3. Descriptive statistics of repeated measurements

When a measurement is repeated under nominally identical conditions, the resulting data set is never perfectly constant. Thermal drift, electrical noise, mechanical vibration, operator variation, and quantization all contribute to scatter. Descriptive statistics compress that scatter into a small set of numbers that summarize location (where the data cluster) and dispersion (how widely they spread). Choosing the right summary — and knowing when a summary is being distorted by outliers — is the foundation of every uncertainty budget that follows.

3.1 Sample Mean and the Arithmetic Center

For a set of n repeated readings x₁, x₂, …, x_n, the sample mean is

x̄ = (1/n) · Σ xᵢ   (i = 1 … n)

The mean is the best single estimate of the true value when errors are random, symmetric, and free of outliers. It uses every data point, so it is efficient — but that same property makes it fragile. A single gross error (a dropped sample, a misread display, a transient spike) pulls the mean toward the corrupted value.

Consider a temperature channel reading a stable bath. Ten samples in °C:

50.12, 50.09, 50.15, 50.11, 50.13,
50.10, 50.14, 50.12, 50.11, 50.13

The sum is 501.20, so x̄ = 50.120 °C. Now suppose the sixth reading was actually a transient: 50.90 instead of 50.10. The new sum is 502.00 and x̄ = 50.200 °C — an 80 m°C shift caused by one sample out of ten. This sensitivity is the practical reason robust summaries exist.

3.2 Sample Variance and Standard Deviation

Dispersion is quantified by the sample variance, which averages the squared deviations from the mean:

s² = Σ (xᵢ − x̄)² / (n − 1)

The denominator n − 1 (Bessel's correction) is used because the deviations are measured from the sample mean rather than the unknown true mean, which removes one degree of freedom. Using n instead of n − 1 systematically underestimates the population variance; for small n the bias is significant (10 % at n = 10).

The sample standard deviation is the square root, s = √s², and carries the same unit as the data — which is why it is preferred for reporting. For the clean ten-sample set above, the deviations are small and s ≈ 0.019 °C. With the 50.90 outlier included, s jumps to ≈ 0.25 °C, more than a tenfold inflation. Variance and standard deviation are even more outlier-sensitive than the mean because deviations are squared.

3.3 Median, Range, and Robust Summaries

The median is the middle value of the sorted data (or the average of the two middle values for even n). It depends only on rank, not magnitude, so a single extreme value cannot drag it far. For the corrupted set, the sorted values place 50.12 at the center, giving a median of 50.12 °C — essentially unchanged from the clean case. The range, R = x_max − x_min, is the simplest dispersion measure but is defined entirely by the two most extreme points and is therefore the least robust of all.

Two robust companions are widely used in engineering data reduction:

• Interquartile range (IQR) = Q3 − Q1, the spread of the middle 50 % of the data. It ignores the tails entirely.
• Median absolute deviation (MAD) = median(|xᵢ − median(x)|). Scaled by 1.4826, it becomes a consistent estimator of σ for Gaussian data.

A practical outlier screen is the 1.5 × IQR rule: flag any point below Q1 − 1.5·IQR or above Q3 + 1.5·IQR. For the corrupted set, Q1 ≈ 50.105, Q3 ≈ 50.135, IQR ≈ 0.030, so the upper fence is ≈ 50.180 — the 50.90 sample is flagged immediately.

3.4 Choosing Among Summaries

The trade-off is explicit: robustness costs efficiency. If the data are genuinely Gaussian and clean, the mean and standard deviation extract more information per sample. If contamination is plausible — and in field measurements it usually is — the median and MAD give a more trustworthy picture at the cost of wider confidence intervals.

3.5 Worked Example: Reducing a Noisy Strain Record

A strain gauge bridge is sampled 12 times at a constant load. Readings in microstrain (µε):

412, 408, 415, 410, 409, 413,
411, 407, 414, 410, 409, 452

Step 1 — Sort: 407, 408, 409, 409, 410, 410, 411, 412, 413, 414, 415, 452.

Step 2 — Mean: sum = 4970, x̄ = 414.17 µε. The 452 sample alone adds ≈ 3.2 µε to the mean.

Step 3 — Median: average of the 6th and 7th values = (410 + 411)/2 = 410.5 µε.

Step 4 — Standard deviation: Σ(xᵢ − x̄)² ≈ 1938, so s² = 1938/11 ≈ 176.2 and s ≈ 13.3 µε. Excluding the 452 point, s drops to ≈ 2.6 µε.

Step 5 — Outlier screen: Q1 = 409, Q3 = 413.5, IQR = 4.5, upper fence = 413.5 + 6.75 = 420.25. The 452 reading exceeds the fence and is flagged.

Step 6 — Decision: Investigate the flagged sample before discarding it. If a loose lead or a load transient is confirmed, remove it and recompute: x̄ = 410.7 µε, s ≈ 2.6 µε. If no cause is found, retain it and report the median (410.5 µε) with the IQR as the dispersion measure, noting the heavy tail.

3.6 Visualizing the Summary

The figure below contrasts how each statistic responds to the same contaminated data set. The clean cluster drives the mean and standard deviation, while the single outlier shifts them dramatically; the median and IQR remain anchored to the bulk of the data.

The visual makes the trade-off concrete: the orange mean line sits well to the right of the data bulk because of one point, while the green median line stays inside the cluster. The wide orange band (±1 s) suggests far more scatter than the clean blue band, even though eleven of twelve samples are tightly grouped. In an uncertainty budget, using the contaminated s would overstate the random contribution by roughly a factor of five.

3.7 Practical Reporting Guidance

For engineering measurement records, report at minimum: n, the mean, the standard deviation, and the standard deviation of the mean. Add the median and IQR whenever the data are skewed or contain flagged points. State the outlier criterion used and the disposition of any excluded samples. This combination lets a reviewer reconstruct the uncertainty contribution and judge whether the summary is trustworthy.