Using Quartiles to Recognize Skew and Concentration

Why Quartile Spacing Matters

A median describes the middle of a set of claim amounts, but it does not show how the values are distributed around that middle. Two portfolios can have the same median and interquartile range while one has a wider spread below the median and the other has a wider spread above it. Comparing the quartiles with the median helps identify that difference and can indicate where a relatively large portion of the observations is concentrated.

For an actuarial analyst, this is a useful first reading of a numerical summary: a quartile pattern can suggest asymmetry, or show that a quarter of the observations lies within a comparatively narrow amount range. Such a pattern is a clue about the observed data, not a complete description of every claim or a guarantee about future claims. The interpretation is aligned with the SOA Exam PA (Predictive Analytics) syllabus.

Definition: The first quartile $Q_1$ is the 25th percentile, the median $Q_2$ is the 50th percentile, and the third quartile $Q_3$ is the 75th percentile. The interquartile range is $IQR=Q_3-Q_1$. Define the lower and upper central spacings as $L=Q_2-Q_1$ and $U=Q_3-Q_2$. These spacings describe the widths of the lower and upper halves of the central 50% of the data.

For a population distribution, each quartile interval contains one quarter of the probability, subject to how percentile boundaries are defined. For a finite sample, quartiles are estimated from ordered observations; interpolation conventions can differ. R’s default quantile() convention is type 7. When interpreting a supplied summary, use the quartiles as reported rather than silently substituting a different convention.

The first comparison is between $L$ and $U$. If $U>L$, the upper half of the central 50% spans a wider range than the lower half. This is evidence of greater spread on the upper side of the median and can indicate rightward asymmetry in the central data. If $L>U$, the corresponding evidence is toward greater spread on the lower side. If the spacings are similar, the central 50% is approximately balanced around the median.

A Quartile-Based Measure of Asymmetry

A numerical summary of the spacing comparison is the Bowley coefficient of skewness, also called the quartile coefficient of skewness. It focuses on the median and the two quartiles, so it describes asymmetry in the central portion of the distribution rather than using the most extreme observations.

B=\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1} =\frac{U-L}{U+L}, \qquad IQR>0.

The coefficient is defined when $Q_3>Q_1$. Its value is between $-1$ and $1$, because $L$ and $U$ are nonnegative and the absolute difference between two nonnegative numbers cannot exceed their sum. A positive value means $U>L$, a negative value means $L>U$, and zero means the two central spacings are equal. The size indicates the relative imbalance: a value near zero signals similar spacings, while a value farther from zero signals a larger difference between them.

Justification: Since $Q_3-Q_1=(Q_3-Q_2)+(Q_2-Q_1)=U+L$, the numerator $Q_3+Q_1-2Q_2$ can be rewritten as $(Q_3-Q_2)-(Q_2-Q_1)=U-L$. Thus the coefficient compares the difference in the two spacings with their total width. If $U=L$, the difference is zero; if $U>L$, it is positive.

This coefficient should be treated as a summary of central asymmetry, not as a universal measure of the shape of a distribution. In particular, it does not use the minimum or maximum, and it cannot by itself establish how long or heavy either extreme tail is. A positive coefficient is evidence that the upper central spacing is wider; it does not prove that every part of the distribution is right-skewed.

Worked Interpretations of Claim Quartiles

Worked Example: A Wide Upper Central Spacing

A hypothetical portfolio has claim amounts summarized in thousands of dollars as follows: minimum $0.3$, $Q_1=1.0$, median $Q_2=1.6$, $Q_3=4.6$, and maximum $38.0$. Assess the central spacing, calculate the Bowley coefficient, and describe what the summary does and does not indicate.

Step 1: Calculate the two central spacings. The lower spacing is $L=Q_2-Q_1=1.6-1.0=0.6$ thousand dollars. The upper spacing is $U=Q_3-Q_2=4.6-1.6=3.0$ thousand dollars. The upper spacing is five times as wide as the lower spacing because $3.0/0.6=5$.

Step 2: Calculate the interquartile range and coefficient. The interquartile range is $IQR=4.6-1.0=3.6$ thousand dollars. Therefore,

B=\frac{U-L}{U+L} =\frac{3.0-0.6}{3.0+0.6} =\frac{2.4}{3.6} =0.6667.

Step 3: Interpret the result. The positive coefficient and the much wider upper spacing indicate greater spread above the median within the central 50% of claims. The maximum is also much farther above $Q_3$ than the minimum is below $Q_1$: $38.0-4.6=33.4$ thousand dollars, compared with $1.0-0.3=0.7$ thousand dollars. This describes the observed endpoints, but the quartiles alone do not show how claims are distributed between $Q_3$ and the maximum.

Method and check. Compare spacings before interpreting the coefficient; its numerator is upper spacing minus lower spacing. The result is positive, as expected from $U>L$. No probability distribution is assumed for this hypothetical summary.

Worked Example: Concentration in a Narrow Quartile Interval

For a different hypothetical set of repair claims, suppose $Q_1=\$2{,}500$, $Q_2=\$10{,}500$, and $Q_3=\$11{,}500$. The minimum is $\$400$ and the maximum is $\$18{,}000$. Identify the narrower central interval and quantify the relative width of the two intervals.

Step 1: Find the widths. The lower central spacing is $L=\$10{,}500-\$2{,}500=\$8{,}000$. The upper central spacing is $U=\$11{,}500-\$10{,}500=\$1{,}000$. The interquartile range is $\$11{,}500-\$2{,}500=\$9{,}000$, and $8{,}000+1{,}000=9{,}000$, which checks the calculation.

Step 2: Compare concentration. Each interval represents approximately one quarter of the observations. The upper interval, from $\$10{,}500$ to $\$11{,}500$, has one eighth the width of the lower interval, since $\$1{,}000/\$8{,}000=0.125$. Equivalently, the lower interval is eight times wider.

Step 3: Interpret cautiously. The upper quarter of the central data is packed into a relatively narrow amount range compared with the lower quarter. A rough observations-per-dollar comparison gives $0.25/1{,}000=0.00025$ per dollar for the upper interval and $0.25/8{,}000=0.00003125$ per dollar for the lower one. Their ratio is $0.00025/0.00003125=8$. These are interval-width comparisons, not exact estimates of a smooth probability density. Ties at quartile boundaries or discrete claim amounts can affect the precise share within an interval.

Step 4: Check the direction of asymmetry. The coefficient is $B=(1{,}000-8{,}000)/(1{,}000+8{,}000)=-7{,}000/9{,}000=-0.7778$, rounded to four decimal places. Its negative sign agrees with the wider lower spacing. No distributional form is assumed.

Worked Example: Symmetric Central Spacings Do Not Settle the Tail Question

A hypothetical claim summary has minimum $\$500$, $Q_1=\$2{,}000$, median $\$5{,}000$, $Q_3=\$8{,}000$, and maximum $\$100{,}000$. Determine what the quartiles say about central asymmetry, then assess whether they establish symmetry of the full data set.

Step 1: Compare the central spacings. $L=\$5{,}000-\$2{,}000=\$3{,}000$ and $U=\$8{,}000-\$5{,}000=\$3{,}000$. Thus $IQR=\$8{,}000-\$2{,}000=\$6{,}000$, and the two spacings sum to the IQR.

Step 2: Calculate the coefficient. $B=(3{,}000-3{,}000)/(3{,}000+3{,}000)=0/6{,}000=0$. The quartiles show equal widths for the lower and upper halves of the central 50%.

Step 3: Check the scope of the conclusion. The distance from $Q_3$ to the maximum is $\$100{,}000-\$8{,}000=\$92{,}000$, while the distance from the minimum to $Q_1$ is $\$2{,}000-\$500=\$1{,}500$. This large difference in observed endpoint distances is a reason not to describe the entire data set as symmetric based on the quartiles. The quartiles establish balance only in the central spacings; they do not describe all observations beyond $Q_1$ and $Q_3$.

Method and check. Use the median-to-quartile distances for the central asymmetry calculation, then separately report what the endpoints show. A zero coefficient is consistent with the equal spacings, but it is not proof of full-distribution symmetry. No probability distribution is assumed.

Common Mistakes and Interpretation Limits

A tempting shortcut is to compare $Q_1$ directly with $Q_3$, or to treat the interquartile range as a measure of skew. The IQR is the total width of the central 50%; by itself, it does not say whether that width lies mainly below or above the median. The useful comparison is $Q_2-Q_1$ against $Q_3-Q_2$.

A sign error can reverse the interpretation. For the first worked example, an incorrect calculation might use $(L-U)/(L+U)=(0.6-3.0)/3.6=-0.6667$, then label the result as evidence of greater lower-side spread. That reverses the defined Bowley coefficient. The correct numerator is $U-L$, so $B=(3.0-0.6)/3.6=0.6667$, consistent with the wider upper spacing.

Quartile spacing also does not say that every claim in a narrow interval is identical, nor does it reveal the exact arrangement of values inside that interval. In small samples, tied values and the selected interpolation convention can matter. State the comparison precisely: for example, “the upper central spacing is narrower than the lower central spacing,” rather than making a stronger claim about every observation or the population that generated the data.

Check the quartiles.
Confirm that $Q_1\leq Q_2\leq Q_3$, and note the units and the convention used to calculate them.

Calculate the spacings.
Find $L=Q_2-Q_1$ and $U=Q_3-Q_2$. Confirm that $L+U=IQR$.

Describe the central pattern.
Compare $L$ with $U$, and, if useful, calculate $B=(U-L)/(U+L)$.

Keep the claim within the evidence.
Describe central asymmetry or relative interval width; do not infer full-tail shape or a probability model from quartiles alone.

Key takeaway: Quartiles divide the data into approximate quarters, while the median separates the two central quartile intervals. Comparing their widths reveals central asymmetry; comparing their widths relative to the equal quarter share indicates where observations are more concentrated per unit of the measured scale. Neither calculation fully describes the tails.

Check Your Understanding

For each hypothetical summary, calculate or interpret the quartile pattern. Answers are provided for checking.

A claim summary has $Q_1=\$12$, $Q_2=\$20$, and $Q_3=\$24$. Find $L$, $U$, and the Bowley coefficient, then state the central direction of asymmetry. (Answer: $L=\$8$, $U=\$4$, $B=-4/12=-0.3333$; the lower central spacing is wider, indicating lower-side central asymmetry.)
For another portfolio, $Q_1=\$3$, $Q_2=\$4$, and $Q_3=\$10$. Which central interval is narrower, and by what factor is the other interval wider? (Answer: the upper interval is narrower; the lower interval is $7/1=7$ times wider.)
A summary has $Q_1=\$1{,}200$, $Q_2=\$1{,}800$, and $Q_3=\$2{,}400$. Calculate $IQR$ and $B$, and state the conclusion’s scope. (Answer: $IQR=\$1{,}200$, $L=U=\$600$, and $B=0$; the central spacings are balanced, but this does not establish symmetry of the full distribution.)
For claims measured in thousands of dollars, $Q_1=4$, $Q_2=7$, $Q_3=16$, minimum $=2$, and maximum $=80$. Calculate $L$, $U$, and $B$, and state one limitation. (Answer: $L=3$, $U=9$, $B=(9-3)/(9+3)=0.5000$; the upper central spacing is wider, but the quartiles do not specify the full upper tail.)
Suppose $Q_1=5$, $Q_2=11$, and the Bowley coefficient is $0.5$. Find $Q_3$. (Answer: let $U=Q_3-11$ and $L=11-5=6$. Then $(U-6)/(U+6)=0.5$, so $U-6=0.5U+3$, $0.5U=9$, and $U=18$. Therefore $Q_3=11+18=29$.)

What You'll Learn