1.19 Descriptive statistics and distribution shape

What this is and why it exists

Before any model touches a dataset, someone must look at it — and looking is a skill with instruments. This topic is the instrument panel: summaries that resist being fooled, plots that expose shape, and the standing rule that a mean quoted without its distribution is a rumour.

The vocabulary

• Mean / median / mode — balance point, middle value, most common value; three "centres" that disagree exactly when it matters.
• Standard deviation / IQR — spread by average squared distance, and spread by the middle 50%'s width; the second shrugs at outliers.
• Quantiles — the sorted data read by fractions: quartiles, percentiles (pr2's quantile, now empirical).
• Boxplot — median, quartile box, whiskers, and outliers as dots: a distribution's one-line summary.
• Robust statistic — one a few wild values cannot drag: median and IQR, versus mean and standard deviation.
• Histogram — counts in bins; the distribution's actual shape, bin width permitting.
• QQ plot — sample quantiles against a theoretical distribution's: a straight line means the shape matches; curved tails confess.

The mental model

The mean is a balance point, and balance points are draggable: one billionaire in the room moves the average income by crores while the median moves not at all. That sensitivity is not a flaw — the mean feeds the CLT machinery of pr5 and st2 — but it makes the mean-median COMPARISON a free diagnostic: agreement suggests symmetry; a gap names the skew and its direction (pr3's story, read from two numbers). Salaries, latencies, city populations: right-skewed, mean above median, median the honest headline. The same split runs through spread: standard deviation squares distances, so outliers dominate it; the IQR measures the middle half and cannot be bribed. Robust statistics are the pair to reach for FIRST on unfamiliar data — switch to moment-based ones after the shape has been seen and trusted.

Then look, in order. HISTOGRAM first: shape, skew, outliers, and — the summary-killer — multiple humps. Two humps mean two populations sharing one table (two campuses, two device types), and every single-number summary of a bimodal distribution describes nobody: the mean falls in the valley between the humps. Try two or three bin widths; too few bins hide humps, too many manufacture noise. BOXPLOTS second, one per group side by side — the fastest honest comparison of several groups on one axis. QQ PLOT third, when a method assumes normality (st2's intervals, st3's tests): sample quantiles against Gaussian quantiles; a straight line blesses the assumption, and tails that peel away from the line are the heavy tails pr3 warned about, caught before they poison an error bar.

The closing rule is cheap and permanent: no summary without a look at the shape it summarises. Anscombe's famous quartet — four datasets, identical means, variances and correlations, wildly different pictures — is the ten-minute demonstration that this rule has teeth.

What you should now be able to explain or do

Choose mean-and-SD versus median-and-IQR for a stated dataset and defend it. Read skew from the mean-median gap. Spot bimodality on a histogram and say why summaries then lie. Read a QQ plot's verdict on normality. Recite Anscombe's lesson.

Check yourself