Standard Deviation of Random Variable

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  Mind on Statistics (with JMP Printed Access Card)

  • Jessica Utts, Robert Heckard(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Standard Deviation for a Discrete Random Variable We first encountered the idea of standard deviation in Chapter 2, where we learned how to compute the standard deviation as a measure of spread for a quantitative vari-able and then used it in the Empirical Rule. Similarly, the standard deviation of a Copyright 2014 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. Random Variables 275 discrete random variable quantifies how spread out the possible values of a discrete random variable might be, weighted by how likely each value is to occur. As with the standard deviation of a set of measurements, the standard deviation of a random vari-able is roughly the average distance the random variable falls from its mean, or ex-pected value, over the long run. F O R M U L A Variance and Standard Deviation of a Discrete Random Variable If X is a random variable with possible values of x 1 , x 2 , x 3 , ... , occurring with proba-bilities p 1 , p 2 , p 3 , ... , and with expected value E ( X ) µ, then, Variance of X V X 2 x i 2 p i Standard deviation of X square root of V X x i 2 p i The sum is taken over all possible values of the random variable X . EXAMPLE 8.13 Stability or Excitement—Same Mean, Different Standard Deviations Suppose you decide to invest $100 in a scheme that you hope will make some money. You have two choices of investment plans, and you must decide which one to choose.

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  Portfolio 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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  Introduction to Statistics and Data Analysis

  • Roxy Peck, Chris Olsen, , Tom Short, Roxy Peck, Chris Olsen, Tom Short(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  7.4 Mean and Standard Deviation of a Random Variable 363 25 will hardly ever be observed, so it won’t contribute much to variability in a long sequence of observations. This is why each squared deviation is multiplied by the probability associ-ated with the value to obtain a measure of variability. DEFINITIONS Variance of a discrete random variable: The variance of a discrete random vari-able x , denoted by s 2 x , is calculated by 1. subtracting the mean from each possible x value to obtain the deviations 2. squaring each deviation 3. multiplying each squared deviation by the probability of the corresponding x value 4. adding these quantities s 2 x  os x 2 m x d 2 p s x d Standard deviation of a discrete random variable: The standard deviation of a discrete random variable x , denoted by s x , is the square root of the variance. all possible x values Example 7.12 Glass Panels Revisited For x  number of flaws in a glass panel from the first supplier in Example 7.11, s 2 x  s 0 2 1 d 2 p s 0 d 1 s 1 2 1 d 2 p s 1 d 1 s 2 2 1 d 2 p s 2 d 1 s 3 2 1 d 2 p s 3 d  s 1 ds 0.4 d 1 s 0 ds 0.3 d 1 s 1 ds 0.2 d 1 s 4 ds 0.1 d  1.0 The standard deviation of x is then s x  Ï s 2 x  Ï 1.0  1.0. For y  the number of flaws in a glass panel from the second supplier, s 2 y  s 0 2 1 d 2 s 0.2 d 1 s 1 2 1 d 2 s 0.6 d 1 s 2 2 1 d 2 s 0.2 d  0.4 Then s y  Ï 0.4  0.632. The fact that s x is greater than s y confirms the impression about the variability in x and y conveyed by the probability distributions shown in Figure 7.12. Example 7.13 More on Apgar Scores Reconsider the distribution of Apgar scores for children born at a certain hospital, introduced in Example 7.10. What is the probability that a randomly selected child’s score will be within 2 standard deviations of the mean score? As Figure 7.13 shows, values of x within 2 standard deviations of the mean are those for which m 2 2 s , x , m 1 2 s From Example 7.10 we already know m x  7.16.

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  Transdisciplinary Engineering Design Process

  • Atila Ertas(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The mean deviation can be calculated using the formula Mean deviation = ∑ n i = 1 | x i − x | n . (7.3) 7.3.3 Standard Deviation While the mean deviation is a useful measure of variability, it may not be practical to use because of the absolute value involved in calculating the mean deviation. Thus, the standard deviation, 448 7 Statistical Decisions which is the most reliable measure of variability, is used in most cases. The standard deviation, S , of a random sample can be calculated using the formula S = √ ∑ n i = 1 ( x i − x ) 2 n − 1 , (7.4) where n is the number of observations. To calculate the standard deviation of a population, x and n are replaced by 𝜇 and N in equation (7.4): ̂ 𝜎 = √ ∑ N i = 1 ( x i − 𝜇 ) 2 N − 1 (7.5) where ̂ 𝜎 is the standard deviation of a population. 7.3.4 Variance The square of the standard deviation is another important measure known as the variance of a random variable; it shows the spread of the distribution. The variances of the random sample and the population, respectively, are given by the formulas S 2 = ∑ n i = 1 ( x i − x ) 2 n − 1 (7.6) and ̂ 𝜎 2 = ∑ N i = 1 ( x i − 𝜇 ) 2 N − 1 . (7.7) Example 7.3 Assume that for a good clearance fit between a pin and bushing, 1 million pin-bushings were manufactured. Suppose clearance tolerances for two sets of pin-bushing samples drawn from the 1 million population are 0.02, 0.02, 0.02, 0.03, 0.03, 0.04, 0.04, 0.04, 0.04, 0.04 (first box) and 0.01, 0.02, 0.02, 0.03, 0.03, 0.03, 0.03, 0.04, 0.05, 0.06 (second box). Calculate the sample mean, standard deviation, and variance for the data. Solution As shown above, both data sets give the same mean of 0.032 mm tolerance. When both sample data are carefully checked, one would have more confidence in a mean value calculated from the first sample data set. To understand which sample truly represents the population mean, we calculate the standard deviation of the samples.

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  Statistical Techniques for Data Analysis

  • John K. Taylor, Cheryl Cihon(Authors)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  2 . The symbol V is sometimes used to designate variance.

  Ordinarily one is not dealing with a population, but rather with a sample of n individuals of the population. The individual measured values may be indicated by the symbols X1 , X2 , Xn .

  The sample mean, (called X bar), calculated as shown in the figure, is hopefully a good estimate of the population mean–that is why the measurements were made in the first place! One can calculate the sample standard deviation, s, using the formula shown in the figure. Likewise one can calculate the sample variance, s2 , as shown. Of course, one can use a simple calculator to do this, as indicated by PUSH BUTTON. This convenience is good because arithmetic is hard work and one may make mistakes. However, one should make a few calculations by the formula just to understand and appreciate what the calculator is doing.

  Remember that s is an estimate of the standard deviation of the population and that it is not σ. It is often called “the standard deviation”, maybe because the term ‘estimate of the standard deviation’ is cumbersome. It is, of course, the sample-based standard deviation but that term is also cumbersome. The standard deviation and its estimates always have the same units as those for X. When considering variability, a dimensionless quantity, the coefficient of variation, cv, is frequently encountered. It is simply

  Figure 4.1. Population values and sample estimates.

  If one knows cv and the level, X, s can be calculated.

  Figure 4.2. Distribution of means.

  Another term frequently used is called the relative standard deviation, RSD, and it is calculated as

  RSD = cv×100

  The relative standard deviation is thus expressed as a percent. There could be room for confusion when results are reported on a percentage basis, as the percentage of sulfur in a coal sample, for example. Here the value for s could be in units of percent. In such cases, one can make the distinction by using the terms % relative and % absolute.

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  Essentials of Business Research Methods

  • 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.

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  Felder's Elementary Principles of Chemical Processes

  • Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The range is the crudest measure of scatter: it involves only two of the measured values and gives no indication of whether or not most of the values cluster close to the mean or scatter widely around it. The sample variance is a much better measure. To define it we calculate the deviation of each measured value from the sample mean, X j  X ð j ¼ 1; 2; . . . ; NÞ, and then calculate Sample Variance: s 2 X ¼ 1 N  1 X 1  X   2 þ X 2  X   2 þ ∙ ∙ ∙ þ X N  X   2 h i (2.5-3) The degree of scatter may also be expressed in terms of the sample standard deviation, by definition the square root of the sample variance: Sample Standard Deviation: s X ¼ ffiffiffiffiffi s 2 X q (2.5-4) The more a measured value (X j ) deviates from the mean, either positively or negatively, the greater the value of ðX j  XÞ 2 and hence the greater the value of the sample variance and sample standard deviation. If these quantities are calculated for the data sets of Figure 2.5-1, for example, relatively small values are obtained for Set (a) (s 2 X ¼ 0:98, s X ¼ 0:99) and large values are obtained for Set (b) (s 2 X ¼ 132, s X ¼ 11:5). For typical random variables, roughly two-thirds of all measured values fall within one standard deviation of the mean; about 88% fall within two standard deviations; and about 99% fall within three standard deviations. 4 A graphical illustration of this statement is shown in Figure 2.5-2. Of the 37 measured values of X, 27 fall within one standard deviation of the mean, 33 within two standard deviations, and 36 within three standard deviations. Values of measured variables are often reported with error limits, such as X ¼ 48:2  0:6. This statement means that a single measured value of X is likely to fall between 47.6 and 48.8.

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  Elementary Principles of Chemical Processes

  • Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The range is the crudest measure of scatter: it involves only two of the measured values and gives no indication of whether or not most of the values cluster close to the mean or scatter widely around it. The sample variance is a much better measure. To define it we calculate the deviation of each measured value from the sample mean, X j  X  j  1; 2; . . . ; N, and then calculate Sample Variance: s 2 X  1 N  1 X 1  X   2  X 2  X   2  ∙ ∙ ∙  X N  X   2 h i (2.5-3) The degree of scatter may also be expressed in terms of the sample standard deviation, by definition the square root of the sample variance: Sample Standard Deviation: s X  ffiffiffiffiffi s 2 X q (2.5-4) The more a measured value (X j ) deviates from the mean, either positively or negatively, the greater the value of X j  X 2 and hence the greater the value of the sample variance and sample standard deviation. If these quantities are calculated for the data sets of Figure 2.5-1, for example, relatively small values are obtained for Set (a) (s 2 X  0:98, s X  0:99) and large values are obtained for Set (b) (s 2 X  132, s X  11:5). For typical random variables, roughly two-thirds of all measured values fall within one standard deviation of the mean; about 88% fall within two standard deviations; and about 99% fall within three standard deviations. 4 A graphical illustration of this statement is shown in Figure 2.5-2. Of the 37 measured values of X, 27 fall within one standard deviation of the mean, 33 within two standard deviations, and 36 within three standard deviations. Values of measured variables are often reported with error limits, such as X  48:2  0:6. This statement means that a single measured value of X is likely to fall between 47.6 and 48.8.

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  Behavioral 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)

  38 Behavioral Research and Analysis If we were dealing with the entire population instead of just a sample or subset, we would cal-culate the population standard deviation, also called the biased estimate of the standard devia-tion. The only difference between this and the aforementioned sample or unbiased estimate of the standard deviation is that n instead of n – 1 is used in the denomination. (Technically the total population should be depicted with a capital N to distinguish it from a lower case n for samples; but because many statistics books simply use a lower case n for both, it is used here in this common form. Elsewhere, the capital N refers specifically to the entire population.) Using the same data, Population SD = = = ∑ -= = ∑ σ σ n X n X n 2 2 144 10 3 795 ( ) . Relation of Standard Deviation to Variance The standard deviation also may be defined as the square root of the variance; the variance is defined as the average squared deviation. Notice again the distinction between variability in data representing the entire population ( N ) composed compared to the subset sample ( n ) that uses n –1 in the denominator. Population Variance = ∑ = ∑ -= = ∑ x n X n X n n 2 2 2 2 2 ( ) σ σ Also = ∑ = ∑ -( ) = = ∑ 2 2 2 2 2 x N x N x N N σ σ Sample Variance = ∑ -= ∑ --= = ∑ -x n X n s X n n 2 2 1 2 2 1 1 2 ( ) σ The variance is a measure of variation but is a square rather than a linear measure. Note that vari-ances are additive and standard deviations are not. This means you can perform statistical opera-tions on variances but not on standard deviations. Averaging Standard Deviations We noted earlier that standard deviations are not additive.

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  Finite Mathematics

  • Stefan Waner, Steven Costenoble(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Variance and Standard Deviation of a Finite Random Variable Recall that the expected value of a random variable X is a prediction of the average of a large sample of values of X . Can we similarly predict the variance of a large sample? Suppose we have a sample x 1 , x 2 , . . . , x n . If n is large, the sample and popu-lation variances are essentially the same, so we concentrate on the population vari-ance, which is the average of the numbers 1 x i 2 x 2 2 . This average can be predicted by using E 13 X 2 m 4 2 2 , the expected value of 1 X 2 m 2 2 . In general, we make the following definition. Variance and Standard Deviation of a Finite Random Variable If X is a finite random variable taking on values x 1 , x 2 , . . . , x n , then the vari-ance of X is s 2 5 E 13 X 2 m 4 2 2 5 1 x 1 2 m 2 2 P 1 X 5 x 1 2 1 1 x 2 2 m 2 2 P 1 X 5 x 2 2 1 c 1 1 x n 2 m 2 2 P 1 X 5 x n 2 5 a i 1 x i 2 m 2 2 P 1 X 5 x i 2 . The standard deviation of X is then the square root of the variance: s 5 s 2 . To compute the variance from the probability distribution of X , first compute the expected value m , and then compute the expected value of 1 X 2 m 2 2 . Quick Example 5. The following distribution has expected value m 5 E 1 X 2 5 2 : x 2 1 2 3 10 P 1 X 5 x 2 .3 .5 .1 .1 The variance of X is s 2 5 1 x 1 2 m 2 2 P 1 X 5 x 1 2 1 1 x 2 2 m 2 2 P 1 X 5 x 2 2 1 c 1 1 x n 2 m 2 2 P 1 X 5 x n 2 5 1 2 1 2 2 2 2 1 .3 2 1 1 2 2 2 2 2 1 .5 2 1 1 3 2 2 2 2 1 .1 2 1 1 10 2 2 2 2 1 .1 2 5 9.2. The standard deviation of X is s 5 ! 9.2 < 3.03 . Note We can interpret the variance of X as the number we expect to get for the vari-ance of a large sample of values of X , and similarly for the standard deviation. ■ Figure 8 Frequency 15 3 9 Approximately 99.7% Approximately 0.15% Life Span Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300

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