Lab 2

1.4 Measurement Error and Propagation of Errors

1.4.1 Mean and Standard Deviation of $R_1$ Measurements

The ten repeated measurements of the paper-resistor $R_1$:

$$R_1=[5.586,5.804,5.809,5.712,5.998,5.734,5.645,5.606,5.708,5.797]\mathrm{M\Omega }.$$

From these we obtain

$${\mu }_{R_1}=5.740\mathrm{M\Omega },{\sigma }_{R_1}=0.121\mathrm{M\Omega }.$$

---

1.4.3 Uncertainty of Direct‐Measured $R_s$

From §1.4.2, the directly measured series resistances have

$${\mu }_{R_s}^{\mathrm{direct}}=9.484\mathrm{M\Omega },{\sigma }_{R_s}^{\mathrm{direct}}=5.362\mathrm{M\Omega }.$$

At 95 % confidence ($\delta =2\sigma$),

$${\delta }_{R_s}^{\mathrm{direct}}=2{\sigma }_{R_s}^{\mathrm{direct}}=2\times 5.362=10.724\mathrm{M\Omega }.$$

---

1.4.4 Propagated Uncertainty for $R_s=R_1+R_2$

The sums $R_{s,i}=R_{1,i}+R_{2,i}$ are:

$i$	$R_{1,i}$ (MΩ)	$R_{2,i}$ (MΩ)	$R_{s,i}$ (MΩ)
1	5.586	1.473	7.059
2	5.804	1.357	7.161
3	5.809	4.512	10.321
4	5.712	3.228	8.940
5	5.998	1.536	7.534
6	5.734	0.936	6.670
7	5.645	11.762	17.407
8	5.606	8.056	13.662
9	5.708	5.844	11.552
10	5.797	1.976	7.773

The individual uncertainties are

$${\delta }_{R_1}=2{\sigma }_{R_1}=2\times 0.121=0.242\mathrm{M\Omega },{\delta }_{R_2}=2{\sigma }_{R_2}=2\times 3.557=7.113\mathrm{M\Omega }.$$

By propagation for sums,

$${\delta }_{R_s}^{\mathrm{prop}}=\sqrt{{\delta }_{R_1}^2+{\delta }_{R_2}^2}=\sqrt{0.242^2+7.113^2}\approx 7.117\mathrm{M\Omega }.$$

---

1.4.5 Comparison of Uncertainties

• Direct measurement:
  ${\delta }_{R_s}^{\mathrm{direct}}\approx 10.724\mathrm{M\Omega }$
• Propagation:
  ${\delta }_{R_s}^{\mathrm{prop}}\approx 7.117\mathrm{M\Omega }$
  Since

$${\delta }_{R_s}^{\mathrm{prop}}<{\delta }_{R_s}^{\mathrm{direct}},$$

the propagated uncertainty is smaller, reflecting only the known variances of $R_1$ and $R_2$ without additional series measurement.

2.2 Histogram