Lesson 5.1 - Descriptive Statistics Review

Target: ~9 min read - 15 min hands-on

Overview

A quick review: mean and median describe the center of a distribution (and disagree when data is skewed); variance and standard deviation describe spread; skewness measures asymmetry; kurtosis measures tail heaviness. These directly inform engineering tolerance decisions - like how wide a spec must be to capture 99% of output.

Why This Matters (Engineering Context)

A skewed distribution - like time-between-failures, or repair durations, mostly short with occasional long ones - means the mean is a misleading summary on its own. When the audience will act on the number (a spares budget, an SLA), report the full shape.

Code-Along

# One roughly symmetric quantity vs one strongly right-skewed quantity
bore_mm = np.random.normal(40.00, 0.05, 500)       # Normal -> symmetric, bell-shaped
gap_hours = np.random.exponential(60, 500)          # Exponential -> long right tail

for name, data in [("Bore diameter (mm)", bore_mm), ("Hours between stops", gap_hours)]:
    print(f"--- {name} ---")
    print(f"  Mean:     {np.mean(data):.3f}")
    print(f"  Median:   {np.median(data):.3f}")           # mean == median only if symmetric
    print(f"  Std dev:  {np.std(data, ddof=1):.3f}")       # ddof=1 -> sample std dev
    print(f"  Skewness: {stats.skew(data):.2f}")           # 0 = symmetric, >0 = right tail
    print(f"  Kurtosis: {stats.kurtosis(data):.2f}")       # 0 = Normal-like tails (excess kurtosis)
    print()

fig, axes = plt.subplots(1, 2, figsize=(11, 4))
axes[0].hist(bore_mm, bins=30, color="steelblue", edgecolor="white")
axes[0].axvline(np.mean(bore_mm), color="red", label="Mean")            # axvline = vertical line
axes[0].axvline(np.median(bore_mm), color="green", linestyle="--", label="Median")
axes[0].set_title("Bore Diameter (roughly symmetric)"); axes[0].legend()
axes[1].hist(gap_hours, bins=30, color="teal", edgecolor="white")
axes[1].axvline(np.mean(gap_hours), color="red", label="Mean")          # note: mean sits right of median
axes[1].axvline(np.median(gap_hours), color="green", linestyle="--", label="Median")
axes[1].set_title("Hours Between Stops (right-skewed)"); axes[1].legend()
plt.tight_layout(); plt.show()

Run it: the bore-diameter histogram is roughly bell-shaped with mean ~ median and skewness near 0. The hours-between-stops histogram has a long right tail, with the mean pulled noticeably higher than the median and a clearly positive skewness - the signature of skewed real-world reliability data.

Practice Exercises

1. Compute the same five statistics for equipment time-to-failure (np.random.weibull(1.5, 500) * 1000 hours) - is it skewed?
2. For the gap_hours data, compute the 95th percentile and compare it to the mean - what does the gap imply about planning maintenance windows on the average alone?
3. Generate a high-kurtosis dataset (np.random.standard_t(df=3, size=500)) and compare its histogram shape to a Normal's.

# Try the practice exercises here

Knowledge Check

1. Why can the mean and median disagree substantially for a skewed dataset?
2. What does positive skewness indicate about a distribution's tail?
3. Why might reporting only the mean of a skewed engineering dataset be misleading?