |
3.28 |
3.22 |
3.15 |
3.12 |
3.08 |
3.04 |
3.01 |
2.97 |
2.93 |
9 |
3.07 |
3.01 |
2.94 |
2.90 |
2.86 |
2.83 |
2.79 |
2.75 |
2.71 |
10 |
2.91 |
2.85 |
2.77 |
2.74 |
2.70 |
2.66 |
2.62 |
2.58 |
2.54 |
11 |
2.79 |
2.72 |
2.65 |
2.61 |
2.57 |
2.53 |
2.49 |
2.45 |
2.40 |
12 |
2.69 |
2.62 |
2.54 |
2.51 |
2.47 |
2.43 |
2.38 |
2.34 |
2.30 |
13 |
2.60 |
2.53 |
2.46 |
2.42 |
2.38 |
2.34 |
2.30 |
2.25 |
2.21 |
14 |
2.53 |
2.46 |
2.39 |
2.35 |
2.31 |
2.27 |
2.22 |
2.18 |
2.13 |
15 |
2.48 |
2.40 |
2.33 |
2.29 |
2.25 |
2.20 |
2.16 |
2.11 |
2.07 |
16 |
2.42 |
2.35 |
2.28 |
2.24 |
2.19 |
2.15 |
2.11 |
2.06 |
2.01 |
17 |
2.38 |
2.31 |
2.23 |
2.19 |
2.15 |
2.10 |
2.06 |
2.01 |
1.96 |
18 |
2.34 |
2.27 |
2.19 |
2.15 |
2.11 |
2.06 |
2.02 |
1.97 |
1.92 |
19 |
2.31 |
2.23 |
2.16 |
2.11 |
2.07 |
2.03 |
1.98 |
1.93 |
1.88 |
20 |
2.28 |
2.20 |
2.12 |
2.08 |
2.04 |
1.99 |
1.95 |
1.90 |
1.84 |
21 |
2.25 |
2.18 |
2.10 |
2.05 |
2.01 |
1.96 |
1.92 |
1.87 |
1.81 |
22 |
2.23 |
2.15 |
2.07 |
2.03 |
1.98 |
1.94 |
1.89 |
1.84 |
1.78 |
23 |
2.20 |
2.13 |
2.05 |
2.01 |
1.96 |
1.91 |
1.86 |
1.81 |
1.76 |
24 |
2.18 |
2.11 |
2.03 |
1.98 |
1.94 |
1.89 |
1.84 |
1.79 |
1.73 |
25 |
2.16 |
2.09 |
2.01 |
1.96 |
1.92 |
1.87 |
1.82 |
1.77 |
1.71 |
26 |
2.15 |
2.07 |
1.99 |
1.95 |
1.90 |
1.85 |
1.80 |
1.75 |
1.69 |
27 |
2.13 |
2.06 |
1.97 |
1.93 |
1.88 |
1.84 |
1.79 |
1.73 |
1.67 |
28 |
2.12 |
2.04 |
1.96 |
1.91 |
1.87 |
1.82 |
1.77 |
1.71 |
1.65 |
29 |
2.10 |
2.03 |
1.94 |
1.90 |
1.85 |
1.81 |
1.75 |
1.70 |
1.64 |
30 |
2.09 |
2.01 |
1.93 |
1.89 |
1.84 |
1.79 |
1.74 |
1.68 |
1.62 |
40 |
2.00 |
1.92 |
1.84 |
1.79 |
1.74 |
1.69 |
1.64 |
1.58 |
1.51 |
60 |
1.92 |
1.84 |
1.75 |
1.70 |
1.65 |
1.59 |
1.53 |
1.47 |
1.39 |
120 |
1.83 |
1.75 |
1.66 |
1.61 |
1.55 |
1.50 |
1.43 |
1.35 |
1.25 |
∞ |
1.75 |
1.67 |
1.57 |
1.52 |
1.46 |
1.39 |
1.32 |
1.22 |
1.00 |
Learn more about the Statistical Inference tools for understanding statistics in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, in his online Intro. to Statistics short course (only $89) or his online Black Belt certification training course ($875).