What is the disadvantage to using the definitional formula?

In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.

Sample variance is used to measure how dispersed scores are from their mean when the dataconsist of less than an entire population of scores.The sample variance (denoted S^2) is defined by the following formula: S^2 = ∑(x-M)^2/n-1

the numerator for sample variance is computed in the same way as that for the populationvariance: the numerator is the SS. It is calculations in the denominator that differ for populationsand samples. To compute the sample variance, we divide the SS by the sample size (n) minus 1.The notations for the mean (M) and the sample size (n) have also changed to account for the factthat the data are from a sample and not a population. To compute sample variance, we againsplit the steps into two parts:oCalculate the SS (numerator)oDivide SS by (n-1)The sample of six scores was 5, 10, 3, 7, 2, and 3. We will follow the steps to calculate the samplevariance (s^2) of these scores.oPart 1: calculate the SS. Like in example above which is SS=46.oPart 2: divide SS by (n-1). The sample variance is S^2= 46/(6-1)=9.20oSample variance =9.20Computations of SS are the same for population variance and sample variance. The change incomputation is whether we divide SS by N (population variance) or by n-1 (sample variance).There are three reasons we square each deviation to compute SS:oThe sum of the differences of scores from their mean is zero.oThe sum of the squared differences of scores from their mean is minimal.oSquaring scores can be corrected by taking the square root.The most straightforward way to measure variability is to subtract each score from its mean andto sum each deviation.The problem is that the sum will always be equal to zero (this was the fourth characteristic of themean listed in chapter 3)The SS produces the smallest possible positive value for deviations of scores from their mean.The SS is computed in the same way for sample variance and population variance.The reason we square each deviation before summing is largely because of Reasons 2 & 3 listedearlier in this section: We want to compute how far scores are from their mean without endingup with a solution equal to zero every timeThis is one reason for squaring deviations: it provides a solution with minimal error.Another reason for squaring is that we can correct for this by taking the square root of thesolution for variance.Th population variance is computed by dividing the SS by the population size (N),whereas thesample variance is computed by dividing the SS by sample size (n) minus 1.The following example will illustrate the reason and demonstrate why (n-1) improves the samplevariance calculation.Ex: Suppose we have a hypothetical population of 3 people (N=3) who scored an 8 (person A), 5(person B), and 2 (person c) on a quiz. This hypothetical population has a variance of 6.oO^2 = (8-5)^2 + (5-5)^2 + (2-5)^2/3 = 9+0+9/3 = 18/3 = 6Abiased estimator is any sample statistic, such as the sample variance when we divide SS by n,

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 48 pages?

Upload your study docs or become a

Course Hero member to access this document

Tags

Standard Deviation, Variance

The term variance refers to a statistical measurement of the spread between numbers in a data set. More specifically, variance measures how far each number in the set is from the mean (average), and thus from every other number in the set. Variance is often depicted by this symbol: σ2. It is used by both analysts and traders to determine volatility and market security.

The square root of the variance is the standard deviation (SD or σ), which helps determine the consistency of an investment’s returns over a period of time.

• Variance is a measurement of the spread between numbers in a data set.
• In particular, it measures the degree of dispersion of data around the sample's mean.
• Investors use variance to see how much risk an investment carries and whether it will be profitable.
• Variance is also used in finance to compare the relative performance of each asset in a portfolio to achieve the best asset allocation.
• The square root of the variance is the standard deviation.

In statistics, variance measures variability from the average or mean. It is calculated by taking the differences between each number in the data set and the mean, then squaring the differences to make them positive, and finally dividing the sum of the squares by the number of values in the data set.

Variance is calculated by using the following formula:

σ 2 = ∑ i = 1 n ( x i − x ‾ ) 2 N where: x i = Each value in the data set x ‾ = Mean of all values in the data set N = Number of values in the data set \begin{aligned}&\sigma^2 = \frac { \sum_{i = 1} ^ { n } \big (x_i - \overline { x } \big ) ^ 2 }{ N } \\&\textbf{where:} \\&x_i = \text{Each value in the data set} \\&\overline { x } = \text{Mean of all values in the data set} \\&N = \text{Number of values in the data set} \\\end{aligned} ​σ2=N∑i=1n​(xi​−x)2​where:xi​=Each value in the data setx=Mean of all values in the data setN=Number of values in the data set​

You can also use the formula above to calculate the variance in areas other than investments and trading, with some slight alterations. For instance, when calculating a sample variance to estimate a population variance, the denominator of the variance equation becomes N − 1 so that the estimation is unbiased and does not underestimate the population variance.

Statisticians use variance to see how individual numbers relate to each other within a data set, rather than using broader mathematical techniques such as arranging numbers into quartiles. The advantage of variance is that it treats all deviations from the mean as the same regardless of their direction. The squared deviations cannot sum to zero and give the appearance of no variability at all in the data.

One drawback to variance, though, is that it gives added weight to outliers. These are the numbers far from the mean. Squaring these numbers can skew the data. Another pitfall of using variance is that it is not easily interpreted. Users often employ it primarily to take the square root of its value, which indicates the standard deviation of the data. As noted above, investors can use standard deviation to assess how consistent returns are over time.

In some cases, risk or volatility may be expressed as a standard deviation rather than a variance because the former is often more easily interpreted.

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 average of these three returns is 5%. The differences between each return and the average are 5%, 15%, and −20% for each consecutive year.

Squaring these deviations yields 0.25%, 2.25%, and 4.00%, respectively. If we add these squared deviations, we get a total of 6.5%. When you divide the sum of 6.5% by one less the number of returns in the data set, as this is a sample (2 = 3-1), it gives us a variance of 3.25% (0.0325). Taking the square root of the variance yields a standard deviation of 18% (√0.0325 = 0.180) for the returns.

Follow these steps to compute variance:

1. Calculate the mean of the data.
2. Find each data point's difference from the mean value.
3. Square each of these values.
4. Add up all of the squared values.
5. Divide this sum of squares by n – 1 (for a sample) or N (for the population).

Variance is essentially the degree of spread in a data set about the mean value of that data. It shows the amount of variation that exists among the data points. Visually, the larger the variance, the "fatter" a probability distribution will be. In finance, if something like an investment has a greater variance, it may be interpreted as more risky or volatile.

Standard deviation is the square root of variance. It is sometimes more useful since taking the square root removes the units from the analysis. This allows for direct comparisons between different things that may have different units or different magnitudes. For instance, to say that increasing X by one unit increases Y by two standard deviations allows you to understand the relationship between X and Y regardless of what units they are expressed in.