Great job with lesson 1.2! You have now learned how to categorize data. The next task is to summarize it. Suppose you have a list of 20,000 claim amounts from last year. You cannot be asked to provide all 20,000 numbers; you must have one representative. Measures of central tendency come in here. Let’s look at the three most significant ones.
The most widely used measure is the mean. The mean is the sum of all the values in the data set divided by the total number of values (n). It is also known as the arithmetic average. The mean is denoted as x̄.
x̄ = (Σ xᵢ) / n
Where: x̄ = sample mean, Σxᵢ = sum of all values, n = number of values.
Note: In reality, actuaries tend to use samples (a subset of all policies). We denote x̄ for sample mean and μ for population mean.
Example: An actuary records the following five claim amounts (in $): 1,000; 2,000; 1,500; 500; 3,000.
x̄ = (1000 + 2000 + 1500 + 500 + 3000) / 5 = 8000 / 5 = 1600
The mean claim amount is $1,600.
Adding one very large claim of $50,000 to the above list would cause a drastic increase in the mean. This is an actual issue in insurance, where huge catastrophic claims can skew the average.
The median is the value in the middle when the data is sorted. Data can be sorted in either ascending or descending order. It is also very helpful in insurance analysis because it is not affected by extreme values (outliers).
Median = value at position (n + 1) / 2
Median = (value at n / 2 + value at (n / 2) + 1) / 2
Examples:
When is it better to use the median? In insurance, when there are a few large claims (outliers) in the data, the median provides a more accurate view of the typical claim than the mean.
The mode is the most common value in a dataset. A dataset may contain a single mode (unimodal), two modes (bimodal), or three modes (trimodal). The term "multi-modal" refers to a dataset with more than one mode. When all values appear only once, the dataset has no mode.
Example: Policy durations (in years): 1, 2, 2, 3, 4, 4, 4, 5. The mode is 4 years because it occurs 3 times, more than the other values.
In actuarial use, mode can help identify the most frequent type of claim, the most common policy tenure, or the most frequently occurring risk category.
In most practical actuarial scenarios, not all data points are equally important. When this happens we use a weighted mean, where each value has a weight based on its significance or frequency.
x̄w = (Σ wᵢ xᵢ) / (Σ wᵢ)
Example: An insurance portfolio has three types of policies:
| Policy type | Average claim | Number of policies |
|---|---|---|
| Type A | $500 | 100 |
| Type B | $800 | 200 |
| Type C | $1,200 | 50 |
x̄w = (500×100 + 800×200 + 1200×50) / (100 + 200 + 50) = 270,000 / 350 ≈ $771.43
The weighted mean claim is $771.43, which is more meaningful than the simple average of $500, $800, and $1,200 = $833.33. The simple average treats each policy type equally and ignores the number of policies in each category.
In this lesson you have learned about the three measures: mean, median, and mode. You also studied the weighted mean, which is helpful in actuarial work when policies vary in size and frequency.
In the coming lesson, we will look at how spread out the data is. This is called measures of dispersion. Just knowing the center point is not enough. As an actuary, you need to understand how much the data varies.
Let's move on!