Human ecology
Mathematics education
Statistics
Psychology of learning mathematics
Kolmogorov - Smirnov test

ref: Siegel "Non-parametric statistics", pp.47-52

(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.
(End of extract from Prof. Disney's course notes)
Example - Convergence and Divergence

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)

Cumulative null hypothesis (F_nought (x)) 2. Arrange the observed frequencies in a cumulative distribution.

History

Cumulative distribution N = 7 + 15 + 17 + 3 + 2 = 44

Cumulative distribution (S_N(x))

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

Cumulative null hypothesis (F_nought (x))

Minus

Cumulative distribution (S_N(x))

Equals

Difference between cumulative distributions.

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

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

Absolute difference between cumulative distributions.

D = 0.20

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.


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Created 19/2/00
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