Calculate Variance of a Dataset in Rust

Understanding the spread or dispersion of data is crucial in statistics. Variance is a key measure that quantifies this spread by indicating how far each number in a set is from the mean (average) and thus from every other number in the set. Implementing variance calculation in Rust will test your ability to handle numerical data, perform floating-point arithmetic, and structure your code effectively.

Problem Description

Your task is to implement a Rust function that calculates the statistical variance of a given dataset of numbers. Variance can be calculated in two primary ways: population variance (when your dataset represents the entire population) and sample variance (when your dataset is a sample of a larger population). For this challenge, you should implement the calculation for sample variance.

The formula for sample variance ($s^2$) is: $s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$

Where:

• $x_i$ is each individual data point.
• $\bar{x}$ is the mean (average) of the dataset.
• $n$ is the number of data points in the dataset.

Your function should:

1. Accept a slice of floating-point numbers (e.g., f64).
2. Calculate the mean of the dataset.
3. Calculate the sum of the squared differences between each data point and the mean.
4. Divide this sum by n-1 (where n is the number of elements in the dataset) to get the sample variance.
5. Handle edge cases gracefully.

Edge Cases to Consider:

• An empty dataset.
• A dataset with only one element.

Examples

Example 1:

Input: &[1.0, 2.0, 3.0, 4.0, 5.0]
Output: 2.5
Explanation:
1. Mean: (1.0 + 2.0 + 3.0 + 4.0 + 5.0) / 5 = 3.0
2. Squared differences from mean:
   (1.0 - 3.0)^2 = 4.0
   (2.0 - 3.0)^2 = 1.0
   (3.0 - 3.0)^2 = 0.0
   (4.0 - 3.0)^2 = 1.0
   (5.0 - 3.0)^2 = 4.0
3. Sum of squared differences: 4.0 + 1.0 + 0.0 + 1.0 + 4.0 = 10.0
4. Sample Variance: 10.0 / (5 - 1) = 10.0 / 4 = 2.5

Example 2:

Input: &[60.0, 58.0, 62.0, 59.0, 61.0]
Output: 2.5
Explanation:
1. Mean: (60.0 + 58.0 + 62.0 + 59.0 + 61.0) / 5 = 60.0
2. Squared differences from mean:
   (60.0 - 60.0)^2 = 0.0
   (58.0 - 60.0)^2 = 4.0
   (62.0 - 60.0)^2 = 4.0
   (59.0 - 60.0)^2 = 1.0
   (61.0 - 60.0)^2 = 1.0
3. Sum of squared differences: 0.0 + 4.0 + 4.0 + 1.0 + 1.0 = 10.0
4. Sample Variance: 10.0 / (5 - 1) = 10.0 / 4 = 2.5

Example 3: Edge Case - Dataset with One Element

Input: &[10.0]
Output: NaN (Not a Number) or an appropriate error/option type.
Explanation: For sample variance, the denominator is n-1. If n=1, then n-1=0, leading to division by zero. This scenario is undefined.

Example 4: Edge Case - Empty Dataset

Input: &[]
Output: NaN (Not a Number) or an appropriate error/option type.
Explanation: Variance is undefined for an empty dataset as there are no data points to calculate a mean or differences from.

Constraints

• The input will be a slice of f64 (double-precision floating-point numbers).
• The dataset can be empty, contain a single element, or contain multiple elements.
• Your function should return an f64 representing the sample variance. For cases where variance is undefined (empty or single-element dataset), it should return f64::NAN or use Rust's Option type to return None.
• Consider potential floating-point precision issues.
• Performance is not a primary concern for this challenge, but an efficient, idiomatic Rust solution is encouraged.

Notes

• You will likely need to calculate the mean as an intermediate step.
• Remember that the denominator for sample variance is n-1.
• For handling the undefined cases, returning f64::NAN is a common approach in numerical computing. Alternatively, you could define your function to return an Option<f64> where None signifies an undefined variance. Choose one consistent approach.
• Ensure your code is well-commented and follows Rust's idiomatic style.