Track 2 · Describing Data · Lesson 8 of 11

Shape: skewness and kurtosis

Describe the shape of a distribution — symmetry, skew, modality and tails — and calculate and interpret skewness and excess kurtosis.

30 minIntermediate8 graded exercisesFree

An insurer's average motor claim is about 14 400 SAR, yet half of all claims are below 6 900 SAR and one claim alone is 480 000 SAR. Pricing, reserving and reporting all depend on understanding that shape: a few huge values in a long tail.

Centre and spread are two numbers; the shape is everything else a histogram shows. Four features cover most of what matters:

Symmetry or skew. A right-skewed (positively skewed) distribution has a long tail of large values — incomes, claim amounts, waiting times. A left-skewed one has a long tail of small values — scores on a very easy test, age at retirement.
Modality. One peak (unimodal), two (bimodal — often a sign of two groups mixed together), or none (uniform).
Tails. How often extreme values occur compared with a normal distribution.
Gaps and outliers. Isolated values or clusters away from the rest.

Skew shows up in the mean–median ordering. The mean is pulled towards the long tail; the median is not. So, as a rule of thumb, mean > median suggests right skew and mean < median suggests left skew. (It is a rule of thumb: discrete or multimodal data can break it, so always look at the histogram as well.)

Insured vehicle values (insurance_claims dataset): right-skewed, with a few very expensive vehicles. Mean 95 505 SAR, median 83 500 SAR.

Show the data as a table

From	To	Count
16500	29875	26
29875	43250	117
43250	56625	175
56625	70000	230
70000	83375	199
83375	96750	167
96750	110125	147
110125	123500	114
123500	136875	85
136875	150250	63
150250	163625	32
163625	177000	38
177000	190375	34
190375	203750	17
203750	217125	12
217125	230500	7
230500	243875	7
243875	257250	7
257250	270625	3
270625	284000	2
284000	297375	5
297375	310750	2
310750	324125	3
324125	337500	1
337500	350875	1
350875	364250	3
364250	377625	0
377625	391000	0
391000	404375	1
404375	417750	0
417750	431125	0
431125	444500	0
444500	457875	0
457875	471250	1
471250	484625	0
484625	498000	0
498000	511375	0
511375	524750	0
524750	538125	0
538125	551500	1

g_1 = \frac{m_3}{m_2^{3/2}}, \qquad G_1 = g_1\,\frac{\sqrt{n(n-1)}}{n-2}, \qquad \text{where } m_k = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^k

Moment skewness g₁ and the adjusted sample skewness G₁ (Excel SKEW, pandas .skew(), and this course's default)

$m_2$: the average squared deviation (the population-style variance)
$m_3$: the average cubed deviation: cubing keeps the sign, so a long right tail makes it positive
$g_1$: skewness without units: 0 for a symmetric distribution, positive for right skew, negative for left skew
$G_1$: a small-sample adjustment of $g_1$ ; the two are almost equal for large $n$

Rough guide: $|G_1| < 0.5$ — fairly symmetric; 0.5 to 1 — moderately skewed; above 1 — strongly skewed. In the (synthetic) student_scores data, exam scores have $G_1 = -0.01$ (symmetric, mean 70.9 ≈ median 71). Vehicle values have $G_1 = 2.10$ .

\text{excess kurtosis } g_2 = \frac{m_4}{m_2^{2}} - 3

Excess kurtosis (the course and Excel KURT report the adjusted version G₂; both are 0 for a normal distribution)

$m_4$: the average fourth power of the deviations — dominated by the most extreme values
$-3$: subtracting 3 (the value of $m_4/m_2^2$ for any normal distribution) makes the normal the reference point: excess kurtosis 0

Kurtosis is about the tails, not about how “peaked” the histogram looks. Positive excess kurtosis means extreme values (far from the mean in SD terms) are more frequent than a normal distribution would produce — heavy tails. Daily investment returns are the classic example: in asset_returns the growth fund has excess kurtosis 11.3, so crash days that “should” almost never happen under a normal model do happen. Negative excess kurtosis (light tails) occurs for flat, bounded distributions such as the uniform (−1.2).

Generate data with chosen skewness and tail heaviness; the widget shows the histogram, the mean and median, and the sample skewness and excess kurtosis.

Quick check

A company reports a mean salary of 18 400 SAR and a median of 13 100 SAR. What does this suggest about the shape?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Practice, difficulty 1 of 3

Name the shape

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For each variable, choose the shape you would expect its distribution to have.

Exercise 2 · Guided, difficulty 1 of 3

A quick skewness index

Not started

A retailer's daily sales (SAR) have mean 790, median 590 and standard deviation 330.

Pearson's second skewness coefficient is $\text{Sk}_2 = 3(\bar{x} - \text{median})/s$ . Calculate it to 2 decimal places and note its sign.

Exercise 3 · Challenge, difficulty 3 of 3

Moment skewness by hand

Not started

Five patients' lengths of hospital stay (days) are 2, 4, 5, 6, 13.

Calculate the moment skewness $g_1 = m_3 / m_2^{3/2}$ , where $m_k$ is the average of the $k$ -th powers of the deviations from the mean (divide by $n$ ). Give it to 3 decimal places.

Exercise 4 · Practice, difficulty 1 of 3

Kurtosis or excess kurtosis?

Not started

An old statistics package reports the kurtosis of a dataset as $m_4/m_2^2 = 7.82$ (it does not subtract 3). What is the excess kurtosis, and are the tails heavier or lighter than a normal distribution's?

Exercise 5 · Application, difficulty 2 of 3

The shape of claim sizes

Not started

In the insurance_claims dataset, most policies have no claim, so total_claims_sar is 0 for them. Restrict to policies with a claim (total_claims_sar > 0). Calculate the mean, the median and the sample skewness ( $G_1$ , as Excel SKEW) of the claim amounts.

Data: insurance_claims · Open in the lab to explore and calculate.

Exercise 6 · Interpretation, difficulty 1 of 3

Reading the mean–median gap

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In a company training course, the final assessment scores have mean 61 and median 74 (out of 100). What is the most likely shape of the distribution?

Exercise 7 · Application, difficulty 2 of 3

Fat tails in fund returns

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Calculate the excess kurtosis ( $G_2$ , as Excel KURT) of the daily returns growth_fund in asset_returns, to 2 decimal places.

Data: asset_returns · Open in the lab to explore and calculate.

Exercise 8 · Interpretation, difficulty 2 of 3

Describe claim sizes for a manager

Not started

Among policies with a claim, claim amounts have mean 14 439 SAR, median 6 884 SAR, skewness 10.4 and one claim of 480 000 SAR. In two or three sentences, describe the shape of the distribution for a non-specialist manager and say which typical value to quote.

What you can now do

Describe a distribution's symmetry, modality, tails and unusual values
Use the mean–median ordering as a quick check of skew
Calculate and interpret skewness and excess kurtosis, and know which version software reports
Recognise how a single extreme value can dominate skewness and kurtosis