quickconverts.org
☰

Variance Formula

Image related to variance-formula

Understanding the Variance Formula: A Simple Guide



Understanding data is crucial in many fields, from finance and science to marketing and social sciences. One of the most important measures of data dispersion, or spread, is variance. It tells us how far individual data points are spread out from the mean (average). A high variance indicates data points are widely scattered, while a low variance means they are clustered closely around the mean. This article will demystify the variance formula, making it accessible to everyone.

1. What is Variance?



Variance measures the average squared deviation from the mean. Why squared deviation? Simply summing the deviations from the mean will always result in zero, as positive and negative deviations cancel each other out. Squaring the deviations ensures all values are positive, providing a meaningful measure of dispersion. The result is then averaged to provide a single, representative value of spread. Larger variance indicates greater variability in the data set.

2. The Population Variance Formula



When you have data for the entire population (e.g., the height of every student in a specific school), you use the population variance formula:

σ² = Σ(xi - μ)² / N

Where:

σ² (sigma squared) represents the population variance.
Σ (sigma) denotes summation (adding up all values).
xi represents each individual data point.
μ (mu) represents the population mean (average).
N represents the total number of data points in the population.

Let's break it down:

1. (xi - μ): This calculates the deviation of each data point (xi) from the population mean (μ).
2. (xi - μ)²: This squares each deviation, ensuring positive values.
3. Σ(xi - μ)²: This sums all the squared deviations.
4. Σ(xi - μ)² / N: This divides the sum of squared deviations by the total number of data points (N), providing the average squared deviation – the variance.

Example: Imagine the heights (in cm) of all five students in a class are: 160, 165, 170, 175, 180. The mean (μ) is 170 cm. Calculating the variance:

1. Deviations: (-10, -5, 0, 5, 10)
2. Squared Deviations: (100, 25, 0, 25, 100)
3. Sum of Squared Deviations: 250
4. Variance (σ²): 250 / 5 = 50 cm²

3. The Sample Variance Formula



More often, we work with a sample of data (e.g., the height of a randomly selected group of students from a large school) to estimate the population variance. In this case, we use the sample variance formula:

s² = Σ(xi - x̄)² / (n - 1)

Where:

s² represents the sample variance.
x̄ (x-bar) represents the sample mean.
n represents the total number of data points in the sample.

Notice the denominator is (n - 1) instead of n. This is called Bessel's correction. It provides an unbiased estimator of the population variance. Using 'n' would underestimate the population variance, especially with small samples.

Example: Let's say we have a sample of three heights: 160, 165, 170. The sample mean (x̄) is 165 cm.

1. Deviations: (-5, 0, 5)
2. Squared Deviations: (25, 0, 25)
3. Sum of Squared Deviations: 50
4. Variance (s²): 50 / (3 - 1) = 25 cm²

4. Standard Deviation: The Square Root of Variance



While variance is a useful measure, its units are squared (cm² in our examples). To get a measure of dispersion in the original units, we calculate the standard deviation. The standard deviation is simply the square root of the variance:

Population Standard Deviation (σ) = √σ²
Sample Standard Deviation (s) = √s²

5. Key Takeaways



Variance measures the average squared deviation from the mean, indicating data spread.
The population variance formula uses 'N' while the sample variance formula uses '(n-1)' (Bessel's correction).
Standard deviation is the square root of the variance, providing a measure of spread in the original units.
High variance signifies greater variability, while low variance indicates data points cluster closely around the mean.

Frequently Asked Questions (FAQs)



1. Why do we square the deviations? Squaring ensures all values are positive, preventing positive and negative deviations from canceling each other out.

2. What is the difference between population and sample variance? Population variance uses data from the entire population, while sample variance uses data from a subset and includes Bessel's correction for unbiased estimation.

3. Why use (n-1) in the sample variance formula? This is Bessel's correction, which provides an unbiased estimate of the population variance, particularly crucial with smaller sample sizes.

4. What is the relationship between variance and standard deviation? Standard deviation is the square root of the variance, expressing the spread in the original units of measurement.

5. Can variance be negative? No, variance is always non-negative because it involves squaring the deviations. A variance of zero indicates all data points are identical.

Links:

Converter Tool

Conversion Result:

=

Note: Conversion is based on the latest values and formulas.

Formatted Text:

96f to c
frank william abagnale jr
480 lbs to kg
55kg in pounds
150 grams is how many ounces
intel i7 7700k transistor count
49 fahrenheit to celsius
13mm is what in inches
69 fahrenheit to celsius
hclo3 lewis structure
144 cm in ft
154 lbs to kg
24 ft to m
52 kg in pounds
123km in miles

Search Results:

为什么样本方差（sample variance）的分母是 n-1？ - 知乎 在统计实践中人们发现，偏差的产生，很多时候也是因为样本数据之间出现了 各种隐含的关联关系，降低了数据之间的独立性。 而解决的策略还很清晰，就是发现其中隐含的关联关系，然后进行校正。 让我们再回到样本方差（Sample Variance）的分母（n-1）上来。

如何理解管理会计中的Flexible-Budget？ - 知乎 Flexible—Budget， 翻译成中文就是弹性预算。 首先第一个问题，什么是弹性预算？ 企业预算体系中各种固定预算的编制是以一定的产销量为基础的。但企业内外部条件的变化往往使实际产销量和预计的产销水平产生较大差异，这样就削弱了以预计产销量为基础编制的固定预算应有的控制作用 …

covariance（协变）和 correlation（相关性）如何理解他们的区 … Covariance 是绝对值，体现了两组合之间绝对相关性的大小； Correlation 是在两组数据基础上的相对值，消除了数据组本身大小对相关性的影响（eliminate the effects of size)，着重描述其相对的相关性,从而使不同规模的数据组之间具有可比性和对照性。

为什么样本方差（sample variance）的分母是 n-1？ - 知乎 如果已知随机变量 的期望为 ，那么可以如下计算方差 ：

Realized Volatility不同数据频率 差异巨大 如何解读这一现象？ - 知乎 第四，不知道你用什么formula计算的realized volatility，你的2 day change是怎么定义的。 正确做法是，把两天的log price process每5min分一个点，然后相差再平方求和，这个是这两天的realized volatility。

怎么理解running mean和running variance？ - 知乎 Batch Normalization的running mean和running variance是什么？

机器学习中的 Bias（偏差）、Error（误差）、Variance（方差） … 首先看Variance的变化，还是举打靶的例子。 假设我把抢瞄准在10环，虽然每一次射击都有偏差，但是这个偏差的方向是随机的，也就是有可能向上，也有可能向下。

如何理解方差膨胀因子（Variance Inflation Factor，VIF）？ 那么我们要怎么找到特征之间的多重共线性呢，其中的一个方法，就是使用方差膨胀因子（Variance Inflation Factor，VIF），在了解 VIF 如何进行计算之前，需要先知道拟合优度的计算方法。

DeepSeek深度思考和联网搜索有什么区别？ - 知乎 5 Feb 2025 · DeepSeek为大家提供了：深度思考R1和联网搜索，两个功能强悍的按钮，但，在和知乎网友的交流过程中，我发现有很多人，不知道这两个强悍的工具如何搭配使用。今天就好好聊聊这个问题。 深度思考模式详解 深度思考模式就像是一个“超级大脑”，当你遇到复杂问题时，它会帮你仔细分析、多角度 ...

(Variance Swap)方差互换是什么？如何理解？ - 知乎 所以Variance Swap最核心的问题就是: 该怎么去定variance strike？ 理论上来说，fair variance strike 是应该等于 risk-neutral expected value of the return variance，所以当我们假设stock是diffusion process,利用Ito's lemma可以得到