Business
Variance and Standard Deviation
Variance and standard deviation are measures of the spread or dispersion of a set of data points. Variance quantifies the average squared difference of each data point from the mean, while standard deviation is the square root of the variance and provides a measure of how much the data deviates from the mean. In business, these measures are used to assess risk and variability in financial data.
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11 Key excerpts on "Variance and Standard Deviation"
- Martin Lee Abbott(Author)
- 2016(Publication Date)
- Wiley(Publisher)
This value is also shown in the specification window itself in the third row of figures. STANDARD DEVIATION AND VARIANCE The range and percentiles that we discussed previously are helpful ways to understand the distribution of a set of raw scores. However, researchers use further measures that are more precise for calculation and understanding of statistical procedures we will cover in this book. Make sure you have a level of comfort with what they represent and how to calculate them before you move on to the further topics we discuss. The standard deviation (SD) and variance (VAR) are both measures of the disper- sion of scores in a distribution. That is, these measures provide a view of the nature and extent of the scatter of scores around the mean. So, along with the mean, skew- ness, and kurtosis, measures of dispersion (i.e., the SD and VAR) provide a fourth way of describing the distribution of a set of scores. With these measures, the researcher can decide whether a distribution of scores is normally distributed. Figure 3.7 shows how scores in a distribution spread out around the mean value. Each score can be thought to have a “deviation amount” or a distance from the mean. Figure 3.7 shows these deviation amounts for four raw scores (X 1 , X 2 , X 3 , and X 4 ). CALCULATING THE Variance and Standard Deviation 61 Deviation amount Deviation amount Deviation amount Deviation amount Mean X 2 X 1 X 3 X 4 Figure 3.7 The components of the SD. The VAR is by definition the square of the SD. Conceptually, the VAR is a global measure of the spread of scores since it represents an average squared deviation. If you summed the squared distances between each score and the mean of a distribution of scores (i.e., if you squared and summed the deviation amounts), you would have a global measure of the total amount of variation among all the scores. If you divided this number by the number of scores, the result would be the VAR, or the average squared distance from the mean.
eBook - PDF- Sally Caldwell(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Measures of Variability or Dispersion 39 measure. The act of squaring the deviations has a way of markedly changing the magnitude of the numbers we’re dealing with—something that happens anytime you square a number. The illustration you encountered in Table 2-12 is a good example. That illustration was based on a distribution of income data, and income data necessar-ily involve some fairly large numbers. Inspection of that illustration reveals how quickly the original values explode in magnitude when deviations are squared. In truth, all values have a way of exploding in magnitude when they’re squared. Whether you’re dealing with single-digit numbers or values in the thousands, the same process is at work. The mere act of squaring numbers can radically alter the values. In the process, you’re apt to lose sight of the original scale of measurement you were working with. Fortunately, there is a fairly easy way to bring everything back in line, so to speak. All you have to do is calculate the variance and then turn right around and take the square root. Indeed, statisticians make a habit of doing just that. Moreover, they have a specific name for the result. It is referred to as the standard deviation. LEARNING CHECK Question: What is the variance? How does it deal with the problem that the sum of the deviations from the mean always equals 0? What is a major limitation of the variance? Answer: The variance is a measure of dispersion. To avoid the problem of the deviations from the mean always summing to 0, the variance is based on squaring the deviations before they are summed. A major limitation of the variance is that squaring the deviations inflates the magnitude of the values in the distribution, which can cause you to lose sight of the original units of measurement. The Standard Deviation Before we get to the business of calculating the standard deviation, let me point out an important distinction.
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
This sta- tistic is used to estimate a population variance, . s The estimated population standard deviation obtained from a sample of scores. This statistic is used to estimate a population standard deviation, . The population variance, a parameter calculated on all scores in a population. The population standard deviation, a parameter calculated on all scores in a population. Table 5.3 presents the definitional and sum of squares formulas for both s and s. Interpreting the Variance and the Standard Deviation In published reports of research, the standard deviation is presented as a descriptive statis- tic more often than the variance. One reason for this greater use is that, because the stan- dard deviation is the square root of the variance, it provides a measure of the variability of scores using the scores’ original units. In Example Problem 5.1, s is 3.5 seconds squared, a value not easily interpreted. The value of s, however, is 1.9 seconds, the same units (i.e., seconds) that the raw scores were measured in. Thus, the focus in this section is on the in- terpretation of s. The variance, however, is used in several statistical hypothesis testing methods and is discussed more fully in Chapter 10. The standard deviation provides a measure of the average of how much the scores in a distribution differ from the mean. If all the scores in a distribution are equal to the mean, such as those of group 1 of Table 5.1, then s will equal zero. Thus, the average amount by which the scores in group 1 differ from the mean is zero. In this case, if you use to pre- dict any of the scores in the distribution, there will be zero error in your prediction. If you see a value of s equal to zero, then you know that all scores in the distribution are equal to the sample mean, and the mean is a perfect predictor of all the scores.
eBook - PDFQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
87 Measures of Variation and Skewness 4.4 Standard Deviation The concept of standard deviation (SD) was introduced by Karl Pearson in 1823. Also known as root mean square deviation, standard deviation provides an average distance for each element from the mean. It is the positive square root of the mean of the squared deviations of values from the arithmetic mean. It is denoted by σ (sigma). It is also known as root mean square devia-tion. It is the square root of the means of the squared deviation from the arithmetic mean. It measures the absolute dispersion or variability of dispersion; the greater the amount of dispersion or variability, the greater the standard deviation. If x is the means of x 1 , x 2 ,…,x n , then σ = ( ) + ( ) + + ( ) { } = ( ) ----= ∑ 1 1 2 2 2 2 2 1 N N N x x x x x x x x n i … i 4.4.1 Individual Series Exercise Find the standard deviation of (Rs.) 7, 9, 16, 24, 26. Solution Calculation of the standard deviation is shown in Table 4.22. Calculation from Actual Mean : x x N Rs x x i i i = = = = -( Σ Σ 82 5 16 40 . . σ ) 2 N σ = = 293 20 7 66 . . . 5 Rs Calculation for Assumed Mean : dx N dx N i i σ σ = - = -Σ Σ 2 2 294 5 2 5 7 66 2 = Rs. . Exercise Table 4.23 shows marks obtained by students in Quantitative Methods. Find the stan-dard deviation. 88 Quantitative Techniques in Business, Management and Finance Solution Calculation of the standard deviation is shown in Table 4.24. x xi N = = = 2245 Σ 526 10 50 6 5 1 . S.D. σ ( ) = - = - = Σ Σ d N d N 2 2 2 1826 10 26 10 13 260 . 4.4.2 Discrete Series Exercise Calculate the standard deviation from the data in Table 4.25. TABLE 4.22 Calculation of Standard Deviation from Actual Mean and Assumed Mean A. Calculate from actual mean Variants, Rs. x i Deviation from Actual Mean (16.40), x = x x i – x i 2 7 –9.4 88.36 9 –7.4 54.76 16 –0.4 0.16 24 7.6 57.76 26 9.6 92.16 Σ x i = 82 Σ x i 2 = 293.20 B.
eBook - PDFEducational Assessment
Tests and Measurements in the Age of Accountability
- Robert J. Wright(Author)
- 2007(Publication Date)
- SAGE Publications, Inc(Publisher)
The scores that are greater than the mean will be can-celled out by those below, and the net will always be zero. A solution to this mathematical cul-de-sac is found by multiplying the subtraction remainders by themselves. Both positive and negative values are both positive after squaring. Squaring the subtraction remainders produces a series of squared difference scores from the mean. The “average” of these squared differences is known as variance. This average is found by dividing the sum of the squared difference (mean – individual score) by the number of subjects. Variance and Standard Deviation Variance is the “average” of the squared deviations around the mean of a data set. The value of variance is always greater than zero and has no upper limit. For most measurement applications, variance is presented in the form of standard deviation. This statistic is the square root of variance. The statistic standard deviation is sometimes presented as the symbol s or SD, and in the case of an entire population, σ . The utility of this statistic is drawn from its relationship to the Gaussian normal curve. This relationship can be seen on Table 3.2. When data are collected from a random sample of a normal population, a distribution of those scores will approximate the nor-mal curve. The approximate proportion of scores that will lie between the mean and one standard deviation s over the mean is a constant 34.13%. The normal distribution is symmetric, so another 34.13% of all the scores in the sample will lie between the mean and one standard deviation s below the mean. Thus, just over 68% of all scores lie between –1 s and + 1 s. By expanding this to almost ± 2 s it is possible to account for 95% of all scores within a nor-mal sample. That represents 47.5% above and 47.5% below the mean. standard deviation = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n i = 1 X i − X 2 n v u u u t variance = X n i = 1 ð X i − X Þ 2 n Chapter 3 The Measurement and Description of Variables – – 97
- Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Step 2: Calculate X. Step 3: Obtain deviation scores by subtracting X from each value of X (as a check on your work, remember that Σ(X − X) should equal zero). Step 4: Square each deviation score Step 5: Sum the values of the squared deviation scores to get the sum of squares (SS). Step 6: Divide the sum of squares by n as in Formula 5.2b. 5.4 Deviational Measures: The Standard Deviation The variance is frequently used in advanced statistical procedures and especially in inferential statistics. However, for basic description, it has a fatal flaw: Its calculated value is expressed in squared units of measurement. (If the scores are weights in pounds, the variance will be a certain 5.5 Calculation of the Variance and Standard Deviation: Raw-Score Method 63 quantity of squared pounds.) Consequently, it is of little use in descriptive statistics. However, this is easily remedied. By taking the square root of the variance, we return to the original units of mea- surement and thereby obtain a widely used index of variability called the standard deviation. standard deviation the square root of the vari- ance Although it is not technically correct, it is easier for beginning students to think of the stan- dard deviation as “the average amount that scores deviate from the mean.” When there is wider scatter among the raw scores, there are bigger deviations from the mean, and thus the “standard” (typical, representative) deviation increases. The defining formulas for the standard deviation are: DEFINITIONAL FORMULA FOR STANDARD DEVIATION OF A POPULATION X = √ ∑ (X − X ) 2 N or √ SS X N (5.3a) DEFINITIONAL FORMULA FOR STANDARD DEVIATION OF A SAMPLE S X = √ ∑ (X − X) 2 n or √ SS X n (5.3b) As you can see, the symbols for the standard deviation of a population () and a sample (S) are symbol for the standard deviation of a population S symbol for the standard deviation of a sample the same as those used for the variance, but without the square root sign.
eBook - PDFBehavioral Research and Analysis
An Introduction to Statistics within the Context of Experimental Design, Fourth Edition
- Max Vercruyssen, Hal W. Hendrick(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
36 Behavioral Research and Analysis deviation scores. A good computational check at this point is to see whether the deviation scores add to zero—the sum of the signed deviation scores about the mean is always equal to zero. The second step is to drop the signs of the deviation scores. The final step is to take the average of the absolute deviation scores. The average deviation, or mean deviation as it is alternately known, is no longer widely used in statistical work, having been replaced by the standard deviation. We considered it here only to make the material about standard deviation easier to understand. Variance Variance is the average squared deviation. Variance equals the standard deviation squared and is represented by the lower case Greek letter sigma squared, σ 2 . Variances are additive, but standard deviations are not. Therefore it is generally more appropriate to apply statistics to the variance rather than the standard deviation in describing changes in distribution variability across conditions. Standard Deviation The standard deviation is the square root of the average squared deviation or root mean square deviation. Of all the measures of variability, it is the one most often used, primarily because it is needed in so many other statistical operations. When referring to the sample standard deviation, the symbols σ , σ n –1 , S , or SD are used. When referring to the standard deviation of a population , the σ or σ n (lower case sigma) is used. A computational example is provided in Table 2.5. To find the sample or unbiased standard deviation, we use the following formula and substitute in our data as shown. Sample SD s x n X n n X n = = = ∑ -= ∑ --= = = -∑ σ 1 2 2 1 1 144 9 16 4 2 ( ) TABLE 2.5 Computing Standard Deviation From Ungrouped Data X x x 2 25 7 49 23 5 25 20 2 4 19 1 1 18 0 0 18 0 0 16 –2 4 15 –3 9 14 –4 16 12 –6 36 Σ X = 180 Σ x = 0 Σ x 2 = 144 X = 18 (Sum of squares) Note: x = X – X x or distance from raw score to mean.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
For the Formula-philes: Definition of the Sample Variance Consider a sample of n observations x 1 , x 2 , x 3 , . . . x n The Arithmetic Mean of the sample, x , is: x x x x n n = + + + x x + 1 2 x x 3 Notationally, the Sample Variance, s 2 , is: Note: the symbol, s 2 is one that is in common usage to portray the Variance of a sample. If this were the variance of a set of data for the entire population, it is common practice to use the abbreviation σ 2 Definition 3.8 Variance of a Sample The Variance of a sample of data taken from the entire population is a measure of the extent to which the sample data is dispersed around its Arithmetic Mean. It is calculated as the sum of squares of the deviations of each individual value from the Arithmetic Mean of all the values divided by the Degrees of Freedom, which is one less than the number of data points in the sample. Definition 3.9 Standard Deviation of a Sample The Standard Deviation of a sample of data taken from the entire population is a measure of the extent to which the sample data is dispersed around its Arithmetic Mean. It is calculated as the square root of the Sample Variance, which is the sum of squares of the deviations of each individual value from the Arithmetic Mean of all the values divided by the Degrees of Freedom, which is one less than the number of data points in the sample. s n x x i n i 2 1 2 1 1 = − ( ) = ∑ Measures of Dispersion and Shape | 91 3.4.2 Coefficient of Variation The main problem with Variance and Standard Deviation as Measures of Dispersion is that of the scale and units of measurement used.
eBook - PDFPortfolio Selection
Efficient Diversification of Investments
- Harry M. Markowitz(Author)
- 1968(Publication Date)
- Yale University Press(Publisher)
CHAPTER IV STANDARD DEVIATIONS AND VARIANCES DEFINITIONS The definition of the variance of a series is illustrated by Table 1. The first column of the table lists returns on a hypothetical security during 5 years. The average of these returns is .01. The second column indicates, TABLE 1 RETURNS AND THEIR VARIANCE Returns .10 -.05 00 .05 -.055 Deviations from Average .09 -.066 -.011 .04 -.066 Squared Deviations .0081 .0036 .0001 .0016 .0036 Sum .05 .00 .0170 Average .01 .00 Variance = .0034. Standard deviation = V.0034 = .058. for each year, the difference between that year's return and the average return. Thus, in the first year, return was .09 greater than the five-year average. The third column presents the squares of the numbers in the 72 .0034 STANDARD DEVIATIONS AND VARIANCES 73 second column. In the first year, for example, the squared deviation from the average was .09 times .09 = .0081. The average of the entries in the third column, .0034, is referred to as the variance of return. The variance of return on a past series, then, is the average of the squared deviations from the average. The variance of a past series can also be computed from a table of relative frequencies. Column 1 of Table 2 lists returns which occurred to TABLE 2 DERIVATION OF VARIANCE FROM RELATIVE FREQUENCIES Returns -.05 .00 .05 .10 SUM Relative Frequencies 2/5 1/5 1/5 1/5 1 Deviations from Average -.06 -.01 .04 .09 Squared Deviations .0036 .0001 .0016 .0081 Frequency times Squares .00144 .00002 .00032 .00162 .00340 our hypothetical security; column 2 shows the relative frequency of each return. The average of the series, .01, can be computed from columns 1 and 2, as explained in the last chapter. Column 3 lists the difference between the return and the average return; column 4 lists the squares of the entries in column 3; column 5 presents the product of the entries in columns 2 and 4. The sum of column 5 is the average of the squared deviations from the average.
eBook - PDF- Perry J. Kaufman(Author)
- 2019(Publication Date)
- Wiley(Publisher)
—Chinese proverb The moments of the distribution describe the shape of the data points, the way they cluster around the mean. There are four moments: mean , variance , skewness , and kurtosis . 1. The mean is the center or average value, around which other points cluster. 2. Variance is the distance of the individual points from the mean. 3. Skewness is the way the distribution leans to the left or right relative to the mean. 4. Kurtosis is the peakedness of the clustering. We have already discussed the mean, so we will start with the 2nd moment. In the following calculations, we will use the bar notation, P , to indicate the average of a list of n prices. The capital P refers to all prices and the small p to individual prices. We’ll use the following form so that the similarity between moments can be easily seen. The mean is calculated as: P = ∑ n i = 1 p i n Variance (2nd Moment) and Standard Deviation Variance (Var) is very similar to mean deviation (MD), which does not square the differences, and is the best estimation of dispersion. Var = ∑ n i = 1 ( p i − P ) 2 n − 1 Notice that the variance is the square of the standard deviation, var = s 2 = 𝜎 2 , one of the most commonly used statistics. In Excel, the variance is the function var and in TradeStation it is variance(series,n) . The standard deviation, most often shown as 𝜎 (sigma), is a special form of mea-suring average deviation from the mean, which uses the root-mean-square: 𝜎 = √ ∑ n i − 1 ( p i − P ) 2 n where the differences between the individual prices and the mean are squared to emphasize the significance of extreme values, and then the total value is reduced using the square root function. This popular measure, also used throughout this book, is the Excel function Stdevp and the TradeStation function, StdDev(price,n) , for n prices.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
■ ■ Formulas for Sample Variance and Standard Deviation The calculations of Variance and Standard Deviation for a sample follow the same steps that were used to find population Variance and Standard Deviation. First, calculate the sum of squared deviations (SS). Second, calculate the variance. Third, find the square root of the variance, which is the standard deviation. Except for minor changes in notation, calculating the sum of squared deviations, SS , is exactly the same for a sample as they were for a population. The changes in notation involve using M for the sample mean instead of m , and using n (instead of N ) for the num-ber of scores. Thus, the definitional formula for SS for a sample is Definitional formula: SS 5 S ( X 2 M ) 2 (4.5) A sample statistic is said to be biased if, on the average, it consis-tently overestimates or underestimates the cor-responding population parameter. Population variability Population distribution Sample X X X X X X X X Sample variability X X F I G U R E 4.4 The population of adult heights forms a normal distribution. If you select a sample from this popula-tion, you are most likely to obtain individuals who are near average in height. As a result, the variabil-ity for the scores in the sample is smaller than the variability for the scores in the population. 112 CHAPTER 4 | Variability Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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