01
Shape
What does the distribution look like?
Look for symmetry, skew, clusters, gaps, tails, modes, and unusual observations before compressing the data into a few numbers.
Shape · center · spread · position
Descriptive Statistics
Descriptive statistics organize observed data without claiming more than the data show. Good description combines distribution shape with measures of center, spread, and relative position, chosen to match the variable and the question.
Distribution reading order
Describe the shape before deciding which summaries deserve trust.
Two datasets can share the same mean and standard deviation while having very different shapes. Statistical summaries are compressions, so the first job is to see what information the compression might erase.
02
Center
Where are typical values located?
Mean, median, mode, and other location summaries answer related but different questions and respond differently to skew and outliers.
03
Spread
How much do observations vary?
Range, IQR, variance, standard deviation, and robust alternatives describe different aspects of dispersion.
04
Position
Where does one observation sit relative to the rest?
Quantiles, percentiles, ranks, and standardized scores locate individual values within a distribution.
Distribution-shape lab
Change the shape of the data and watch which summaries move with it.
3observed value11
Mean and median sit close together because the distribution is balanced around its center.
Interpretation principles
A summary statistic is useful only when you know what it summarizes.
Context, units, data quality, and distribution shape determine whether a numerical summary clarifies the data or hides its most important structure.
01Plot first
A summary can hide multimodality, skew, truncation, data-entry errors, or subgroups. Visual structure should inform which numerical summaries are appropriate.
02Units stay attached
Means, medians, IQRs, and standard deviations inherit the variable’s units; variance has squared units.
03Resistance is a tradeoff
Median and IQR resist extreme observations, while mean and standard deviation use every value and can be more efficient under suitable symmetric models.
04Description stops at the observed data
Descriptive statistics organize what was measured. Generalizing to a population or causal process requires additional design and inferential reasoning.