Thống kê đếm, độ không chắc chắn và mô hình chiều
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| standard deviation/ˈstændəd ˌdiːvɪˈeɪʃn/ | A measure of spread equal to the square root of the variance. |
| dimensional analysis/daɪˈmenʃənl əˈnæləsɪs/ | Checking or building relations by matching powers of mass, length and time. |
| Poisson/ˈpɔɪsn/ | The distribution of counts of independent random events, with variance equal to its mean. |
Ten percent is not exactly ten
- A detector with 20% efficiency sees 100 photons. It does not record exactly 20: repeated runs scatter with standard deviation 标准差 4.
- GRE lab questions test counting statistics, error propagation, precision versus accuracy and dimensional analysis 量纲分析.
Counting statistics
- $N$ independent trials, probability $p$: binomial mean $Np$, variance $Np(1-p)$, standard deviation $\sqrt{Np(1-p)}$.
- Large $N$, small $p$: Poisson 泊松 with mean $\lambda = Np$ and standard deviation $\sqrt\lambda$. $N = 200$, $p = 0.05$: exact $\sqrt{9.5} \approx 3.08$ versus Poisson $\sqrt{10} \approx 3.16$.
400 particles hit a detector of efficiency 0.10. Standard deviation of the count?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$\sqrt{400 \times 0.1 \times 0.9}$.
A Poisson count has mean 9. Its standard deviation?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A Poisson distribution has variance equal to its mean, so the standard deviation is $\sqrt9 = 3$.
For a Poisson count with mean 25, the standard deviation is 5.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. For a Poisson distribution the variance equals the mean, 25, so the standard deviation is √25 = 5.
Propagating uncertainty
- Independent inputs: $\sigma_y^2 \approx \sum (\partial y/\partial x_i)^2\sigma_i^2$. For $y = x^a$, relative error is $|a|$ times that of $x$.
- $K = \tfrac12mv^2$ with a 4% speed uncertainty and exact mass: 8%. With 3% in $m$ and 2% in $v$: $\sqrt{3^2 + 4^2} = 5\%$.
How does the relative error scale?
Use y = xᵃ to propagate a relative uncertainty.
Mass uncertain by 3%, speed by 2% (independent). Relative uncertainty of $\tfrac12mv^2$ in %?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$\sqrt{3^2 + (2\times2)^2}$.
Precision, accuracy, averaging
- Averaging $n$ independent readings shrinks the random standard error by $1/\sqrt n$; a shared calibration bias stays.
- Adding contributions linearly gives a worst-case bound, not the standard uncertainty.
Dimensional analysis
- Match powers of M, L, T. $[G] = \text{L}^3\text{M}^{-1}\text{T}^{-2}$, $[\hbar] = \text{ML}^2\text{T}^{-1}$, $[c] = \text{LT}^{-1}$.
Length from $G^a\hbar^bc^d$: M: $-a + b = 0$; L: $3a + 2b + d = 1$; T: $-2a - b - d = 0$. So $a = b = \tfrac12$, $d = -\tfrac32$: the Planck length $\sqrt{G\hbar/c^3}$. Dimensions cannot fix a pure number like $2\pi$.
Which combination has dimensions of time?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The Planck length divided by c.
Recap
- Binomial sd $\sqrt{Np(1-p)}$; Poisson sd $\sqrt\lambda$.
- Powers multiply relative errors; independent errors add in quadrature.
- Averaging cannot remove bias; dimensions fix exponents, not constants.
- dimensional analysis: Checking or building relations by matching powers of mass, length and time.
- Poisson: The distribution of counts of independent random events, with variance equal to its mean.