Uncertainty in parenthesis

Question

In a physics text book I read the following:

$$e/m=1.758820150(44) ×10^{11} \mathrm{C/kg} $$ In this expression, $(44)$ indicates the likely uncertainty in the last two digits, $50$.

How should I understand this uncertainty? Does it mean $\pm 44$ on the last two digits?

Answer 1

The digits in parentheses are the uncertainty, to the precision of the same number of least significant digits. (The meaning of the uncertainty is context-dependent but generally represents a standard deviation, or a 95% confidence interval.) So: $$e/m=1.758\,820\,1\color{blue}{50}\,\color{magenta}{(44)}×10^{11} \ \mathrm{C/kg}=\left(1.758\,820\,1\color{blue}{50}×10^{11} \pm 0.000\,000\,0\color{magenta}{44}×10^{11}\right) \ \mathrm{C/kg}$$

Answer 2

The uncertainty is equal to the standard deviation $\sigma = 0.000000044×10^{11} C/kg = 4400 C/kg$. It means you are

• approx. 68% confident that the true value $m_e$ lies within the interval $[\hat m_e-\sigma, \hat m_e+\sigma]$, where $\hat m_e$ is the measure value.
• and approx. $95\%$ confident that the true value $m_e$ lies within the interval $[\hat m_e-2\sigma, \hat m_e+2\sigma]$.