| x |
|
xi,
individual or average,
i = 1..n. • |
Computes the
weighted variance,
σ2 W(x),
of a vector of values,
x = (xi, i = 1..n),
and corresponding weights,
w = (wi, i = 1..n),
assuming successively that
(a) x are individual values with weights w, and
(b) x are averages (below, a) from samples of sizes w,
in order to estimate the underlying (individual)
μ and σ.
Case '(b)' relates to estimating
from sums of unequal (or equal) size samples,
assuming that the individual values are not known.
The estimation is done through a linear weighting criterion,
a conjecture. Thus,
σ2 ^ = nW
s²W(a). (Below.)
(→ Excel file.xlsx)
Example with T = 4 trucks, and procedure
(as mentioned, only nt and Lt
are assumed known):
| t |
nt |
Lt |
at |
wt |
wt(at −
xbarW)² |
| 1 | 10 | 250.1 | 25.01 | 0.03571 |
4.81E-6 |
| 2 | 20 | 499.4 | 24.97 |
0.07143 | 57.58E-6 |
| 3 | 100 | 2502 | 25.02 |
0.35714 | 166.74E-6 |
| 4 | 150 | 3748 | 24.987 |
0.53571 | 69.53E-6 |
| Sum: |
1 | S = 298.76E-6 |
at = Lt ⁄
nt;
wt = nt ⁄
Σnt;
xbarW,
weighted average of at =
Σwt at = 24.998 kg;
weighted size, nW =
Σwt nt = 117.86;
weighted variance of a, s²W(a) =
[T ⁄ (T − 1)] S = 398.0E-6 kg²
σ²^ = nW .
s²W(a) = 117.86 × 398.0E-6 =
0.04693 kg² → σ^ = 0.2166 kg.
From the base data, simulated from a population
with μ = 25 and σ = 0.2, the method gives
('wIstdev' =) 0.2166 ≅ σ .
Draws a simple plot of x and
the estimated average (lower and upper lines are
this average −/+ 'wIstdev', est. σ.). |
