Simulates,
via Monte Carlo, a sample of n random integers between
mLO and mHI and examines
the behaviour of their standard deviation. Plots the pdf and cdf
of the standard deviation. In a questionnaire,
m would be the response in a Likert-type scale (m-point item),
and n would be the number of respondents to a certain question.
Notes For m = 1..n, it is
s = √[n(n+1) ⁄ 12]. The maximum s is
smax = √[n ⁄ (n − 1)]
(mHI − mLO) ⁄ 2,
for n even and (same value) for n−1 (odd). E.g.,
for m = 1..4
and n = 15, smax is the value for 16, i.e.,
√(16 ⁄ 15) (4 − 1) ⁄ 2 = 1.549... .
Also [Samuelson's ineq.], it is
(xi − x–)² ≤
(n − 1) s²(n). |
|
• Wikipedia: standard deviation
• Wikipedia: Likert scale
• Google search: "Samuelson's inequality".
• Jensen, Shane, and G. P. H. Styan, 1997,
"Some comments and a bibliography on the Laguerre-Samuelson inequality
with extensions and applications in Statistica and matrix theory"
(=.pdf).
• Mathew, Thomas, and Kenneth Nordström, 1997,
"An Inequality for a Measure of Deviation in Linear Models",
The American Statistician, 51(4), 344–349 (=.pdf, p 346).
• Samuelson, P. A., 1968, "How deviant can you be ?",
Journal of the American Statistical Association, 63, 1522–1525,
in Mathew & Nordström.
• 1755-04-27: Parseval des Chênes, Marc Antoine
(1836-08-16). |
