Mathematics
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much the values differ from the mean of the set. A higher standard deviation suggests greater variability, while a lower standard deviation indicates that the values are closer to the mean.
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12 Key excerpts on "Standard Deviation"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Technically, the Standard Deviation of a statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler though practically less robust than the expected deviation or average absolute deviation. A useful property of Standard Deviation is that, unlike variance, it is expressed in the same units as the data. Note, however, that for measurements with percentage as unit, the Standard Deviation will have percentage points as unit. In addition to expressing the variability of a population, Standard Deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected Standard Deviation in the results if the same poll were to be conducted multiple times. The reported margin of error is typically about twice the Standard Deviation–the radius of a 95% confidence interval. In science, researchers commonly report the Standard Deviation of experimental data, and only effects that fall far outside the range of Standard Deviation are considered ________________________ WORLD TECHNOLOGIES ________________________ statistically significant—normal random error or variation in the measurements is in this way distinguished from causal variation. Standard Deviation is also important in finance, where the Standard Deviation on the rate of return on an investment is a measure of the volatility of the investment. When only a sample of data from a population is available, the population standard de-viation can be estimated by a modified quantity called the sample Standard Deviation, explained below.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
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. In simple terms, the Standard Deviation provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered. Although the concept of Standard Deviation is straightforward, the actual equations tend to be more complex. Therefore, we begin by looking at the logic that leads to these equa-tions. If you remember that our goal is to measure the standard, or typical, distance from the mean, then this logic and the equations that follow should be easier to remember. The first step in finding the standard distance from the mean is to determine the deviation , or distance from the mean, for each individual score. By definition, the deviation for each score is the difference between the score and the mean. Deviation is distance from the mean: deviation score 5 X 2 µ For a distribution of scores with µ 5 50, if your score is X 5 53, then your deviation score is X 2 µ 5 53 2 50 5 3 If your score is X 5 45, then your deviation score is X 2 µ 5 45 2 50 5 2 5 Notice that there are two parts to a deviation score: the sign ( 1 or 2 ) and the number. The sign tells the direction from the mean—that is, whether the score is located above ( 1 ) or below ( 2 ) the mean. The number gives the actual distance from the mean. For example, a deviation score of 2 6 corresponds to a score that is below the mean by a distance of 6 points. Because our goal is to compute a measure of the standard distance from the mean, the obvi-ous next step is to calculate the mean of the deviation scores.
- 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)
Moreover, it does so in a way that is free of the problems associated with the variance. Remember: The big problem with the variance is that values are magnified as a result of the squaring process. So, what does the Standard Deviation really tell you about a distribution? Suppose you were told that the Standard Deviation for a distribution has a value of 15.5. This value of 15.5 may mean 15.5 dollars or 15.5 pounds Table 2-14 Calculating the Standard Deviation of a Population Scores/Values Deviations Squared Deviations ( N = 5) ( X ) 5 10 12 14 19 Mean = 12 ( X – Mean) 5 − 12 10 − 12 12 − 12 14 − 12 19 − 12 − 7 − 2 0 + 2 + 7 49 4 0 4 49 106 Sum of Squared Deviations = 106 106/5 = 21.2 (5 is Number of Cases) Square Root of 21.2 = 4.60 Standard Deviation = 4.60 © Cengage Learning 2013 Copyright 2012 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. Measures of Variability or Dispersion 43 or 15.5 test points, depending on what variable you’re looking at and the nature of the data you’ve collected. But still it’s reasonable to ask: So what? So what does the Standard Deviation (or variance, for that matter) really tell us? From my perspective, there are at least three answers to that question. First, you can think of the Standard Deviation as a measure that tells you (sort of) how far scores or values (in general) deviate from the mean. In short, the Standard Deviation tracks along with the overall variability in a distribution. When there is more variability in a distribution, the Standard Deviation increases.
eBook - ePub- Joe F. Hair Jr., Michael Page, Niek Brunsveld(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
variance . It is useful for describing the variability of the distribution and is a good index of the degree of dispersion. The variance is equal to 0 if each and every respondent in the distribution is the same as the mean. The variance becomes larger as the observations tend to differ increasingly from one another and from the mean.Standard Deviation
The variance is used often in statistics, but it does have a major drawback. The variance is a unit of measurement that has been squared. For example, if we measure the number of colas consumed in a day and wish to calculate an average for the sample of respondents, the mean will be the average number of colas, and the variance will be in squared numbers. To overcome the problem of having the measure of dispersion in squared units instead of the original measurement units, we use the square root of the variance, which is called the Standard Deviation. The Standard Deviation describes the spread or variability of the sample distribution values from the mean and is perhaps the most valuable index of dispersion.To obtain the squared deviation, we square the individual deviation scores before adding them (squaring a negative number produces a positive result). After the sum of the squared deviations is determined, the result is divided by the number of respondents minus 1. The number 1 is subtracted from the number of respondents to help produce an unbiased estimate of the Standard Deviation. If the estimated Standard Deviation is large, the responses in a sample distribution of numbers do not fall very close to the mean of the distribution. If the estimated Standard Deviation is small, you know that the distribution values are close to the mean.
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 - 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 - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
In contrast, a larger value of the Standard Deviation for a data set indicates that the values of that data set are spread over a relatively larger range around the mean. The Standard Deviation is obtained by taking the positive square root of the variance. The variance calculated for population data is denoted by σ 2 (read as sigma squared), 2 and the vari- ance calculated for sample data is denoted by s 2 . Consequently, the Standard Deviation calculated for population data is denoted by σ, and the Standard Deviation calculated for sample data is denoted by s. Following are what we will call the basic formulas that are used to calculate the variance and Standard Deviation. 3 σ 2 = ∑ ( x − μ) 2 N and s 2 = ∑ ( x − x ) 2 n − 1 σ = B ∑ ( x − μ) 2 N and s = B ∑ ( x − x ) 2 n − 1 where σ 2 is the population variance, s 2 is the sample variance, σ is the population Standard Deviation, and s is the sample Standard Deviation. The quantity x − μ or x − x in the above formulas is called the deviation of the x value from the mean. The sum of the deviations of the x values from the mean is always zero; that is, ∑ ( x − μ) = 0 and ∑ ( x − x ) = 0. For example, suppose the midterm scores of a sample of four students are 82, 95, 67, and 92, respectively. Then, the mean score for these four students is x = 82 + 95 + 67 + 92 4 = 84 The deviations of the four scores from the mean are calculated in Table 3.5. As we can observe from the table, the sum of the deviations of the x values from the mean is zero; that is, ∑ ( x − x ) = 0. For this reason we square the deviations to calculate the variance and Standard Deviation. Table 3.5 x x - x 82 82 − 84 = −2 95 95 − 84 = +11 67 67 − 84 = −17 92 92 − 84 = +8 ∑ ( x − x ) = 0 From the computational point of view, it is easier and more efficient to use short-cut formulas to calculate the variance and Standard Deviation. By using the short-cut formulas, we reduce the computation time and round-off errors.
eBook - PDFMathematics
A Practical Odyssey
- David Johnson, , Thomas Mowry, , David Johnson, Thomas Mowry(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 274 CHAPTER 4 Statistics FINDING THE VARIANCE AND Standard Deviation: SINGLE VALUES George bowled six games, and his scores were 185, 135, 200, 185, 250, and 155. Find the Standard Deviation of his scores. To find the Standard Deviation, we must first find the variance. The mean of the six data points is 185. The necessary calculations for finding variance are shown in Figure 4.72. Data ( x ) Deviation ( x 2 185) Deviation Squared ( x 2 185) 2 135 2 50 ( 2 50) 2 5 2,500 155 2 30 ( 2 30) 2 5 900 185 0 (0) 2 5 0 185 0 (0) 2 5 0 200 15 (15) 2 5 225 250 65 (65) 2 5 4,225 Sum 5 7,850 Figure 4.72 Finding variance. variance 5 sum of the squares of the deviations n 2 1 s 2 5 7,850 6 2 1 5 7,850 5 5 1,570 The variance is s 2 5 1,570. Taking the square root, we have s 5 Ï 1,570 5 39.62322551 . . . It is customary to round off s to one place more than the original data. Hence, the Standard Deviation of George’s bowling scores is s 5 39.6 points. Because they give us information concerning the spread of data, variance and Standard Deviation are called measures of dispersion . Standard Deviation (and variance) is a relative measure of the dispersion of a set of data; the larger the Standard Deviation, the more spread out the data. Consider George’s stan-dard deviation of 39.6. This appears to be high, but what exactly constitutes a “high” Standard Deviation? Unfortunately, because it is a relative measure, there is no hard-and-fast distinction between a “high” and a “low” Standard Deviation.
eBook - PDF- L S Blake(Author)
- 1994(Publication Date)
- CRC Press(Publisher)
1/30 Mathematics and statistics frequently saves effort. However, this method is not recom- mended for use on computers because of the danger of loss of accuracy when n is large and x~ has several significant figures. 1.12.1.5 Coe~cient of variation The coefficient of variation is the Standard Deviation expressed as a percentage of the mean. This is useful for dealing with properties whose Standard Deviation rises in proportion to the mean, for instance the strengths of concrete as measured by compressive tests on cubes. 1.12.1.6 Standard error The standard error is the Standard Deviation of the mean (or of any other statistic). If in repeated samples of size n from a population the sample means are calculated, the Standard Deviation calculated from these means is expected to have a value: Sm=a/,dn (1.71) where tr is the Standard Deviation of the population. An important result is that whatever the distribution of the parent population (normal or not) the distribution of the sample mean tends rapidly to normal form as the sample size increases. 1.13 Samples and population 1.13.1 Representations 1.13.1.1 Frequency The number of observations having values between two speci- fied limits. It is often convenient to group observations by dividing the range over which they extend into a number of small, equal, intervals. The number of observations falling in each interval is then the frequency for that interval. This allows a convenient representation of the information by means of a histogram. 1.13.1.2 Histogram or bar chart A diagram in which the observations are represented by rec- tangles or bars with one side equal to the interval over which the observations occurred and the other equal to the frequency of occurrence of observations within that range (Figure 1.45). 1.13.1.3 Distribution curve The result of refining a histogram by reducing the size of the intervals and correspondingly increasing the total number of observations.
eBook - PDFPortfolio Selection
Efficient Diversification of Investments
- Harry M. Markowitz(Author)
- 1968(Publication Date)
- Yale University Press(Publisher)
76 PORTFOLIO SELECTION The variance of a random variable, then, is the expected value of the squared deviation from the expected value. The variance of a random variable can be computed from a table of probabilities. The entries in the second column of Table 2 can be inter-preted as the probabilities associated with the random variable on the outer ring of the wheel in Figure 3. The variance of the random variable is computed from these probabilities exactly as the variance of return of the past series was computed from its table of relative frequencies. Figure 3. The variance of a random variable. The outer ring generates the random variable r. The second ring generates the random variable r', equal to r minus the expected value of r. The third ring generates (r') 2 . The expected value of (r') 2 is the variance of r. The Standard Deviation of a random variable is the square root of its variance. The Standard Deviation of a random variable measures how close the random variable is likely to be to its expected value. The variance of an uncertain future event is defined in terms of proba-bility beliefs exactly as variance was defined, in terms of objective probabilities, for a random variable. If the entries in the second column of Table 2 represented probability beliefs, then .0034 would be the variance based on these probabilities. As before, Standard Deviation is the square root of variance. Standard Deviation, in this case, measures the degree of uncertainty associated with the future event. In Chapter XIII the Standard Deviation is compared with other measures of risk and variability. For most of the measures considered, the efficient Standard DeviationS AND VARIANCES 77 portfolios produced by using Standard Deviation are to be preferred. Some measures which seem reasonable offhand produce completely unsatisfactory portfolios. One of the measures considered, the semi-deviation, produces efficient portfolios somewhat preferable to those of the Standard Deviation.
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