(From the course notes of Prof. Mike Disney)

(To do: create web pages with past study details for Cardiff and Warwick - 19/2/00)

1. Specify the theoretical cumulative distribution function under the "null" hypothesis, i.e. the hypothesis that nothing interesting is to be expected (call it F_nought(x))

2. Arrange the observed frequencies in a cumulative distribution (call it S_N(x))

3. For each step in the distribution subtract S_N(x) from F_nought(x)

4. Find D the maximum difference between the two accumulated distributions ie.

   D = maximum ( absolute_value ( F_nought(x) - S_N(x) ) )

5. With D so found, and N = total number of observations, enter the Kolmogorov - Smirnov table to find level of significance.

The sixth form subjects chosen by convergers and divergers (n = 267). (Hudson, 1966, p.180)

1. Null hypothesis

"The converger is the boys who is substantially better at the intelligence test than he is at the open-ended tests; the diverger is the reverse. In addition are the all-rounders , the boys who are more or less equally good (or bad) on both types of test. As a matter of convenience, I define 30 per cent of my usual schoolboy sample as convergers, 30 per cent as divergers, and leave the remaining 40 per cent in the middle as all-rounders." (Hudson, 1966, p.55)

Null hypothesis (per cent)

• Extreme divergers: 10% = 0.1
• Mild divergers: 20% = 0.2
• All-rounders: 40% = 0.4
• Mild convergers: 20% = 0.2
• Extreme convergers: 10% = 0.1

• Extreme divergers: 0.1
• Mild divergers: 0.3
• All-rounders: 0.7
• Mild convergers: 0.9
• Extreme convergers: 1.0

History

• Extreme divergers: 7
• Mild divergers: 15
• All-rounders: 17
• Mild convergers: 3
• Extreme convergers: 2

• Extreme divergers: 7
• Mild divergers: 22
• All-rounders: 39
• Mild convergers: 42
• Extreme convergers: 44

Cumulative distribution (S_N(x))

• Extreme divergers: 0.16
• Mild divergers: 0.5
• All-rounders: 0.87
• Mild convergers: 0.95
• Extreme convergers: 1.00

Cumulative null hypothesis (F_nought (x))

• Extreme divergers: 0.1
• Mild divergers: 0.3
• All-rounders: 0.7
• Mild convergers: 0.9
• Extreme convergers: 1.0

Cumulative distribution (S_N(x))

• Extreme divergers: 0.16
• Mild divergers: 0.5
• All-rounders: 0.87
• Mild convergers: 0.95
• Extreme convergers: 1.00

Difference between cumulative distributions.

• Extreme divergers: -0.06
• Mild divergers: -0.20
• All-rounders: -0.17
• Mild convergers: -0.05
• Extreme convergers: 0.00

Absolute difference between cumulative distributions.

• Extreme divergers: 0.06
• Mild divergers: 0.20
• All-rounders: 0.17
• Mild convergers: 0.05
• Extreme convergers: 0.00

5. With D so found, and N = total number of observations, enter the Kolmogorov - Smirnov table to find level of significance.

D = 0.2, N = 44 from table roughly 5% significance.

• Astrophysics
• Non-parametric statistics
• Convergence and Divergence