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September 19, 2022

# sampling Is a sample covariance matrix always symmetric and positive definite? Cross Validated

Discover how the popular chi-square goodness-of-fit test works. @Zen’s answer plus @whuber’s comment to @Konstantin’s answer provide a complete proof. Nevertheless, I’ll rephrase the proof by trying to place more statistical emphasis. This equation doesn’t change if you switch the positions of $x$ and $y$. So variance is affected by outliers, and an extreme outlier can have a huge effect on variance .

Sample variance is a type of variance by means of which metrics are examined and quantified through a systemic process of any particular sample data. Different algebraic formulae are utilized for the analytical process. Variance is expressed in much larger units (e.g., meters squared). If there’s higher between-group variance relative to within-group variance, then the groups are likely to be different as a result of your treatment. If not, then the results may come from individual differences of sample members instead. The main idea behind an ANOVA is to compare the variances between groups and variances within groups to see whether the results are best explained by the group differences or by individual differences. Sure–I have answered this question in some detail at stats.stackexchange.com/a/3904. Briefly, there are infinitely many such functionals–but they must all asymptotically converge to the variance. Portfolio variance is the measurement of how the actual returns of a group of securities making up a portfolio fluctuate. Here’s a hypothetical example to demonstrate how variance works. Let’s say returns for stock in Company ABC are 10% in Year 1, 20% in Year 2, and −15% in Year 3. The differences between each return and the average are 5%, 15%, and −20% for each consecutive year.

An outlier can have a drastic effect on both mean and variance. Range is in linear units, while variance is in squared units. For example, when the mean of a data set is negative, the variance is guaranteed to be greater than the mean .

## Why Is Standard Deviation Often Used More Than Variance?

Another is the importance in decision theory of minimizing quadratic loss. The square root of the variance is called the Standard Deviation σ. Note that σ is the root mean squared of differences between the data points and the average. In general, the risk of an asset or a portfolio is measured in the form of the standard deviation of the returns, where standard deviation is the square root of variance. Variance weights outliers more heavily than data very near the mean due to the square. While in sample variance in addition to dividing the resultant value by the total number in the data set you also have to subtract one from the data set. The variance is calculated by taking the square of the standard deviation. When a square of any value is taken, either its positive or a negative value it always becomes a positive value.

“Squaring always gives a positive value, so the sum will not be zero.” and so does absolute values. Mean is in linear units, while variance is in squared units. Note that this also means the standard deviation will be greater than 1. The reason is that if a number is greater than 1, its square root will also be greater than 1.

The “unique solution” argument is quite weak, it really means there is more than one value well supported by the data. Additionally, penalisation of the coefficients, such as L2, will resolve the uniqueness problem, and the stability problem to a degree as well. Yes, as I stated, “if your population is normally distributed.”

Both accounting and finance faculty should help finance majors understand variance analysis from a practitioner’s standpoint. To make sure students understand the practitioner’s viewpoint, we use corporate business simulations that are more operationally focused, as opposed to being academic in tone. We prefer the squared differences when calculating a measure of dispersion because we can exploit the Euclidean distance, which gives us a better discriptive statistic of the dispersion. When there are more relatively extreme values, the Euclidean distance accounts for that in the statistic, whereas the Manhattan distance gives each measurement equal weight. Regardless of the distribution, the mean absolute deviation is less than or equal to the standard deviation. MAD understates the dispersion of a data set with extreme values, relative to standard deviation.

## Population vs. sample variance

A common one is about the sign of variance, so we’ll start there. All the products of deviations, then are added up altogether. Then the deviation of each value of x and y from their respective means, i.e. (xi – x) and (yi – y), respectively is calculated. Then, each value we get after the subtraction process, has to be squared . When the variance is zero, then the same value will probably apply to all entries.

• Variance cannot be negative, but it can be zero if all points in the data set have the same value.
• Users often employ it primarily to take the square root of its value, which indicates the standard deviation of the data.
• The variance is calculated by taking the square of the standard deviation.
• You could say that SD implicitly assumes a symmetric distribution because of its equal treatment of distance below the mean as of distance above the mean.
• Faculty who use case studies should always include a case specific to variance analysis tools.

In a way, the measurement you proposed is widely used in case of error analysis — then it is called MAE, “mean absolute error”. The standard deviation is always positive precisely because of the agreed on convention you state – it measures a distance from the mean. Variance is a measurement of the spread between numbers in a data set. Is always positive and semi-definite because it can be seen as the variance of a suitable univariate variable, which is always non-negative. An outlier are data points that are far outside of the expected range of values, or ones that lie far away from other data points.

Thus, the sum of the squared deviations will be zero and the sample variance will simply be zero. @sesqu Standard deviations did not become commonplace until Gauss in 1809 derived his eponymous deviation using squared error, rather than absolute error, as a starting point. Thus the SD became a natural omnibus measure of spread advocated in Fisher’s 1925 “Statistical Methods for Research Workers” and here we are, 85 years https://cryptolisting.org/ later. After you learn how to calculate variance and what it means (it is related to the spread of a data set!), it is helpful to know the answers to some common questions that pop up. Since the units of variance are much larger than those of a typical value of a data set, it’s harder to interpret the variance number intuitively. That’s why standard deviation is often preferred as a main measure of variability.

I think the absolute value of the difference would only express the difference from the mean and would not take into account the fact that large differences are doubly disruptive to a normal distribution. It is zero when all the samples $x$ are equal, and otherwise its magnitude measures variation. Squaring however does have a problem as a measure of spread and that is that the units are all squared, whereas we might prefer the spread to be in the same units as the original data .

## Can Variance Be Greater Than Mean?

Variance tells us about how spread out the values in a data set are. A high variance means the data is spread out far from the mean . The product of the deviation of x and deviation of y is then calculated. It is done by taking the difference between two values of x, the difference between the two values of y and multiplying both the variables i.e. (xi – µx) × (yi – µy). This shows that if the values of one variable match those of another, it is said that the positive covariance is present between them. And a lot of distributions and real data are an approximately normal. The mean absolute deviation is about .8 times (actually $\sqrt$) the size of the standard deviation for a normally distributed dataset. The reason that we calculate standard deviation instead of absolute error is that we are assuming error to be normally distributed. You need to be able to understand how the degree to which data values are spread out in a distribution can be assessed using simple measures to best represent the variability in the data. It measures the degree of variation of individual observations with regard to the mean.

## Step 4: Find the sum of squares

Covariance is the measurement of two random variables in a directional relationship. This means, how much two random variables differ together is measured as covariance. In population variance calculation, the last step constitutes dividing the summed results by the data set’s total number. You have why is variance always positive become familiar with the formula for calculating the variance as mentioned above. Now let’s have a step by step calculation of sample as well as population variance. It’s important to note that doing the same thing with the standard deviation formulas doesn’t lead to completely unbiased estimates.

The squared formulation also naturally falls out of parameters of the Normal Distribution. Compare this to distances in euclidean space – this gives you the true distance, where what you suggested is more like a manhattan distance calculation. One way you can think of this is that standard deviation is similar to a “distance from the mean”. Squaring emphasizes larger differences, a feature that turns out to be both good and bad . The “rules” will be implicit in that definition if the definition is any good. Connect and share knowledge within a single location that is structured and easy to search.

In fact, if every squared difference of data point and mean is greater than 1, then the variance will be greater than 1. Note that this also means that the standard deviation is zero, since standard deviation is the square root of variance. Every point in the data set has the same value, which is also the value of the mean.