Describe the distribution before choosing a story

Do not begin with the average. Begin with the values.

Year 2 has many shorter observations and one much longer observation. Any single center will compress that tension.

Center answers a positional or balancing question

For Year 1, the sum is 510 days and (n=16):

[ \bar{x}_1=510/16=31.875\text{ days}. ]

The middle values are 31 and 32, so the median is 31.5 days. Year 2 has a mean of 29.375 days and median of 26.5 days. The right-minus-left differences are −2.5 mean days and −5 median days.

Those are descriptive sample differences. The larger gap in medians tells us the 78-day value pulls the Year 2 mean upward; it does not tell us which center is the “correct” business result.

Spread needs a denominator and a unit

The typed calculation uses n - 1 because these are samples. Year 1 has sum of squared deviations 513.75 day²:

[ s_1^2=513.75/15=34.25\text{ day}^2, \qquad s_1=5.852349955\text{ days}. ]

Year 2 sample variance is 190.783333333 day² and sample standard deviation is 13.812434012 days. The standard deviation returns to days, but it still gives the tail substantial influence. A complete population would divide squared deviations by (N), not n - 1.

Flag the 78-day invoice

Under the declared median-of-halves convention, split the 16 ordered values into two groups of eight. The middle pair of the lower half is 22 and 23, so (Q_1=(22+23)/2=22.5). The middle pair of the upper half is 30 and 31, so (Q_3=(30+31)/2=30.5). Thus (IQR=8), the upper inner fence is 42.5 days, and 78 lies beyond it.

Open the tail investigation. The evidence supports a flag, not a cause. Trace dates, payments, disputes, terms, joins, and identifiers. Correct a verified source error; retain a valid record; or apply a predeclared exclusion rule. Never delete it because the interval gets narrower.

Make a bounded comparison

Complete the distribution comparison, then write three sentences:

1. one about centers;
2. one about spread and tail shape; and
3. one naming the population and causal claims the two samples do not establish.

Depth checkpoint: use the already computed median as a tail-resistant companion to the mean, then explain both sides of the choice. It preserves a view of typical position when 78 pulls the mean, but it can hide the economically important delay represented by that same invoice. A companion summary adds a view; it does not license silent deletion.