Variance is a statistical measure used to quantify the dispersion of numbers within a dataset. It indicates how far each value deviates from the mean and one another. Variance is widely applicable across various fields, including project management, in which it can help assess process volatility.
To calculate variance, subtract the mean from each number in the dataset, square the resulting differences, and then divide the sum of these squares by the total number of values in the dataset.
Variance Formula
The variance formula measures the dispersion of a set of values. It is calculated differently for a population and a sample:
Population Variance
Population variance measures the spread of an entire population’s data. It is calculated by finding the average of the squared differences between each data point and the population mean.
Formula:
Where N is the total number of data points in the population.
Sample Variance
Sample variance estimates the spread of a subset (sample) of data taken from a larger population. To account for the smaller size and better represent the population, the formula divides by n -1 instead of n.
Formula:
Where n is the number of data points in the sample.
Population variance uses the actual population size divided by N. Sample variance adjusts for a smaller dataset by dividing by n -1 to avoid underestimating the entire population’s true variance.
How to Calculate Variance
Here’s a step-by-step process to calculate variance:
Step 1: Find the Mean (i.e., Average)
Add all the numbers in your data set. Divide the sum by the number of numbers in the data set.
Step 2: Subtract the Mean from Each Data Point
Subtract the mean from each number in the data set. These are called “deviations.”
Deviation = Data point – Mean
Step 3: Square Each Deviation
Take each deviation (the result from Step 2) and square it (multiply it by itself). This removes negative signs and ensures all differences are positive.
Squared Deviation = (Deviation)2
Step 4: Find the Average of the Squared Deviations
Add up all the squared deviations from Step 3. Divide by the number of data points (for sample variance, divide by n – 1; for population variance, divide by n.
Variance = (Sum of squared deviations)/[n or (n – 1) for sample]
Variance Example
Let’s go through a simple example of variance:
Imagine you have the following test scores for 5 students: 3, 6, 9, 12, and 15.
Step 1. Find the Mean (i.e., Average)
Mean = (3+6+9+12+15)/5 = 9
Step 2. Subtract the Mean from Each Number and Square the Result
For 3: (3 – 9)2 = 36
For 6: (6 – 9) 2 = 9
For 9: (9 – 9) 2 = 0
For 12: (12 – 9) 2 = 9
For 15: (15 – 9) 2 =36
Step 3. Find the Average of the Squared Differences
Variance = (36+9+0+9+36)/5 = 18
The variance of the given dataset is 18.
Variance Applications
Some practical applications of variance are:
1. Finance
- Risk Assessment: Investors use variance to understand the volatility of stock prices or investment returns. A higher variance indicates that returns are more spread out and thus riskier. For example, if one stock has a high variance as compared to another, its price fluctuates more, thus indicating more risk.
2. Quality Control
- Product Consistency: In manufacturing, variance can help you monitor product consistency. If the variance in product dimensions is high, it suggests inconsistency in the production process. For example, a car manufacturer may track the variance in the thickness of car parts. A high variance would indicate the need for tighter quality control.
3. Sports
- Player Consistency: Coaches and analysts use variance to assess how consistent an athlete’s performance is over time. A lower variance indicates consistent performance, while a high variance may indicate erratic or inconsistent results. For example, if a basketball player’s score has low variance, they consistently score similar points in each game.
Variance Vs Standard Deviation
Variance and standard deviation measure the data spread in a set, but they do so differently.
Variance calculates the average of the squared differences between each data point and the mean. By squaring the differences, variance eliminates negative values but also results in units that are different from the original data. For example, if your data is in meters, the variance will be in square meters, making it difficult to interpret.
Standard deviation is simply the square root of the variance, bringing the result back to the original units of the data. This makes it easier to understand in practical terms.
The formula for standard deviation is:
Why is Standard Deviation More Useful?
- Interpretability: Standard deviation is in the same units as the original data, which makes it easier to relate to real-world applications.
- Context: Standard deviation directly shows how much, on average, data points deviate from the mean, which makes it easier to compare variability across different datasets.
While variance and standard deviation measure variability, standard deviation is more practical for interpretation and comparison.
Advantages and Disadvantages of Using Variance
Pros
- Displays Data Spread: Variance can help you understand how much the numbers in a data set differ from the average. If the variance is high, then the numbers are more spread out.
- Compares Data Sets: It is useful when comparing two or more data sets to see which has more variation or inconsistencies.
- Used in Other Important Calculations: Variance is a key part of more advanced methods (e.g., standard deviation), which are used in many fields (e.g., statistics, finance, and science).
Cons
- Difficult to interpret: Since variance uses squared differences, the units of variance differ from the original data, which makes it harder to understand in real-world terms.
- Sensitive to Extreme Values: Variance can be heavily affected by high or low numbers in a data set, which may distort the results.
- Complex Manual Calculations: Calculating variance can be tricky and time-consuming for beginners, as compared to simpler measures (e.g., range).
Frequently Asked Questions
1. Can Variance Be Negative?
No, variance cannot be negative because it is the average of squared differences, and squaring any number results in a non-negative value.
2. Are Variance and Standard Deviation the Same?
No, they are related but different. Variance measures the average squared deviations from the mean, while standard deviation is the square root of the variance.
3. Why is Variance So Important?
Variance is important because it quantifies the data spread, helping identify how much values deviate from the average, which is crucial for assessing risk, consistency, and variability.
4. What is a Good Variance?
A “good” variance depends on the context. Generally, low variance is preferred when consistency is desired, while high variance might indicate significant variability or risk.
5. When Should I Use Variance?
Use variance to measure the spread or dispersion of data points in a dataset—especially when comparing the variability between multiple datasets or assessing risks.
Summary
Variance measures the differences between each value and the average to help you understand how spread out the data is. It is an important tool in many fields (e.g., finance, quality control, and project management) because it allows you to assess consistency and risk.
However, since variance uses squared units, it can be hard to interpret directly. That is why standard deviation—the square root of variance—is often more practical for real-world use. Understanding variance can help you make informed decisions by showing how much things can change or deviate from your expectations.
Further Readings:
References:
This topic is important from a PMP exam point of view.
I am Mohammad Fahad Usmani, B.E. PMP, PMI-RMP. I have been blogging on project management topics since 2011. To date, thousands of professionals have passed the PMP exam using my resources.
