Applying statistical methods to experimental data: quantifying error types, computing descriptive statistics, and comparing sensors using hypothesis testing.
Consistent, repeatable error that shifts all readings in the same direction. Cannot be reduced by averaging. Causes: poor calibration, zero offset, EMI, loading effects. Corrected by calibration.
Unpredictable scatter about the mean. Can be reduced by averaging more readings. Causes: electrical noise, turbulence, vibration, operator variability. Quantified by standard deviation.
A slow systematic change in sensor output over time, often temperature-related. Cannot be reduced by averaging — requires recalibration or temperature compensation.
The inability to return the same output for the same input under identical conditions applied consecutively. A subset of precision error measured under controlled, short-interval conditions.
For a dataset of N measurements x1, x2, ..., xN:
Sample Mean: x̄ = (1/N) ∑ x_i
Sample Variance: s² = (1/(N-1)) ∑(x_i - x̄)²
Std Deviation: s = √s²
Coeff of Variation: CV = (s / x̄) × 100% (dimensionless precision measure)
Random measurement errors are well-approximated by the normal distribution when many independent noise sources are present (Central Limit Theorem). Key intervals:
When the population standard deviation is unknown and the sample is small (N < 30), the t-distribution must be used instead of the standard normal. It has heavier tails, reflecting greater uncertainty from the small sample. The t-distribution approaches the normal distribution as N increases.
The t-value depends on both the confidence level and the degrees of freedom (df = N − 1). Critical values are found in t-tables or computed via software.
CI: x̄ ± t_(α/2, N-1) · (s / √N)
where t_(α/2, N-1) is the two-tailed critical t-value
for confidence level (1-α) and df = N-1.
A 95% confidence interval means: if this experiment were repeated many times and a CI computed each time, approximately 95% of those intervals would contain the true population mean.
To determine if two sensors have statistically different mean outputs for the same measurand, use the two-sample (Welch's) t-test.
t = (x̄_1 - x̄_2) / √(s_1²/N_1 + s_2²/N_2)
Degrees of freedom (Welch): complex formula
Simplified: df ≈ min(N_1, N_2) - 1 (conservative)
Decision: Reject H_0 (no difference) if |t| > t_critical
The null hypothesis H0 is that the two sensors measure the same mean value. Rejecting H0 at the 5% significance level means there is a 95% probability the difference is real, not due to random chance.
Sum = 198.5+199.2+197.8+200.1+199.5+198.9+200.3+199.0 = 1593.3
Mean x̄ = 1593.3 / 8 = 199.163 kPa
Deviations squared: 0.437, 0.001, 1.796, 0.876, 0.110, 0.069, 1.295, 0.027. Sum = 4.611
s² = 4.611 / 7 = 0.659 s = √0.659 = 0.811 kPa
CV = (0.811 / 199.163) × 100% = 0.407%
x̄ = 199.16 kPa, s = 0.81 kPa, CV = 0.41%
CI = x̄ ± t · (s / √N) = 199.163 ± 2.365 · (0.811 / √8)
= 199.163 ± 2.365 · 0.2867 = 199.163 ± 0.678
95% CI: [198.49 kPa, 199.84 kPa]
This is a systematic (bias) error with a drift component linked to ambient temperature. The thermocouple's reference junction compensation circuit is likely affected by the elevated room temperature, introducing a consistent offset.
Systematic/drift error. Correction: cold-junction compensation, isothermal block, or recalibration at operating temperature.