Can Variance Be Negative? No (See Why)

In one word: No.

Why Variance Can’t Be Negative

The reason is that the way variance is calculated makes a negative result mathematically impossible.

Variance is the average squared deviation from the mean. Notice the word “squared”.

Why Variance Calculation Can’t Give Negative Result

To calculate variance, you:

  1. Take each observation (number) in the data set.
  2. Calculate the differences between the individual numbers and the mean of the data set. Some of these differences can be and – unless all the numbers are exactly the same – will be negative.
  3. Then you square each of the differences (multiply it by itself). This is precisely the moment which eliminates the possibility of variance (and therefore also standard deviation) being negative. Something (negative or positive number) squared is always a positive number, except zero squared which is still zero. You can’t get a negative number when squaring something.
  4. Then you find the average (mean) of all the squared numbers from the previous step. Because the squared deviations are all positive numbers or zeroes, their smallest possible mean is zero. It can’t be negative. This average of the squared deviations is in fact variance. Therefore variance can’t be negative.

Smallest Possible Variance Value

The smallest value variance can reach is exactly zero. This is when all the numbers in the data set are the same, therefore all the deviations from the mean are zero, all squared deviations are zero and their average (variance) is also zero.

If there are at least two numbers in a data set which are not equal, variance must be greater than zero.

Related pages

how to calculate intrinsic value of a stockultra short etfsample deviation calculatorexcel cdfleveraged etf listcalculating emaexample of stock indexfinding the median calculatorwhat is the cboe volatility indexfifo ending inventory calculators&p 500 etf listhow to calculate percent deviationpopulation standard deviation in excelstd dvharmonic mean formula examplerules for multiplying square rootsblack scholes puthow do i calculate variance in excelyahoo finance stock quotes in excelsharpe ratio formula exampleexcel function stdevbuffett 13fskew equationuses of skewnessarithmetic mean return formulaoptions profit and loss calculatorwhats arithmetichow to calculate standard deviation using a calculatorcall options for dummiesblack scholes formula excelsq root of 12finding covariance in excelhistorical volatility formulagreeks for optionsrsi indicator pdfcontinuous compounded returninterpreting skewness and kurtosishow to calculate weighted average in excel with percentagesfxa etfhedge fund techniquesbarclays ipathlist of all etfss&p 500 calculationetf short s&phow is the dow jones calculatedvix options chainannualized standard deviation excelshorting putsdelta vega gammacoefficient of kurtosisitm callvix trading strategiesfutures tick sizecalculate population meanmacd emaapple stock history calculatornotional value of option3x etf listprotective put payoffkurtosis meaningquote vixhistorical volatility chartshort straddle option strategycontango chartarithmetic mean formula in statisticsmsft stock historyeuro stoxx 50 wikiblack scholes option pricing calculatorstdev.svariance excel formuladefine kurtosis and skewnessinferential statistics calculatorsec 13f liststandard deviancecboe holidays