Variance vs Standard Deviation

Variance is a method to find or obtain the measure of how the variables differ from one another. In contrast, standard deviation shows us how the data set or the variables differ from the mean or the average value of the data set.

Variance helps to find the distribution of data in a population from a mean, and standard deviation also helps to know the data distribution in a population. Still, standard deviation gives more clarity about the deviation of data from a mean.

Below are the formulas of variance and standard deviation.

Whereas

• σ2 is variance
• X is the variable
• μ is mean
• N is the total number of variables.

Standard Deviation is the square root of the variance.

Imagine a game that works like this:

Case-1

You draw one card from an ordinary deck of cards:

1. If you draw 7, you will win ₹ 2,000/-
2. If you choose another card except 7, you will give ₹ -100 /-

Case-2

1. If you draw 7, you will win ₹1,22,000/-
2. If you choose another card except 7, you will give ₹10,100/-

Assume that you played a game 52,000 times.

For a discrete random variable, the variance is

Where Pi is the probability of the outcome.

The average profit per game for both cases is ₹61.54. Which game would you like to play well? A certain instrument helps to make the decision, i.e., we have to calculate variance and standard deviation.

We need to measure the normal deviation from the expected value, and one common measure is variance. The variance of Case -1 is much less than that of Case -2, which means that the data in Case -2 spread the average value, i.e., ₹64.54, so the Case-1 Game is less risky than the Case-2 Game.

In finance, we talked about the volatility of stocks, meaning that large shocks follow large shocks in financial assets' return, and small shocks in financial assets' return tend to be followed by small shocks.

Let's see the top differences between Variance vs. Standard Deviation.

The key differences are as follows:

• The variance gives an approximate idea of data volatility. 68% of values are between +1 and -1 standard deviation from the mean. That means Standard Deviation gives more details.
• One uses variance to know about the planned and actual behavior with a certain degree of uncertainty. One uses standard deviation for the statistical test to know the relationship between two sets of variables.
• Variance measures data distribution in a population around the central value. Standard deviation measures the distribution of data relative to the central value.
• Sum of two variances (var(A + B ) ≥ var(A) + var(B ) .therefore variance is not coherent. Sum of two standard deviations sd(A + B ) ≤ sd(A) + sd(B ), so the standard deviation is coherent. It gives the idea of the skewness of the data. The skewness value of symmetric distribution lies between -1>0>1.
• The geometric mean is more sensitive to variance than the arithmetic means. One uses a geometric standard deviation to find the bounds of the confidence interval in a population.

Example of determination of oil pricing:

• What will the oil price be in one year? Not one price estimate. A probability of it being low or high
• Variations in delays, variation in scrap/repair, variation in-flight hours actual vs. planned
• Does the next value move back to the average, or does it only depend on the last value?
• Does the next amount of demand move back to average, or does it only depend on the last amount of demand?

A forecasted amount for a number of periods (oil price for 20 months)

*The graph is made by considering one year's data. However, the data shown in the table is only for 6 months. Therefore, the randomly chosen value may not be the same as market data on oil prices.

Both variance and standard deviation measure the spread of data from its mean point. It helps determine the risk in the mutual fund investment, stock, etc. In addition, it is a useful tool used in weather forecasting for temperature variation during the period and in Monte Carlo Simulation to assess the risk of the project.