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Part A - Repeated measurements¶
- a representative value, using the samples mean
- the measurements variability, using the sample standard deviation
In [2]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
vSamples = np.array([4.98, 5.01, 5.00, 5.03, 4.99, 5.02, 4.97, 5.01])
vSamples
Out[2]:
In [11]:
vMean = vSamples.mean() # np.mean(vSample)
vSTD = vSamples.std(ddof=1) #sample standar deviation
print(f"Mean voltage is = {vMean:.4f} V")
print(f"Sample standard deviation is = {vSTD:.4f} V")
In [26]:
plt.figure(figsize=(8,4))
plt.plot(vSamples, 'ok', label="Samples")
plt.axhline(vMean, linestyle="--", linewidth=2, label=f"Mean = {vMean:.4f}")
plt.xlabel("Measurements")
plt.ylabel("Voltage [V]")
plt.ylim([4.9,5.1])
plt.grid(alpha=0.3)
plt.legend()
plt.show()
- What is the estimated voltage?
- What does the sample standard deviation tell us?
- Are the measurements relatively precise or widely scattered?
Part B - Propagation of uncertainty of one variable¶
Example area of a cricle
In [32]:
r = 10.0
sigmar = 0.2
Area = np.pi*r**2
sigmaA = sigmar*(2*np.pi*r)
print(f"Radius: r = {r:.4f} +/- {sigmar:.4f} mm")
print(f"Area: A = {Area:.4f} +/- {sigmaA:.4f} mm^2")
In [46]:
sigmarValues = np.linspace(0.01, 0.5, 10)
sigmaAValues = (np.pi*2*r)*sigmarValues
plt.figure(figsize=(6,4))
plt.plot(sigmarValues, sigmaAValues, 'ok')
plt.title("Propagation of uncertainty")
plt.xlabel("Uncertainty in radius [mm]")
plt.ylabel("Uncertainty in area [mm^2]")
plt.grid(alpha=0.3)
plt.savefig("uncertainty.png", dpi=300, bbox_inches="tight")
plt.show()
Part c - Propagation of uncertainty for several variables¶
Example: Resistance from voltage and current.
- $V=12.0 \pm 0.2$
- $I=2.0 \pm 0.05$
In [48]:
V = 12.0
sigmaV = 0.2
I = 2.0
sigmaI = 0.05
R = V/I
sigmaR = np.sqrt((sigmaV/I)**2 + (sigmaI*V/I**2)**2)
print(f"Resistance: R = {R:.4f} +/- {sigmaR:.4f} ohms")
In [49]:
termV = (sigmaV / I)**2
termI = ((V * sigmaI) / (I**2))**2
total = termV + termI
percent_V = 100 * termV / total
percent_I = 100 * termI / total
print(f"Contribution from voltage uncertainty: {percent_V:.4f}%")
print(f"Contribution from current uncertainty: {percent_I:.4f}%")