What is Variance?

The Story That Explains Varianceवह कहानी जो Variance समझाती है一个讲清方差的故事

Imagine two pizza delivery services in your city — QuickSlice and PizzaPal. Both say their average delivery time is 30 minutes. You order from both for ten days and write down the times.

अपने शहर की दो pizza delivery services की कल्पना करें — QuickSlice और PizzaPal. दोनों कहती हैं कि उनका औसत delivery time 30 minutes है। आप दस दिन दोनों से order करते हैं और समय लिखते हैं।

想象你所在城市有两家披萨外卖 —— QuickSlice 和 PizzaPal。两家都说平均送达时间是 30 分钟。 你连续十天从两家各点一次，并记下送达时间。

Both have a mean of exactly 30 minutes. But QuickSlice always arrives within 28–32 minutes, while PizzaPal arrives anywhere from 5 to 65 minutes. The mean cannot show this difference. Variance can.

दोनों का mean ठीक 30 minutes है। पर QuickSlice हमेशा 28–32 मिनट में आता है, जबकि PizzaPal 5 से 65 मिनट में कभी भी आ सकता है। Mean यह अंतर नहीं दिखा सकता। Variance दिखा सकता है।

两家的均值都正好是 30 分钟。但 QuickSlice 总是在 28–32 分钟送到，而 PizzaPal 可能在 5 到 65 分钟之间的任何时间送到。 均值看不出这个差别。方差可以。

Variance in Pictures — Understand It Before the Formulaतस्वीरों में Variance — Formula से पहले समझें图解方差 — 先看懂，再看公式

Darts: both players aim at the centre and both are "right on average", but one is far more consistent. Variance measures that consistency.
Squares: draw a square from each point to the mean. Its side is the distance, its area is the squared distance. Variance is (roughly) the average area of these squares — big squares mean big spread.

Darts: दोनों खिलाड़ी केंद्र पर निशाना लगाते हैं और दोनों "औसतन सही" हैं, पर एक कहीं ज़्यादा consistent है। Variance इसी consistency को मापता है।
Squares: हर point से mean तक एक square बनाएँ। उसकी भुजा दूरी है, उसका क्षेत्रफल squared दूरी है। Variance (लगभग) इन squares का औसत क्षेत्रफल है — बड़े squares मतलब ज़्यादा फैलाव।

飞镖：两位选手都瞄准中心，平均来看都“打中了”，但其中一位稳定得多。方差衡量的正是这种稳定性。
正方形：从每个点到均值画一个正方形，边长是距离，面积就是距离的平方。方差（大致）就是这些正方形的平均面积 —— 正方形越大，越分散。

The FormulaFormula公式

import numpy as np

# Why divide by n - 1? Draw many small samples from a population with a known variance.
rng = np.random.default_rng(42)
population = rng.normal(30, 10, 1_000_000)      # true variance ≈ 100
print(f"True population variance: {population.var():.1f}")

samples = rng.choice(population, size=(100_000, 5))   # 100,000 samples of size n = 5
print(f"Average of variances dividing by n    : {samples.var(axis=1, ddof=0).mean():.1f}")
print(f"Average of variances dividing by n - 1: {samples.var(axis=1, ddof=1).mean():.1f}")

With n = 5, dividing by n gives on average 80 instead of 100 — exactly (n − 1)/n = 4/5 of the truth. Dividing by n − 1 gives 100. The bigger the sample, the smaller this gap.

n = 5 पर, n से भाग देने पर औसतन 100 की जगह 80 मिलता है — ठीक (n − 1)/n = 4/5 हिस्सा। n − 1 से भाग देने पर 100 मिलता है। Sample जितना बड़ा, यह gap उतना छोटा।

n = 5 时，除以 n 平均得到 80 而不是 100 —— 正好是真值的 (n − 1)/n = 4/5。除以 n − 1 得到 100。样本越大，这个差距越小。

Step-by-Step CalculationStep-by-Step गणना分步计算

Let us calculate the variance for QuickSlice: [29, 31, 30, 28, 32, 30, 29, 31, 30, 30] — mean = 30

चलिए QuickSlice का variance निकालें: [29, 31, 30, 28, 32, 30, 29, 31, 30, 30] — mean = 30

我们来计算 QuickSlice 的方差：[29, 31, 30, 28, 32, 30, 29, 31, 30, 30] —— 均值 = 30

Now PizzaPal: [10, 55, 8, 62, 15, 48, 20, 65, 12, 5] — mean = 30

अब PizzaPal: [10, 55, 8, 62, 15, 48, 20, 65, 12, 5] — mean = 30

再看 PizzaPal：[10, 55, 8, 62, 15, 48, 20, 65, 12, 5] —— 均值 = 30

Python ImplementationPython ImplementationPython 实现

Manual calculationहाथ से गणना手动计算

quickslice = [29, 31, 30, 28, 32, 30, 29, 31, 30, 30]
pizzapal   = [10, 55,  8, 62, 15, 48, 20, 65, 12,  5]

def sample_variance(data):
    n    = len(data)
    mean = sum(data) / n
    return sum((x - mean) ** 2 for x in data) / (n - 1)

print(f"QuickSlice variance: {sample_variance(quickslice):.2f}")
print(f"PizzaPal variance:   {sample_variance(pizzapal):.2f}")
print(f"PizzaPal is {sample_variance(pizzapal) / sample_variance(quickslice):.0f}x more variable")

Using the statistics modulestatistics module से使用 statistics 模块

import statistics

quickslice = [29, 31, 30, 28, 32, 30, 29, 31, 30, 30]
pizzapal   = [10, 55,  8, 62, 15, 48, 20, 65, 12,  5]

# Sample variance (divides by n - 1)
print(statistics.variance(quickslice))
print(statistics.variance(pizzapal))

# Population variance (divides by n)
print(statistics.pvariance(quickslice))
print(statistics.pvariance(pizzapal))

Using NumPy and pandasNumPy और pandas से使用 NumPy 和 pandas

import numpy as np
import pandas as pd

pizzapal   = [10, 55,  8, 62, 15, 48, 20, 65, 12,  5]

# NumPy: ddof=0 (population) is the DEFAULT
print(np.var(pizzapal))            # divides by n
print(np.var(pizzapal, ddof=1))    # divides by n - 1

# pandas: ddof=1 (sample) is the DEFAULT — the opposite of NumPy!
s = pd.Series(pizzapal)
print(s.var())                     # divides by n - 1
print(s.var(ddof=0))               # divides by n

Two Things to Watch: Squared Units and Outliersदो बातों का ध्यान रखें: Squared Units और Outliers需要注意的两件事：平方单位与离群点

Variance is powerful, but its unit is squared. If your data is in minutes, variance is in minutes squared, which is hard to picture.

Variance ताक़तवर है, पर इसकी unit squared होती है। अगर data minutes में है, तो variance minutes squared में होगा, जिसकी कल्पना करना मुश्किल है।

方差很有用，但它的单位是平方。数据以分钟为单位，方差的单位就是平方分钟，很难想象。

Outliers: one value can take overOutliers: एक value हावी हो सकती है离群点：一个值就能主导一切

import numpy as np

data = [29, 31, 30, 28, 32, 30, 29, 31, 30, 30]   # QuickSlice
print(f"Original          : var = {np.var(data, ddof=1):8.2f}   sd = {np.std(data, ddof=1):6.2f}")

# One late delivery: 30 minutes becomes 90 minutes
late = data[:-1] + [90]
print(f"One late delivery : var = {np.var(late, ddof=1):8.2f}   sd = {np.std(late, ddof=1):6.2f}")

# Units: minutes → seconds multiplies every value by 60
secs = [x * 60 for x in data]
print(f"In seconds        : var = {np.var(secs, ddof=1):8.2f}   sd = {np.std(secs, ddof=1):6.2f}")

# Adding a constant (+5 minutes to every order) does not change the spread
plus5 = [x + 5 for x in data]
print(f"Every order +5 min: var = {np.var(plus5, ddof=1):8.2f}   sd = {np.std(plus5, ddof=1):6.2f}")

One late delivery (90 minutes instead of 30) raises the variance from 1.33 to 361 — about 270 times. Converting minutes to seconds multiplies the variance by 60² = 3,600 (1.33 → 4,800), while adding 5 minutes to every order changes nothing.

एक देर वाली delivery (30 की जगह 90 मिनट) variance को 1.33 से 361 कर देती है — लगभग 270 गुना। Minutes को seconds में बदलने से variance 60² = 3,600 से गुणा हो जाता है (1.33 → 4,800), जबकि हर order में 5 मिनट जोड़ने से कुछ नहीं बदलता।

一次迟到的配送（90 分钟而不是 30 分钟）把方差从 1.33 推到 361 —— 约 270 倍。把分钟换成秒，方差乘以 60² = 3,600（1.33 → 4,800）；而每单都加 5 分钟，方差完全不变。

Where Variance Is Used in Data ScienceData Science में Variance कहाँ इस्तेमाल होता है方差在数据科学中的应用

Golden RulesGolden Rules黄金法则