Why Measure Dispersion?
Two hypothetical insurance portfolios can have the same expected loss but differ substantially in how variable their outcomes are. Conversely, two portfolios can have the same variability while one has a much higher expected loss. Variance and standard deviation describe dispersion around the mean; neither one, by itself, tells the full story about the absolute level of loss.
For a loss \(X\) with a finite second moment, variance is the expected squared distance from its mean. Standard deviation is the square root of variance. These measures are useful for describing how spread out a loss distribution is, and their behavior under fixed shifts and proportional scaling is especially important in actuarial calculations.
The two variance formulas are equivalent when \(E[X^2]\) is finite. The moment formula is often quicker for a discrete distribution, while the deviations formula makes the meaning of variance more visible. Variance is expressed in squared units: if \(X\) is measured in dollars, \(\operatorname{Var}(X)\) is in dollars squared. Standard deviation is in dollars.
Calculating Variance from a Loss Distribution
For a finite discrete distribution, the probabilities must sum to one. A reliable calculation finds the mean first, then calculates either the weighted squared deviations or the second moment and subtracts the squared mean. The result must be nonnegative; a negative answer usually signals an arithmetic or formula error.
Worked Example: A Three-Outcome Property Loss
A hypothetical insurer models one claim's loss \(X\) as follows: \(X=0\) with probability \(0.60\), \(X=5{,}000\) with probability \(0.25\), and \(X=12{,}000\) with probability \(0.15\). Calculate the expected loss, variance, and standard deviation. No interest or mortality assumption is relevant; the stated discrete distribution is the complete model.
Method. Use the discrete moment formula. First check the probabilities: \(0.60+0.25+0.15=1.00\).
The mean is \(E[X]=0(0.60)+5{,}000(0.25)+12{,}000(0.15)=0+1{,}250+1{,}800=\$3{,}050.\)
The second moment is \(E[X^2]=0^2(0.60)+5{,}000^2(0.25)+12{,}000^2(0.15)\). The three contributions are \(0\), \(25{,}000{,}000(0.25)=6{,}250{,}000\), and \(144{,}000{,}000(0.15)=21{,}600{,}000\). Thus \(E[X^2]=27{,}850{,}000\) dollars squared.
Therefore, \(\operatorname{Var}(X)=27{,}850{,}000-(3{,}050)^2=27{,}850{,}000-9{,}302{,}500=18{,}547{,}500\) dollars squared. The standard deviation is \(\sqrt{18{,}547{,}500}=\$4{,}306.68\), rounded to the nearest cent.
Check. Variance is positive, and the standard deviation is in dollars, not dollars squared. The method avoids treating the second moment itself as the variance: the squared mean must be subtracted.
Translation and Scaling
A fixed addition changes the level of every loss by the same amount, while multiplication changes the size of deviations from the mean. These differences produce simple transformation rules.
Here is why these rules hold. Write \(\mu=E[X]\). The mean of \(aX+b\) is \(a\mu+b\), so the deviation of the transformed loss from its mean is \((aX+b)-(a\mu+b)=a(X-\mu)\). Squaring this deviation and taking its expectation gives \(a^2E[(X-\mu)^2]\), which is \(a^2\operatorname{Var}(X)\). Taking the nonnegative square root gives \(|a|\sigma_X\). The absolute value matters for a negative multiplier, although insurance loss scaling is ordinarily nonnegative.
Translation does not mean that the loss is unchanged: its mean moves by \(b\). It means only that the distances of outcomes from their mean remain the same. By contrast, multiplying the loss also multiplies every deviation, which changes dispersion.
Worked Example: Fixed Expense and Proportional Inflation
A hypothetical loss \(X\) equals \(\$0\) with probability \(0.75\) and \(\$2{,}000\) with probability \(0.25\). A revised cost measure is \(Y=1.2X+\$500\): losses are scaled by 1.2 and then a fixed \(\$500\) expense is added. Find the means, variances, and standard deviations of \(X\) and \(Y\). There is no interest or mortality assumption; the distribution of \(X\) is as stated.
Method. Calculate the moments of \(X\), then use the scaling and translation rules. The probabilities sum to \(0.75+0.25=1\).
For \(X\), the mean is \(E[X]=0(0.75)+2{,}000(0.25)=\$500\). Its second moment is \(E[X^2]=0^2(0.75)+2{,}000^2(0.25)=1{,}000{,}000\) dollars squared. Therefore, \(\operatorname{Var}(X)=1{,}000{,}000-500^2=750{,}000\) dollars squared, and \(\sigma_X=\sqrt{750{,}000}=\$866.03\), rounded to cents.
For \(Y\), the mean is \(E[Y]=1.2(500)+500=\$1{,}100\). Its variance is \(\operatorname{Var}(Y)=1.2^2(750{,}000)=1.44(750{,}000)=1{,}080{,}000\) dollars squared. Thus \(\sigma_Y=\sqrt{1{,}080{,}000}=\$1{,}039.23\), rounded to cents.
As a direct check, \(Y\) equals \(\$500\) with probability \(0.75\) and \(\$2{,}900\) with probability \(0.25\). Its mean is \(500(0.75)+2{,}900(0.25)=1{,}100\), consistent with the transformed mean. The fixed \(\$500\) addition raises the mean but does not affect dispersion; the factor 1.2 raises standard deviation by 20% and variance by 44%.
Common mistake avoided. Adding \(\$500\) to the standard deviation would be incorrect. A constant shift changes every loss and the mean equally, so it cancels from each deviation from the mean.
Dispersion Is Not Absolute Loss Level
A variance or standard deviation alone does not indicate whether a portfolio has a high expected loss. A deterministic loss has zero variance because it never departs from its mean, even if that fixed loss is large. Similarly, shifting an entire distribution upward raises its expected loss without changing its standard deviation.
| Quantity | What it describes | Effect of adding a fixed amount |
|---|---|---|
| Mean \(E[X]\) | Expected absolute loss level | Increases by the amount added |
| Variance \(\operatorname{Var}(X)\) | Expected squared deviation from the mean | Unchanged |
| Standard deviation \(\sigma_X\) | Square root of expected squared deviation | Unchanged |
Worked Example: Equal Dispersion, Different Expected Loss
Two hypothetical coverage options have the following loss models. Option A has loss \(A=\$1{,}000\) or \(\$5{,}000\), each with probability \(0.50\). Option B has loss \(B=\$4{,}000\) or \(\$8{,}000\), each with probability \(0.50\). A third contract has a fixed loss \(C=\$4{,}500\) with probability 1. Compare their expected losses, variances, and standard deviations. There is no interest or mortality assumption; these finite distributions are the models.
Method. Calculate each mean and the deviations from that mean. Each pair of probabilities sums to one, and the fixed contract has probability one.
For A, \(E[A]=1{,}000(0.50)+5{,}000(0.50)=\$3{,}000\). Each outcome is \(\$2{,}000\) from the mean, so \(\operatorname{Var}(A)=0.50(2{,}000^2)+0.50((-2{,}000)^2)=4{,}000{,}000\) dollars squared, and \(\sigma_A=\$2{,}000\).
For B, \(E[B]=4{,}000(0.50)+8{,}000(0.50)=\$6{,}000\). Its outcomes are also \(\$2{,}000\) from their mean, so \(\operatorname{Var}(B)=0.50(2{,}000^2)+0.50((-2{,}000)^2)=4{,}000{,}000\) dollars squared, and \(\sigma_B=\$2{,}000\).
For C, \(E[C]=\$4{,}500\), and its deviation from its mean is always zero. Therefore \(\operatorname{Var}(C)=0\) and \(\sigma_C=\$0\).
Check and interpretation. B is A shifted upward by \(\$3{,}000\), so their equal standard deviations agree with the translation rule. B has a greater expected loss than A despite the same dispersion. C has the smallest dispersion but still has a positive, guaranteed loss of \(\$4{,}500\). Thus zero variance does not mean zero loss.
Common mistake avoided. Ranking C as the least costly option solely because its standard deviation is zero confuses variability with loss level. The mean and dispersion answer different questions and should be read separately.
Common Calculation Traps
- Forgetting to subtract the squared mean. \(E[X^2]\) is not variance. Use \(\operatorname{Var}(X)=E[X^2]-(E[X])^2\).
- Using the wrong scaling power. If losses are multiplied by \(a\), variance is multiplied by \(a^2\), not \(a\); standard deviation is multiplied by \(|a|\).
- Adding a fixed amount to the standard deviation. A constant shift changes the mean but leaves deviations, variance, and standard deviation unchanged.
- Mixing units. Variance is in squared loss units, while standard deviation is in the original loss units.
- Using dispersion as a measure of absolute level. Compare means as well when the question concerns expected loss, and recognize that a fixed loss can have zero variance.
Check Your Understanding
For each discrete model, verify that its probabilities sum to one before calculating. Round standard deviations to cents where needed.
- A hypothetical claim loss is \(\$0\) or \(\$400\), each with probability \(0.50\). Find its mean, variance, and standard deviation. (Answer: mean \(\$200\); variance \(40{,}000\) dollars squared; standard deviation \(\$200.00\).)
- A hypothetical loss \(X\) is \(\$100\) with probability \(0.20\) and \(\$500\) with probability \(0.80\). Find \(E[X]\), \(\operatorname{Var}(X)\), and \(\sigma_X\). (Answer: mean \(\$420\); variance \(25{,}600\) dollars squared; standard deviation \(\$160.00\).)
- A loss \(X\) is \(\$100\) or \(\$900\), each with probability \(0.50\), and \(Y=2X+\$1{,}000\). Find the mean and standard deviation of \(Y\). (Answer: mean \(\$2{,}000\); standard deviation \(\$800.00\).)
- Contract P has loss \(\$1{,}000\) or \(\$5{,}000\), each with probability \(0.50\); contract Q always has loss \(\$2{,}500\). Find each contract's mean, variance, and standard deviation. Which has the higher expected loss? (Answer: P has mean \(\$3{,}000\), variance \(4{,}000{,}000\) dollars squared, and standard deviation \(\$2{,}000\); Q has mean \(\$2{,}500\), variance \(0\), and standard deviation \(\$0\); P has the higher expected loss.)
- A loss \(X\) has mean \(\$1{,}200\) and variance \(360{,}000\) dollars squared. Define \(Y=0.5X+\$700\). Find the mean, variance, and standard deviation of \(Y\). (Answer: mean \(\$1{,}300\); variance \(90{,}000\) dollars squared; standard deviation \(\$300.00\).)