Understanding Skew and Kurtosis
(Why Extremes Matter)

In Blog 1, we met Carl Friedrich Gauss (1777–1855), the mathematician who taught the world how to measure uncertainty.

At the beginning of the 19th century—around 1800 to 1820—Gauss was trying to solve a practical problem. Astronomers kept measuring the same stars and planets and getting different answers. Instead of arguing over which measurement was “right,” Gauss realized something radical for his time: the variation itself followed a pattern.

From that insight came the normal distribution, variance, and standard deviation.

Gauss answered an essential question:

How wide is the uncertainty around the average?

For astronomy and physics in the early 1800s, that was enough. Measurement errors tended to be small, symmetric, and well-behaved. The bell curve worked beautifully.

But statistics did not stay in observatories.

As data moved into economics, biology, manufacturing, and later business and finance—roughly the late 19th century—a new problem emerged.

Real-world data refused to behave.

Some datasets leaned heavily to one side.

Others looked calm most of the time—until rare, violent events dominated everything.
Many shared the same average and standard deviation, yet carried completely different risk.

Gauss’s mathematics wasn’t wrong.

It was incomplete.

As Karl Pearson later put it:

“The object of statistical methods is the reduction of data.”

That gap is where Karl Pearson (1857–1936) enters the story—nearly 80 years after Gauss’s most influential work.

Working in the late 1800s and early 1900s, Pearson studied data that did not resemble neat bell curves at all. He saw distributions that were lopsided, stretched, and distorted by extremes. Pearson realized that knowing the center and the spread was not enough. You also needed to understand the shape.

He introduced two ideas to do exactly that:

  • Skewness — which side of the data has the long tail
  • Kurtosis — how extreme the rare events really are


Skew and kurtosis were created for a specific reason:
to measure the kinds of behavior Gauss’s assumptions quietly ignored.

A generation later, Ronald Fisher (1890–1962) carried Pearson’s ideas forward. Working in the early to mid-20th century, Fisher refined skewness and kurtosis into the modern statistical tools we use today, formalizing them in terms of higher-order moments and making them practical for real-world samples.

According to Wikipedia, Sir Ronald Aylmer Fisher was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. He has been described as “a genius who almost single-handedly created the foundations for modern statistical science” and “the single most important figure in 20th-century statistics.


Together, across more than 150 years of statistical evolution, Gauss, Pearson, and Fisher answered increasingly deeper questions:

  • Gauss asked: Where is the center, and how wide is the spread?
  • Pearson asked: Is the data balanced—or dangerously lopsided?
  • Fisher asked: How do we measure this rigorously when data is incomplete?


That brings us to the question this blog explores:

Not just how much does the data move—but how does it move, and why do extremes matter so much?

Variance and standard deviation tell us how wide the data spreads.
Skew and kurtosis tell us how that spread is shaped.

And shape is where hidden risk lives.

Meet Our Data: Three Personalities

In this blog, we’ll use one simple dataset:

Daily sales over two full years for three types of products.

Each has 730 days, so we’re comparing apples to apples.

  • STABLE
    Calm. Predictable. Boring in a good way.
  • MODERATE
    Moves around a bit. Some good days, some bad days.
  • VOLATILE
    Roller coaster. Big wins. Big losses. Drama.


All examples use the same table: SALES_BEHAVIOR_LARGE

Step 1: Look at the Data Before Measuring It

Before you use any statistics, please always look.

Counts, minimums, maximums, and averages give us context. They don’t tell the whole story, but they tell us where to start.

As statistician John Tukey once said:

“Far better an approximate answer to the right question than an exact answer to the wrong question.”

Before measuring shape, we need to remind ourselves what kind of data we are actually dealing with.

Numbers don’t give answers by themselves. They give clues.

Query:

SELECT
 SALES_CLASS
,COUNT(*) AS DAY_COUNT
,MIN(DAILY_SALES) AS MIN_SALES
,MAX(DAILY_SALES) AS MAX_SALES
,AVG(DAILY_SALES) AS AVG_SALES
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY SALES_CLASS;
SALES_CLASS  DAY_COUNT	 MIN_SALES   MAX_SALES	  AVG_SALES
MODERATE	   730	     70.20	125.35	  97.50445205
STABLE	           730	     99.60	100.40	  100.00000000
VOLATILE	   730	   -145.76	406.24	  80.05095890


We already know:

  • STABLE barely moves
  • MODERATE moves with structure
  • VOLATILE has extreme highs and lows


Now we want to understand whether those extremes are balanced or lopsided.

STEP 2 – SKEW – Which Direction Does the Data Lean?

SKEW measures asymmetry.

In simple terms:

Does the data lean more to one side?

QUERY:

SELECT
   SALES_CLASS
  ,SKEW(DAILY_SALES) AS DATA_SKEW
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY DATA_SKEW;

RESULTS:

SALES_CLASS	 DATA_SKEW
MODERATE	-0.0006609
STABLE	                 0
VOLATILE	 1.7899832

What This Tells Us

HOW TO READ SKEW

  • SKEW ≈ 0
    Data is balanced on both sides of the average
  • Positive SKEW
    A longer tail on the high side (big spikes)
  • Negative SKEW
    A longer tail on the low side (big drops)


Think of skew like a seesaw.
If one side carries more weight, it tips.

Here, STABLE and MODERATE are balanced.
VOLATILE clearly tips to the high side.

STEP 3 — SKEW IN BUSINESS TERMS

QUERY:

SELECT
   SALES_CLASS
  ,AVG(DAILY_SALES)  AS AVG_SALES
  ,SKEW(DAILY_SALES) AS DATA_SKEW
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY SALES_CLASS;

RESULTS:

SALES_CLASS	AVG_SALES	DATA_SKEW
MODERATE	97.50445205	-0.0006609
STABLE	        100.00000000	         0
VOLATILE	80.05095890	 1.7899832

What This Tells Us

Extreme events pull the average.

SKEW tells us WHICH direction those extremes pull.

VOLATILE data has large upside spikes that inflate the average, even though most days perform worse.


This is one reason averages can lie.

STEP 4 – KURTOSIS – How Extreme Are the Extremes?

KURTOSIS measures tail heaviness.

In plain English: “How violent are the rare events?”


QUERY:

SELECT
   SALES_CLASS
  ,KURTOSIS(DAILY_SALES) AS DATA_KURTOSIS
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY DATA_KURTOSIS;

RESULTS:

SALES_CLASS	DATA_KURTOSIS
STABLE	        -1.300683481195
MODERATE	-0.744698756762
VOLATILE	9.248898708550

RESULT EXPLANATION — KURTOSIS

Negative kurtosis – Fewer extreme events than normal.

Positive kurtosis – Rare but very large shocks.

Kurtosis is not about how often things go wrong.
It is about how bad they are when they do.

As Mike Tyson famously put it:

“Everyone has a plan until they get punched in the mouth.”

STEP 5 — SKEW AND KURTOSIS TOGETHER

In the real world, analysts don’t look at one metric at a time. They look at several together.

Query:

SELECT
   SALES_CLASS
  ,SKEW(DAILY_SALES)     AS DATA_SKEW
  ,KURTOSIS(DAILY_SALES) AS DATA_KURTOSIS
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY SALES_CLASS;

RESULTS:

SALES_CLASS	DATA_SKEW	  DATA_KURTOSIS
MODERATE	-0.0006609	 -0.744698756762
STABLE	                 0	 -1.300683481195
VOLATILE	 1.7899832	 9.248898708550


Two datasets can share the same average and still have completely different risk.

SHAPE TELLS THE STORY

STABLE
Balanced and well-behaved

MODERATE
Structured, controlled movement

VOLATILE
Driven by rare, extreme events

STEP 6 — WHY SHAPE MATTERS BEFORE MODELING

QUERY:

SELECT
   SALES_CLASS
  ,AVG(DAILY_SALES)         AS AVG_SALES
  ,STDDEV_POP(DAILY_SALES)  AS STDDEV
  ,SKEW(DAILY_SALES)        AS DATA_SKEW
  ,KURTOSIS(DAILY_SALES)    AS DATA_KURTOSIS
FROM SALES_BEHAVIOR_LARGE
GROUP BY SALES_CLASS
ORDER BY SALES_CLASS;

RESULTS:

SALES_CLASS  AVG_SALES	   STDDEV       DATA_SKEW    DATA_KURTOSIS
MODERATE     97.50445205   12.1798536  -0.0006609    -0.744698756762
STABLE	     100.00000000   0.2828427	        0    -1.300683481195
VOLATILE     80.05095890   67.1654646   1.7899832    9.248898708550

RESULT SUMMARY — WHAT SHAPE TEACHES US


Looking at the numbers together makes the story unavoidable.

STABLE

  • SKEW = 0
  • KURTOSIS = -1.3
  • STDDEV = 0.28


This data is balanced, calm, and uneventful.
There are no meaningful extremes.
Most days look like every other day.
Forecasts work because surprises are rare and mild.



MODERATE

  • SKEW = -0
  • KURTOSIS = -0.74
  • STDDEV = 12.18


This data moves, but it moves symmetrically.
Upside and downside are roughly equal.
Extremes exist, but they are limited.
Risk is present, but manageable.



VOLATILE

  • SKEW = 1.78
  • KURTOSIS = 9.24
  • STDDEV = 67.16


This data is dominated by rare, extreme events.
Large upside spikes pull the average upward,
even though most days perform worse.
The high kurtosis tells us that when things go wrong,
they go very wrong.


This is the key lesson:

  • Variance tells us how much data moves
  • Skew tells us which direction risk hides
  • Kurtosis tells us how damaging rare events can be


Two datasets can share the same average.
Two datasets can share the same standard deviation.
And still carry a completely different risk.

If you remember one thing:

Shape determines whether averages can be trusted.


As Mark Twain once said,

“It ain’t what you don’t know that gets you into trouble.

It’s what you know for sure that just ain’t so.”

Nexus applies these same statistical ideas directly to result sets, allowing analysts to examine consistency without re-querying the database.

Download a free trial at www.CoffingDW.com.

Posted in Blog