Mathematics education

Statistics

Psychology of learning mathematics

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)

- 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))
- Arrange the observed frequencies in a cumulative distribution
(call it S_N(x))
- For each step in the distribution subtract S_N(x) from
F_nought(x)
- Find D the maximum difference between the two accumulated
distributions ie.
D = maximum ( absolute_value ( F_nought(x) - S_N(x) ) )

- 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.

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Created 19/2/00

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