Distributions Field Guide, Week 6

Before you summarize any column of data, ask it four questions: where is the center, how wide is the spread, what is the shape, and where does a value sit by position. The shape question is the one most people skip, and it is the one that decides whether the average you are about to quote is honest. This guide names the five shapes you will meet in accounting data, shows you how to spot each one, and tells you what each implies for the decision in front of you.

The habit this page is teaching: name the shape, state the spread, flag the outlier. A number reported without its shape is a half-truth. The mean of a right-skewed column can describe almost nobody in the data.

The four questions Box-and-whisker Normal Right-skewed Long-tailed Bimodal Pareto Tell them apart The empirical rule

0The four questions: center, spread, shape, position

Every descriptive statistic answers one of four questions. Reach for the right tool by knowing which question it answers.

Question	The tools	The rule of thumb
CENTER where is the middle	mean (average), median (the middle value), mode (most common value)	Median for skewed data, mean for symmetric data, mode for categorical or most-common questions. When the mean and median disagree, the data is telling you it is skewed.
SPREAD how wide is it	range (max minus min), IQR (the middle 50%), standard deviation, CV (coefficient of variation, `sd / mean`)	A number without its spread is a half-truth. Use the IQR when outliers would distort the range or the standard deviation. Use CV to compare spread across columns measured on different scales. The box-and-whisker plot below is the picture of spread.
SHAPE what does the curve look like	skewness (lopsidedness), kurtosis (how long the tails run)	Mean greater than median means a right skew (a long tail of large values). Long tails (high kurtosis) mean the tails run wider than a normal bell, so more extreme values turn up than a bell curve would predict. This is the bridge to the field guide below.
POSITION where does one value sit	percentiles, quartiles, the 5-number summary (min, Q1, median, Q3, max), z-score (how many spreads from the center)	"Top decile" and "90th percentile" are position statements. A z-score of 3 means three standard deviations above the mean. The empirical rule lives here, with the caveat below.

The bridge: the SHAPE question is what the rest of this page is about. Once you can name the shape, you know which CENTER to report, which SPREAD is honest, and whether the empirical rule applies at all.

BThe box-and-whisker plot: the picture of spread

The box plot draws the 5-number summary in one figure. It makes spread, skew, and outliers visible at a glance, and it stacks cleanly side by side so you can compare groups without doing any arithmetic in your head.

The boxRuns from Q1 to Q3, so it spans the IQR, the middle 50 percent of the values. The line inside is the median.

The whiskersReach out to the smallest and largest values that are still within 1.5 times the IQR. Anything past them is plotted as a dot.

The dotsOutliers. The box plot is where they declare themselves, no separate test needed.

Read it: a median pushed to one end of the box means the data is skewed. A median near the left of the box (as drawn) with a long right whisker is a right skew, the same story the mean-greater-than-median rule tells, drawn instead of computed.

Use it: draw one box per group side by side (commercial versus consumer, by region, by month) and the comparison is instant. Whose spread is wider, whose median is higher, who has the outliers, all in one glance and with no averages to mislead you.

1Normal: the symmetric bell

Spot itSymmetric, both sides mirror each other, the mean sits right on top of the median.

In accountingMeasurement error, large-sample averages, standardized test-like scores. Rare in raw financial amounts.

It impliesThe data is typical and predictable. The mean is a fair summary, and the empirical rule applies, so 68 percent sits within one standard deviation, 95 percent within two.

The trap: assuming financial data is normal when it almost never is. Revenue, balances, and losses are skewed or long-tailed far more often than they are bell-shaped. Do not default to the bell just because it is the one shape everyone learned in school.

2Right-skewed: the long right tail

Spot itA tail stretching to the right, and the mean noticeably greater than the median.

In accountingRevenue, firm size, transaction amounts, executive pay, account balances. The most common shape in financial data.

It impliesMost observations are small, a few are huge. Report the median, because it describes the typical case. The mean is dragged up by the big accounts in the tail.

The trap: quoting the mean. The tail inflates it until it describes almost nobody, "the average customer balance is 18,000 dollars" when 80 percent of customers owe under 5,000 and a handful owe millions. Lead with the median, then mention the tail.

3Long-tailed (also called heavy-tailed): sharp peak, tails that run far out (leptokurtic)

Spot itA sharp central peak plus tails that run far out, so extreme values turn up more often than a bell would predict. The tails are longer and wider than the dashed bell behind it, which shows what "normal" would look like.

In accountingStock returns, audit errors, fraud losses, daily cash swings.

It impliesExtreme events are more common than they look. Plan for them. The "once in a century" loss shows up more like once a decade.

Long-tailed versus Pareto: long-tailed is about how far a single measure can swing (how big one daily return or one loss can get); Pareto (below) is about how a few categories dominate a total (how a few customers hold most of the overdue dollars). Both involve a "tail," but they answer different questions, so keep them apart.

The trap: sizing risk off the normal curve and being blindsided by the "that shouldn't happen" event that does. If the tails run wider than a bell, a two-standard-deviation cushion is not the 95 percent of safety the bell promised.

4Bimodal: two peaks

Spot itTwo humps with a dip between them. The data has two centers, not one.

In accountingTwo customer segments (commercial versus consumer, the Week 5 receivables split), manual versus automated journal entries (the Week 2 lab), two product lines.

It impliesYou have two populations stacked on top of each other, not one. Split them and summarize each on its own.

The trap: reporting one average that describes nobody in either group. If commercial invoices average 40,000 and consumer invoices average 400, the blended "average invoice" sits in the empty valley between them and describes no real customer.

5Pareto: the 80/20 rule, drawn as a Pareto chart

Spot itRanked categories where a few dominate the total, then a long flat run of small ones. Sort descending and the bars drop fast while the cumulative line shoots up.

In accountingRevenue by customer, sales by SKU, overdue dollars by account, late-arrival causes.

It impliesFocus effort on the vital few. Chasing the top handful of overdue accounts recovers most of the dollars; the long tail is rounding error by comparison.

Overdue dollars by customer. The bars (left axis) rank the balances; the line (right axis) is the running cumulative percent. It crosses 80 percent at C3, so the first three customers are the vital few.

Read the chart: the cumulative line crosses the 80 percent mark at the third bar. C1 through C3 hold roughly 82 percent of all the overdue dollars, while C4 through C8, more than half the accounts, hold the remaining 18 percent. That crossing point is the line between the vital few and the trivial many.

Pareto-squared (80% of the 80%): the concentration recurses. The top 20 percent of the vital few usually drive about 80 percent of that group, so roughly the top 4 percent of all items can drive about 64 percent of the whole (0.8 times 0.8). In the chart, C1 alone holds 45 percent of every overdue dollar. When you have to triage hard and cannot work even the vital few, go after the few-of-the-few first: the single biggest account often outweighs everything below the top three combined.

The trap: treating all items as equally worth your time. Spreading collection effort evenly across 500 accounts wastes it; a small fraction of accounts holds most of the overdue balance. The Pareto chart tells you where to point.

★Tell them apart at a glance

When you are staring at a fresh column and not sure which shape it is, run down this short list. The fastest tell for each is the first thing to check.

Shape	The fastest tell	What to report
Normal	Mean and median are about equal, and the histogram is symmetric.	The mean is fair. The empirical rule applies.
Right-skewed	Mean is clearly above the median, one long tail to the right.	Report the median, then mention the tail.
Long-tailed	Center is fine, but extreme values show up far more often than a bell would allow.	The center is not the story; the tail risk is. Plan for extremes.
Bimodal	Two separate humps in the histogram, a dip between them.	Split into two groups and summarize each on its own.
Pareto	Rank the categories; a few hold most of the total and the cumulative line shoots up fast.	Work the vital few first; the long tail can wait.

The two "tail" shapes are the easy ones to confuse. Long-tailed asks how far one measure can swing (one return, one loss). Pareto asks how a few categories split a total (which customers own the overdue balance). Skew and long-tailed both bend the empirical rule; bimodal and Pareto are telling you the data is really several groups, not one.

6The empirical rule, and when it breaks

For a normal distribution, the spread follows a fixed pattern. This is the 68-95-99.7 rule, and it is genuinely useful for the rare normal column.

68%

within 1 standard deviation of the mean

95%

within 2 standard deviations

99.7%

within 3 standard deviations

Highlight the bands and watch the rule hold, then break. Pick a distribution and a band. The shaded area is the mean plus or minus that many standard deviations. The readout compares what the rule promises against what the data actually does.

Distribution

Band

Invoice amounts in dollars. The crimson line is the mean; the shaded band is the mean plus or minus the chosen number of standard deviations.

Where it fails: the empirical rule holds for normal data, and most accounting data is not normal. On right-skewed data (revenue, balances) the mean is not the center the rule assumes, so the bands are lopsided and misleading. On long-tailed data (returns, losses) far more than 0.3 percent of values land beyond three standard deviations, so the rule badly understates how often extremes occur. Check the shape before you trust the bands. Name the shape first, then decide whether the rule even applies.

The honest move: when the shape is not normal, describe the spread with the 5-number summary and the IQR instead of standard deviations. They make no assumption about the shape, so they stay honest on skewed and long-tailed data.

Carry this off the page: the four questions (center, spread, shape, position) work on any dataset in any tool. Naming the shape is what turns "the average is 18,000" into "the balances are right-skewed, so the typical account is 4,000 and a few large accounts pull the mean up." The second sentence is the one a CFO can act on.

ACCTG 5150, Summer 2026, Week 6. The companion Analytical Prompt Library shows how to ask an AI to name these shapes for you and turn the finding into a story.