Question

1. 5 different students are measuring the length of a pencil to be: 5.7cm, 5.5cm, 5.6cm, 5.5cm, 43cm

a. calculate the average:
b. calculate the random error:
c. value and uncertainty:
d. percentage of uncertainty:

Explanation:

Step1: Calculate the average

The average (arithmetic - mean) $\bar{x}$ of a set of data $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $x_1 = 5.7$, $x_2 = 5.5$, $x_3 = 5.6$, $x_4 = 5.5$, $x_5 = 43$, and $n = 5$. But the value $43$ is an out - lier. We will calculate the average without the out - lier. $\sum_{i=1}^{4}x_i=5.7 + 5.5+5.6 + 5.5=22.3$, and $\bar{x}=\frac{22.3}{4}=5.575$ cm.

Step2: Calculate the random error

First, find the differences from the average for each non - outlier measurement: $|5.7 - 5.575|=0.125$, $|5.5 - 5.575| = 0.075$, $|5.6 - 5.575|=0.025$, $|5.5 - 5.575| = 0.075$. The random error is the average of these differences. $\text{Random Error}=\frac{0.125 + 0.075+0.025 + 0.075}{4}=\frac{0.25}{4}=0.0625$ cm.

Step3: Determine value and uncertainty

The value is the average we calculated, $5.575$ cm, and the uncertainty is the random error, $0.0625$ cm. So the measurement can be written as $5.575\pm0.0625$ cm.

Step4: Calculate percentage of uncertainty

The percentage of uncertainty is given by $\text{Percentage Uncertainty}=\frac{\text{Uncertainty}}{\text{Value}}\times100\%$. Substituting the values, we get $\text{Percentage Uncertainty}=\frac{0.0625}{5.575}\times100\%\approx1.12\%$.

Answer:

a. $5.575$ cm
b. $0.0625$ cm
c. $5.575\pm0.0625$ cm
d. $1.12\%$