Question

A student repeatedly measured the length of a simple pendulum and recorded the results in centimetre as: 36.9, 36.7, 36.8 and 36.6. What is the precision index of this measurement in cm?

Answer 1

Answer on Question#39761 – Physics – Other

A student repeatedly measured the length of a simple pendulum and recorded the results in centimetre as: 36.9, 36.7, 36.8 and 36.6. What is the precision index of this measurement in cm?

Solution:

Therefore, it is quite common to forego the complete information provided by the error distribution and instead to describe the errors by an error or precision index. We typically write:

$$x_{\text{exact}} = x_{\text{observed}} \pm \Delta x$$

where $\Delta x$ is the precision index or error. Note that the definition of $\Delta x$ can be ambiguous. It is a single number used to characterize the actual distribution of errors. Some choose to define $\Delta x$ in terms of the standard deviation of the distribution, $s$:

$$s = \frac{1}{n - 1} \sum_{i=1}^{n} [x_i - \bar{x}]^2$$ $$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$

Table with the results of experiment:

$$(2): \bar{x} = \frac{1}{4} \sum_{i=1}^{4} x_i = \frac{1}{4} (36.9 + 36.7 + 36.8 + 36.6) = 36.75$$

(3)in(1):

$$\begin{aligned} s &= \frac{1}{4 - 1} \sum_{i=1}^{4} [x_i - \bar{x}]^2 \\ &= \frac{1}{4 - 1} (36.9 - 36.75)^2 (36.75 - 36.7)^2 (36.8 - 36.75)^2 (36.75 - 36.6)^2 = 1 \times 10^{-9} \text{cm} \end{aligned}$$

The magnitude of $\Delta x$ can then be defined as some multiple of $s$. So a measurement might be reported as:

$$x_{\text{exact}} = \bar{x} \pm 2s \Rightarrow$$ $$\Delta x = 2s = 2 \times 10^{-9} \text{cm}$$

Solution: precision index of this measurement is equal to $2 \times 10^{-9}$ cm.