Population and Sample Variance

/**
 * Fintech Math Bootcamp · Lesson 045 of 120
 * Population and Sample Variance
 * Module 05: Dispersion, Shape, and Robust Statistics
 *
 * Scenario: Variance as a ledger of squared deviations
 * Rule:     population: Σd²/n; sample: Σd²/(n−1)
 *
 * Try it:   Why can two correct variance implementations disagree here?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson045() {
  const x = [1,2,2,4,9];
  if (x.length < 2) throw new Error("Sample variance needs n >= 2");
  const mean = x.reduce((s,v)=>s+v,0)/x.length;
  const ss = x.reduce((s,v)=>s+(v-mean)**2,0);
  const result = {ss, population: ss/x.length,
    sample: ss/(x.length-1)};
  return result;
}

export const checkedResult = {"ss":41.2,"population":8.24,"sample":10.3};

// Run this file directly: npx tsx lessons/05-dispersion-shape-and-robust-statistics/045-population-and-sample-variance.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson045(), null, 2));
}

# Fintech Math Bootcamp · Lesson 045 of 120
# Population and Sample Variance
# Module 05: Dispersion, Shape, and Robust Statistics
#
# Scenario: Variance as a ledger of squared deviations
# Rule:     population: Σd²/n; sample: Σd²/(n−1)
#
# Try it:   Why can two correct variance implementations disagree here?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.

import json


def lesson045() -> dict:
    x = [1, 2, 2, 4, 9]
    if len(x) < 2:
        raise ValueError("Sample variance needs n >= 2")
    mean = sum(x) / len(x)
    ss = sum((v - mean) ** 2 for v in x)
    return {"ss": ss, "population": ss / len(x), "sample": ss / (len(x) - 1)}


if __name__ == "__main__":
    print(json.dumps(lesson045(), indent=2))

/**
 * Fintech Math Bootcamp · Lesson 045 of 120
 * Population and Sample Variance
 * Module 05: Dispersion, Shape, and Robust Statistics
 *
 * Scenario: Variance as a ledger of squared deviations
 * Rule:     population: Σd²/n; sample: Σd²/(n−1)
 *
 * Try it:   Why can two correct variance implementations disagree here?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

import java.util.ArrayList;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;

public class Main {
    static Map<String, Object> lesson045() {
        double[] x = {1, 2, 2, 4, 9};
        if (x.length < 2) throw new IllegalArgumentException("Sample variance needs n >= 2");
        double mean = 0;
        for (double v : x) mean += v;
        mean /= x.length;
        double ss = 0;
        for (double v : x) ss += (v - mean) * (v - mean);
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("ss", ss);
        result.put("population", ss / x.length);
        result.put("sample", ss / (x.length - 1));
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson045(), ""));
    }

    // --- Minimal JSON printer: maps keep insertion order, 2-space indent. ---

    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return "\"" + value + "\"";
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        if (value instanceof double[]) {
            List<Object> items = new ArrayList<Object>();
            for (double v : (double[]) value) items.add(v);
            return toJson(items, indent);
        }
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                sb.append(inner).append('"').append(entry.getKey()).append("\": ")
                  .append(toJson(entry.getValue(), inner))
                  .append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append('}').toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner))
              .append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append(']').toString();
    }

    static String formatNumber(double v) {
        if (Double.isNaN(v) || Double.isInfinite(v)) return "null";
        if (v == Math.rint(v) && Math.abs(v) < 1e15) return Long.toString((long) v);
        return Double.toString(v);
    }
}

// Fintech Math Bootcamp · Lesson 045 of 120
// Population and Sample Variance
// Module 05: Dispersion, Shape, and Robust Statistics
//
// Scenario: Variance as a ledger of squared deviations
// Rule:     population: Σd²/n; sample: Σd²/(n−1)
//
// Try it:   Why can two correct variance implementations disagree here?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

package main

import (
	"encoding/json"
	"errors"
	"fmt"
	"os"
)

// Variance holds the sum of squared deviations and both variance denominators.
type Variance struct {
	SS         float64 `json:"ss"`
	Population float64 `json:"population"`
	Sample     float64 `json:"sample"`
}

func lesson045() (Variance, error) {
	x := []float64{1, 2, 2, 4, 9}
	if len(x) < 2 {
		return Variance{}, errors.New("Sample variance needs n >= 2")
	}
	n := float64(len(x))
	mean := 0.0
	for _, v := range x {
		mean += v
	}
	mean /= n
	ss := 0.0
	for _, v := range x {
		ss += (v - mean) * (v - mean)
	}
	return Variance{SS: ss, Population: ss / n, Sample: ss / (n - 1)}, nil
}

func main() {
	result, err := lesson045()
	if err != nil {
		fmt.Fprintln(os.Stderr, err)
		os.Exit(1)
	}
	out, _ := json.MarshalIndent(result, "", "  ")
	fmt.Println(string(out))
}

/**
 * Fintech Math Bootcamp · Lesson 045 of 120
 * Population and Sample Variance
 * Module 05: Dispersion, Shape, and Robust Statistics
 *
 * Scenario: Variance as a ledger of squared deviations
 * Rule:     population: Σd²/n; sample: Σd²/(n−1)
 *
 * Try it:   Why can two correct variance implementations disagree here?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

#include <charconv>
#include <cmath>
#include <iostream>
#include <numeric>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

struct Json {
    enum class Kind { Null, Number, Text, Array, Object };
    Kind kind = Kind::Null;
    double number = 0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values
};

Json num(double v) { Json j; j.kind = Json::Kind::Number; j.number = v; return j; }
Json str(const std::string& s) { Json j; j.kind = Json::Kind::Text; j.text = s; return j; }
Json arr(const std::vector<Json>& values) { Json j; j.kind = Json::Kind::Array; j.items = values; return j; }

Json arr(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(num(v));
    return arr(items);
}

Json obj(const std::vector<std::pair<std::string, Json>>& fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (const auto& [key, value] : fields) { j.keys.push_back(key); j.items.push_back(value); }
    return j;
}

std::string formatNumber(double v) {
    if (!std::isfinite(v)) return "null";
    char buf[64];
    auto end = std::to_chars(buf, buf + sizeof buf, v).ptr;  // shortest round-trip form
    return std::string(buf, end);
}

void writeJson(std::ostream& out, const Json& j, const std::string& indent) {
    switch (j.kind) {
    case Json::Kind::Null: out << "null"; return;
    case Json::Kind::Number: out << formatNumber(j.number); return;
    case Json::Kind::Text: out << '"' << j.text << '"'; return;
    default: break;
    }
    bool isObject = j.kind == Json::Kind::Object;
    if (j.items.empty()) { out << (isObject ? "{}" : "[]"); return; }
    std::string inner = indent + "  ";
    out << (isObject ? "{\n" : "[\n");
    for (size_t i = 0; i < j.items.size(); ++i) {
        out << inner;
        if (isObject) out << '"' << j.keys[i] << "\": ";
        writeJson(out, j.items[i], inner);
        out << (i + 1 < j.items.size() ? ",\n" : "\n");
    }
    out << indent << (isObject ? '}' : ']');
}

// --- Lesson ---

Json lesson045() {
    const std::vector<double> x = {1, 2, 2, 4, 9};
    if (x.size() < 2) throw std::invalid_argument("Sample variance needs n >= 2");
    double n = static_cast<double>(x.size());
    double mean = std::accumulate(x.begin(), x.end(), 0.0) / n;
    double ss = 0;
    for (double v : x) ss += (v - mean) * (v - mean);
    return obj({{"ss", num(ss)}, {"population", num(ss / n)}, {"sample", num(ss / (n - 1))}});
}

int main() {
    writeJson(std::cout, lesson045(), "");
    std::cout << '\n';
}

// Fintech Math Bootcamp · Lesson 045 of 120
// Population and Sample Variance
// Module 05: Dispersion, Shape, and Robust Statistics
//
// Scenario: Variance as a ledger of squared deviations
// Rule:     population: Σd²/n; sample: Σd²/(n−1)
//
// Try it:   Why can two correct variance implementations disagree here?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

#[allow(dead_code)]
enum Json {
    Null,
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

#[allow(dead_code)]
fn nums(values: &[f64]) -> Json {
    Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
}

#[allow(dead_code)]
fn obj(fields: Vec<(&str, Json)>) -> Json {
    Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
}

fn format_number(v: f64) -> String {
    if !v.is_finite() {
        return "null".to_string();
    }
    if v.fract() == 0.0 && v.abs() < 1e15 {
        return format!("{}", v as i64);
    }
    format!("{:?}", v) // shortest round-trip form
}

impl Json {
    fn render(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Num(v) => format_number(*v),
            Json::Str(s) => format!("\"{}\"", s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let lines: Vec<String> = items.iter().map(|v| format!("{}{}", inner, v.render(&inner))).collect();
                format!("[\n{}\n{}]", lines.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let lines: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}\"{}\": {}", inner, k, v.render(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", lines.join(",\n"), indent)
            }
        }
    }
}

// --- Lesson ---

fn lesson045() -> Result<Json, String> {
    let x: [f64; 5] = [1.0, 2.0, 2.0, 4.0, 9.0];
    if x.len() < 2 {
        return Err("Sample variance needs n >= 2".to_string());
    }
    let n = x.len() as f64;
    let mean = x.iter().sum::<f64>() / n;
    let ss: f64 = x.iter().map(|&v| (v - mean).powi(2)).sum();
    Ok(obj(vec![
        ("ss", Json::Num(ss)),
        ("population", Json::Num(ss / n)),
        ("sample", Json::Num(ss / (n - 1.0))),
    ]))
}

fn main() {
    match lesson045() {
        Ok(result) => println!("{}", result.render("")),
        Err(message) => {
            eprintln!("{}", message);
            std::process::exit(1);
        }
    }
}

/**
 * Fintech Math Bootcamp · Lesson 045 of 120
 * Population and Sample Variance
 * Module 05: Dispersion, Shape, and Robust Statistics
 *
 * Scenario: Variance as a ledger of squared deviations
 * Rule:     population: Σd²/n; sample: Σd²/(n−1)
 *
 * Try it:   Why can two correct variance implementations disagree here?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/dispersion-shape-and-robust-statistics/population-and-sample-variance/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

using System;
using System.Linq;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson045(), options));

static object Lesson045()
{
    double[] x = { 1, 2, 2, 4, 9 };
    if (x.Length < 2) throw new ArgumentException("Sample variance needs n >= 2");
    double mean = x.Average();
    double ss = x.Sum(v => (v - mean) * (v - mean));
    return new { ss, population = ss / x.Length, sample = ss / (x.Length - 1) };
}