# Variance

# Variance

A variance has several meanings in business. In an accounting sense, a variance is the difference between an actual amount and a pre-determined standard amount or the amount budgeted. In a statistical sense, a variance is a measure of the amount of spread in a distribution. It is computed as the average squared deviation of each number from its mean. Finally, variance has a meaning related to land use called a zoning variance. A zoning variance is an administrative exception to land use regulations.

## ACCOUNTING VARIANCES

In accounting, a variance could be a cost variance, where actual costs may be different from the estimated standards for costs. Variances can be favorable or unfavorable. A variance from standard cost is considered favorable if the actual cost is less than the standard or budgeted cost, and it is considered unfavorable if the actual cost is more than was budgeted. It is also possible to break down the cost variance into the factors that may have caused it to occur—such as a quantity variance, or the difference between the actual quantity and the standard quantity; and a price variance, or the difference between the actual price and the standard price.

When a variance occurs, like the cost variance in this example, top management should examine the circumstances to determine the factors that created it. By doing so, management should be able to identify who or what was responsible for the variance and take steps to correct the problem. For example, assume that the standard material cost for producing 1,000 units of a product is $8,000, but that materials costing $10,000 were actually used. The $2,000 unfavorable variance may have resulted from paying a price for the material that was higher than the standard price. Alternatively, the process may have used a greater quantity of material than standard. Or, there may have been some combination of these factors.

The purchasing department is usually responsible for the price paid for materials. Therefore, if the variance was caused by a price higher than standard, responsibility for explaining the problem rests with the purchasing manager. On the other hand, the production department is usually responsible for the amount of material used. Thus, the production department manager is responsible for explaining the problem if the process used more than the standard amount of materials. However, the production department may have used more than the standard amount of material because its quality did not meet specifications, with the result that more waste was created. Then the purchasing manager is responsible for explaining why the inferior materials were acquired. On the other hand, the production manager is responsible for explaining what happened if the analysis shows that the waste was caused by inefficiencies.

Thus variances—like the cost variance in the example above—trigger questions to be answered within the organization. These questions call for answers that, in turn, should lead to changes designed to correct the problem and minimize or eliminate the variances for the next reporting period. In studying variances in expenditures, a company may find that the assumptions upon which its budgets were made were in error. These too should be corrected so that the budget will more accurately reflects the likely outcome in the next period.

A performance report may identify the existence of the problem, but it can do no more than point the direction for further investigation of what can be done to improve future results. Other common variances in accounting include overhead rate and usage variances.

## STATISTICAL VARIANCES

In statistics, a variance is also called the mean squared error. The variance is one of several measures that statisticians use to characterize the dispersion among the measures in a given population. To calculate the variance, it is necessary to first calculate the mean or average of the scores. The next step is to measure the amount that each individual score deviates or is different from the mean. Finally, you square that deviation by multiplying the number by itself. Numerically the variance equals the average of the squared deviations from the mean.

## BIBLIOGRAPHY

Hunter, Katharine. "Variances: The Three-Step Method." *Accountancy*. January 1995.

Larson, K. D. *Fundamental Accounting Principles*. Sixteenth Edition. McGraw-Hill, 2004.

Wolinski, John. "John Wolinski Considers the Use of Budgeting in a Business and How Variances Can Be Analyzed." *Business Review*. November 2004.

*Hillstrom, Northern Lights*

*updated by Magee, ECDI*

# Variance

# Variance

Variance, especially in probability theory and statistics, is a mathematical expression concerning how data points are spread across a data set. Such expressions are known as measures of dispersion since they indicate how values are dispersed throughout a population. The variance is the average, or mean, of the squares of the distance each data point in a set is from the mean of all the data points in the set. Mathematically, variance is represented as ζ^{2}, according to the equation: are the values of specific variables; µ is the mean, or average, of all the values; and n is the total number of values. Variance is commonly replaced in applications by its square root, which is known as the standard deviation or ζ.

Variance is one of several measures of dispersion that are used to evaluate the spread of a distribution of numbers. Such measures are important because they provide ways of obtaining information about data sets without considering all of the elements of the data individually. Most mathematicians consider that British statistician Sir Ronald Aylmer Fisher (1890-1962) first used the word variance in his 1918 paper *The Correlation Between Relatives on the Supposition of Mendelian Inheritance.*

To understand variance, one must first understand something about other measures of dispersion. One measure of dispersion is the average of deviations. This value is equal to the average, for a set of numbers, of the differences between each number and the set’s mean. The mean (also known as the average) is simply the sum of the numbers in a given set divided by the number of entries in the set. For the set of eight test scores: 7 + 25 + 36 + 44 + 59 + 71 + 85 + 97, the mean is 53. The deviation from the mean for any given value is that value minus the value of the mean. For example, the first number in the set, 7, has a deviation from the mean of -46; the second number, 25, has a deviation from the mean of -28; and so on. However, the sum of these deviations across the entire data set will be equal to 0 (since by definition the mean is the middle value with all other values being above or below it.) A measure that will show how much deviation is involved without having these deviations add up to zero would be more useful in evaluating data. Such a nonzero sum can be obtained by adding the absolute values of the deviations. This average is the absolute mean deviation. However, for reasons that will not be dealt with here, even this expression has limited application.

A still more informative measure of dispersion can be obtained by squaring the deviations from the mean, adding them, and dividing by the number of scores; this value is known as the average squared deviation or variance. For example, in the series of test scores cited above, the variance can be calculated as follows:

Theoretically, the value of ζ^{2} should relate valuable information regarding the spread of data. However, in order for this concept to be applied in practical situations (one cannot talk about squared test scores), one may elect to use the square root of the variance. This value is called the standard deviation of the scores. For this series of test scores, the standard deviation is the square root of 825.38 or 28.73. In general, a small standard deviation indicates that the data are clustered closely around the mean; a large standard deviation shows that the data are more spread apart.

While modern computerization reduces the need for laborious statistical calculations, it is still necessary to understand and interpret the concept of variance and its daughter, standard deviation, in order to digest the statistical significance of data. For example, teachers must be thoroughly familiar with these statistical tools in order to properly interpret test data.

*See also* Set theory; Statistics.

## KEY TERMS

**Absolute deviation from the mean** —The sum of the absolute values of the deviations from the mean.

**Average deviation from the mean** —For a set of numbers, the average of the differences between each number and the set’s mean value.

**Measure of dispersion** —A mathematical expression that provides information about how data points are spread across a data set without having to consider all of the points individually.

**Standard deviation** —The square root of the variance.

## Resources

### BOOKS

Burton, David M. *The History of Mathematics: An Introduction.* New York: McGraw-Hill, 2007.

Facade, Harold P., Cummins, Kenneth B. *The Teaching of Mathematics from Counting to Calculus.* Columbus, OH: Charles E. Merrill Publishing Co., 1970.

Haenisch, Siegfried. *Mathematics Concepts.* Circle Pines, MN: AGS Publishing, 2005.

Lloyd, G.E.R. Early Greek Science: Thales to Aristotle. New York: W.W. Norton and Company, 1970.

Reid, Constance. *From Zero to Infinity: What Makes Numbers Interesting.* Wellesley, MA: A.K. Peters, 2006.

Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

Treff, August V., and Donald H. Jacobs. *Basic Math Skills.* Circle Pines, MN: American Guidance Service, 2003.

Randy Schueller

# Variance

# Variance

The variance is a measure of variability among scores. In describing any set of data, one uses three characteristics: the form of the distribution, the mean or central tendency, and the variability. All three are required because they are generally independent of one another. In other words, the mean indicates nothing about the variability.

Variability is a characteristic of all measures. In the social sciences the people or groups that are studied may be exposed to the same treatment or conditions but will show different responses to that treatment or those conditions. In other words, all the scores of the people would be different. The goal of the scientist is to explain why the scores are different. One can think of a score as having two parts: the mean and a deviation from the mean (Hays 1973). If everyone were exactly the same, the variability among scores (deviations from the mean) would be zero.

In manipulating conditions or treatments it is possible to explain different scores between groups by looking at the variability between the groups and also within each group. If the groups are different enough, it is said that they are statistically significantly different. Statistical significance is determined by looking at the probability (odds) that a difference could have occurred by chance. One analyzes the data by looking at the variability of the data and breaking down that variability into its component parts.

There are several methods for measuring variability among scores. The range is the difference between the highest and lowest scores and is limited because it is based on only two scores. It is not very useful because it is influenced easily by extremes among the scores. The mean deviation score sometimes is used as a measure of variability but also is limited in terms of its usefulness in additional mathematical calculations. The standard deviation is used commonly; it is simply the square root of the variance.

The variance is the most useful of these methods. It is defined as the sum of the squared deviations from the mean divided by the number of squared deviations. The equation for the variance is , where var is the variance, *X* is a raw score, *M* is the mean of the scores, *X–M* represents the deviation of a score from the mean, and *N* is the number of scores. Many different symbols have been used to represent the elements of this equation, but in all cases the variance is the average of the squared deviations from the mean.

A simple numerical example can illustrate the variance. There is a small set of five scores: 8, 7, 6, 5, and 4. The sum of those scores is 30, and the mean is 6. The numerator in the variance equation is called the sum of the deviation scores squared (often abbreviated SS for sum of squares). One takes each score and subtracts the mean (i.e., 8 – 6 = 2; 7 – 6 = 1; 6 – 6 = 0; 5 – 6 = –1; and 4–6 = –2). Next, the deviation scores are squared to eliminate the minus signs (2^{2} = 4; 1^{2} = 1; 0^{2} = 0; –1^{2} = 1; and -2^{2} = 4). Then one simply adds the deviation scores squared (4; 1; 0; 1; and 4) to get the SS, which is equal to 10. Finally, the variance is calculated by dividing the SS by the number of scores (*N* ), which in this example is 10/5 = 2. The variance for this small set of five scores is 2.

This example illustrates an important point: The variance is not a clear indicator of variability. Consider the scores in the example above as inches. The variance is 2 squared inches. How would one make sense out of a variability of squared inches? In the social sciences how would one interpret a variability of squared IQ, conformity, opportunity costs, and so on? The variance has no simple or particularly useful explanation in everyday language or in the technical jargon of the social sciences.

However, the variance is an essential mathematical step in describing parametric variability. The variance has the advantage of being additive, something that is not true for its square root, the standard deviation (Games and Klare 1967). This means that in working with more than one group and looking for a measure of pooled or average variability among the groups, one can add variances. This is done in many inferential parametric statistical tests.

The variance is an essential element of much social science data analysis but is not easily interpretable. The standard deviation, which is the square root of the variance, is used commonly to provide a more readily understandable indicator of variability.

**SEE ALSO** *Regression; Regression Analysis; Standard Deviation; Statistics; Test Statistics; Variation*

## BIBLIOGRAPHY

Games, Paul A., and George R. Klare. 1967. *Elementary Statistics: Data Analysis for the Behavioral Sciences*. New York: McGraw-Hill.

Hays, William Lee. 1973. *Statistics for the Social Sciences*. 2nd ed. New York: Holt, Rinehart and Winston.

*Samuel K. Rock Jr.*

# Variance

# Variance

Variance is a mathematical expression of how data points are spread across a data set. Such expressions are known as measures of dispersion since they indicate how values are dispersed throughout a population. The variance is the average or **mean** of the squares of the **distance** each data point in a set is from the mean of all the data points in the set. Mathematically, variance is represented as σ2, according to the equation: σ2 = [(x1-μ)2 + (x2-μ)2 + (x3-μ)2 +...(xn-μ)2]/n; where x1,2,3,.....n are the values of specific variables; μ is the mean, or average, of all the values; and n is the total number of values. Variance is commonly replaced in applications by its **square root** , which is known as the standard deviation or σ.

Variance is one of several measures of dispersion which are used to evaluate the spread of a distribution of numbers. Such measures are important because they provide ways of obtaining information about data sets without considering all of the elements of the data individually.

To understand variance, one must first understand something about other measures of dispersion. One measure of dispersion is the "average of deviations." This value is equal to the average, for a set of numbers, of the differences between each number and the set's mean. The mean (also known as the average) is simply the sum of the numbers in a given set divided by the number of entries in the set. For the set of eight test scores: 7 + 25 + 36 + 44 + 59 + 71 + 85 + 97, the mean is 53. The deviation from the mean for any given value is that value minus the value of the mean. For example, the first number in the set, 7, has a deviation from the mean of -46; the second number, 25, has a deviation from the mean of -28; and so on. However, the sum of these deviations across the entire data set will be equal to 0 (since by definition the mean is the "middle" value with all other values being above or below it.) A measure that will show how much deviation is involved without having these deviations add up to **zero** would be more useful in evaluating data. Such a nonzero sum can be obtained by adding the absolute values of the deviations. This average is the absolute mean deviation. However, for reasons that will not be dealt with here, even this expression has limited application.

A still more informative measure of dispersion can be obtained by squaring the deviations from the mean, adding them, and dividing by the number of scores; this value is known as the average squared deviation or "variance." For example, in the series of test scores cited above, the variance can be calculated as follows:

Theoretically, the value of σ2 should relate valuable information regarding the spread of data. However, in order for this concept to be applied in practical situations (we cannot talk about squared test scores) we may elect to use the square root of the variance. This value is called the standard deviation of the scores. For this series of test scores the standard deviation is the square root of 825.38 or 28.73. In general, a small standard deviation indicates that the data are clustered closely around the mean; a large standard deviation shows that the data are more spread apart.

While modern computerization reduces the need for laborious statistical calculations, it is still necessary to understand and interpret the concept of variance and its daughter, standard deviation, in order to digest the statistical significance of data. For example, teachers must be thoroughly familiar with these statistical tools in order to properly interpret test data.

See also Set theory; Statistics.

## Resources

### books

Dunham, William. *Journey Through Genius.* New York: John Wiley & Sons Inc., 1990.

Facade, Harold P., and Kenneth B. Cummins. *The Teaching of**Mathematics from Counting to Calculus.* Columbus, OH: Charles E. Merrill Publishing Co., 1970.

Lloyd, G.E.R. *Early Greek Science: Thales to Aristotle.* New York: W.W. Norton and Company, 1970.

Randy Schueller

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Absolute deviation from the mean**—The sum of the absolute values of the deviations from the mean.

**Average deviation from the mean**—For a set of numbers, the average of the differences between each number and the set's mean value.

**Measure of dispersion**—A mathematical expression which provides information about how data points are spread across a data set without having to consider all of the points individually.

**Standard deviation**—The square root of the variance.

# Variance

# VARIANCE

*The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial.*

*In*zoning*law, an official permit to use property in a manner that departs from the way in which other property in the same locality can be used.*

The term *variance* is used both in litigation and in zoning law. In both instances it has the general meaning of a difference or divergence.

A party to a civil lawsuit or a prosecutor in a criminal trial must prove the allegations set forth in a complaint, indictment, or information. If there is a substantial difference or discrepancy between the allegations and the proof offered in support, a *variance* exists. For example, if the crime of robbery is alleged and the crime of burglary is proved instead, the failure of proof on the robbery charge constitutes a variance that will lead to the dismissal of the case.

Most U.S. communities have zoning laws that control and direct the development of property within their borders according to its present and potential uses. Typically, a community is divided into zoning districts based on the type of use permitted: residential, commercial, and industrial. Additional restrictions may limit population density and building height within these districts. A *variance* is an exception to one or more of the zoning restrictions on a piece of property.

A variance is different from a nonconforming use, which permits existing structures and uses to continue when zoning is first instituted. Once a zoning plan has been established, a property owner who wishes to diverge from it must seek a variance from the municipal government. The variance will be granted when "unnecessary hardship" would result to the landowner if it were denied. Although other forms of administrative relief from zoning restrictions are available, such as rezoning the area, variances are most frequently used.

There are two types of variances: area variances and use variances. An area variance is usually not controversial because it is generally granted due to some odd configuration of the lot or some peculiar natural condition that prevents normal construction in compliance with zoning restrictions. For example, if the odd shape of a lot prevents a house from being set back the minimum number of feet from the street, the municipality will usually relax the requirement.

Use variances are more controversial because they attempt a change in the permitted use. For example, if a lot is zoned single-family residential, a person who wishes to build a multi-family dwelling must obtain a variance. Residents of an area will generally object to applications for variances that seek to change the character of their neighborhood. Although the municipality may heed these objections, it will likely grant the variance if it believes unnecessary hardship would result without the variance. If, however, the owner seeking a variance for a multifamily dwelling bought the property with notice of the current zoning restrictions, the variance will probably be denied. Applicants for a variance cannot argue hardship based on actions they commit that result in self-induced hardship.

If many use variances are sought in a particular area on the basis of unique or peculiar circumstances, it may be a sign that the entire neighborhood needs to be rezoned rather than forcing property owners to seek variances in a piecemeal fashion. Properly used, variances provide a remedy for hardships affecting a single lot or a relatively small area.

## cross-references

# variance

var·i·ance / ˈve(ə)rēəns/ • n. the fact or quality of being different, divergent, or inconsistent: *her light tone was at variance with her sudden trembling.* ∎ the state or fact of disagreeing or quarreling:

*they were*∎ chiefly Law a discrepancy between two statements or documents. ∎ Law an official dispensation from a rule or regulation, typically a building regulation. ∎ Statistics a quantity equal to the square of the standard deviation. ∎ (in accounting) the difference between expected and actual costs, profits, output, etc., in a statistical analysis.

**at variance with**all their previous allies.# variance

**variance ( mean square)** In statistics, a measure of the spread of the data about the mean. In a set of data, the square of the mean variation about the mean value. It is calculated as the mean of the squared data points minus the mean squared; the standard deviation of a data set is the square root of the variance:

*s*

^{2}= (Σ

*(*

_{i=1}^{i=n}*x*− x̄)

_{i}^{2})/

*n*, where

*s*

^{2}is the variance, χ

*is the mean value of the*

_{i}*i*th measurement of χ, x̄ is the mean value of χ, and

*n*is the number of measurements.

# variance

**variance** See CAUSAL MODELLING; SEQUENCE ANALYSIS; VARIATION (STATISTICAL).

# variance

**variance ( vair-i-ăns) n.** (in statistics) see standard deviation.

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