Summarize variation without hiding extremes

Try

Compare the synthetic delivery times 2,4,4,10 minutes. The mean is 20/4=5 minutes, while the median is 4. Neither calculation is wrong. They summarize different aspects of the same distribution. Predict which better describes a typical central observation when a few deliveries take exceptionally long.

Predict

Replace 10 by 100 before recalculating. The mean becomes 110/4=27.5, but the median remains 4. Predict the effects on range and standard deviation. This controlled change isolates outlier sensitivity; it does not justify discarding a long delivery merely because it changes the summary.

Experiment

For the separate sample 1,3,5 minutes, build the deviation ledger.

Value	Deviation from mean 3	Squared deviation
1	−2	4
3	0	0
5	2	4

The signed deviations sum to zero, so simply averaging them cannot measure spread. Squaring avoids cancellation. The squared-deviation total is 8. Sample variance is s²=8/(3−1)=4 square minutes, and sample standard deviation is s=2 minutes. If these three values are the entire population of interest, its variance instead divides by 3, giving 8/3.

Understand

The mean x̄ is the sum of the n observed values divided by n. The median is the central ordered value, or the average of the central pair for an even-sized sample. A mean responds to every numerical magnitude; a median responds chiefly to ordering near the center. “Robust” means relatively insensitive to a small fraction of extreme observations, not invulnerable to all changes.

Range is maximum minus minimum. Interquartile range is Q3−Q1, measuring the middle half under a stated quantile convention. Standard deviation measures root-mean-squared distance using the appropriate denominator. Always distinguish sample variance s² from population variance σ². With independent identically distributed sampling and finite variance, the n−1 denominator makes sample variance unbiased for population variance. It does not repair biased recruitment, and sample standard deviation itself is not exactly unbiased merely because s² is.

Variance has squared units; standard deviation has the original units. Translating all values by a constant changes the mean and median but leaves spread unchanged. Multiplying all values by a constant multiplies standard deviation by its absolute value and variance by its square. A negative scale reverses ordering, but cannot make spread negative.

A comparison should include context, center, spread and shape. Two groups can have equal means but very different consistency. Values 19,20,21 have mean20 and sample SD1; values5,20,35 have mean20 and sample SD15. If long delays matter, a tail summary or service threshold can be more relevant than the mean alone.

There is no universally best summary independent of the question. Report several complementary measures where needed, and inspect unusual observations before deciding whether they represent errors or valid experiences.

Apply

Write a short comparison of the two delivery groups that includes units and avoids claiming stable population reliability from three observations. Then explain to a colleague why the squared-deviation total uses n for a complete population but n−1 for the usual sample-variance estimator. The denominator reflects the inferential target, not a preference for a larger answer.

Continue

Next express distance from a reference mean in standard-deviation units. Standardizing is a numerical transformation; it does not make a nonnormal distribution normal.