Mean Value and Standard Deviation

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  Fundamental Concepts and Applications of Averages and Means

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ The calculation of the sum of squared deviations can be related to moments calculated directly from the data. The standard deviation of the sample can be computed as: The sample standard deviation can be computed as: For a finite population with equal probabilities on all points, we have Thus, the standard deviation is equal to the square root of (the average of the squares less the square of the average). Interpretation and application A large standard deviation indicates that the data points are far from the mean and a small standard deviation indicates that they are clustered closely around the mean. For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. In a loose sense, the standard deviation tells us how far from the mean the data points tend to be. It will have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population {1000, 1006, 1008, 1014} may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters. Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements should give the precision of those measurements. When deciding whether measurements agree with a theoretical prediction the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised.

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  Averages and Means (Concepts & Applications)

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  In a certain sense, the standard deviation is a natural measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation ________________________ WORLD TECHNOLOGIES ________________________ from the mean is smaller than from any other point. The precise statement is the following: suppose x 1 , ..., x n are real numbers and define the function: Using calculus or by completing the square , it is possible to show that σ( r ) has a unique minimum at the mean: The coefficient of variation of a sample is the ratio of the standard deviation to the mean. It is a dimensionless number that can be used to compare the amount of variance between populations with means that are close together. The reason is that if you compare popula-tions with same standard deviations but different means then coefficient of variation will be bigger for the population with the smaller mean. Thus in comparing variability of data, coefficient of variation should be used with care and better replaced with another method. Often we want some information about the precision of the mean we obtained. We can obtain this by determining the standard deviation of the sampled mean. The standard deviation of the mean is related to the standard deviation of the distribution by: where N is the number of observation in the sample used to estimate the mean. This can easily be proven with: hence Resulting in: ________________________ WORLD TECHNOLOGIES ________________________ Worked example The standard deviation of a discrete random variable is the root-mean-square (RMS) deviation of its values from the mean. If the random variable X takes on N values (which are real numbers) with equal probability, then its standard deviation σ can be calculated as follows: 1. Find the mean, , of the values. 2. For each value calculate its deviation from the mean. 3. Calculate the squares of these deviations. 4. Find the mean of the squared deviations.

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  Statistics for The Behavioral Sciences

  • 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.

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  Pharmaceutical Statistics

  Practical and Clinical Applications, Fifth Edition

  • Sanford Bolton, Charles Bon(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  As discussed later in this chapter (sect. 1.5.4), the standard deviation of a mean will decrease with larger sample sizes. 1.5.3 Coefficient of Variation The variability of data may often be better described as a relative variation rather than as an absolute variation, such as that represented by the standard deviation or range. One common way of expressing the variability, which takes into account its relative magnitude, is the ratio of the standard deviation to the mean, s . d ./ X . This ratio, often expressed as a percentage, is called the coefficient of variation , abbreviated as CV, or RSD, the relative standard deviation. A CV of 0.1 or 10% means that the s.d. is one-tenth of the mean. This way of expressing variability is useful in many situations. It puts the variability in perspective relative to the magnitude of the measurements and allows a comparison of the variability of different kinds of measurements. For example, a group of rats of average weight 100 g and s.d. of 10 g has the same relative variation (CV) as a group of animals with average weight 70 g and s.d. of 7 g. Many measurements have an almost constant CV, the magnitude of the s.d. being proportional to the mean. In biological data, the CV is often between 20% and 50%, and one would not be surprised to see an occasional CV as high as 100% or more. The relatively large CV observed in biological experiments is due mostly to “biological variation,” the lack of reproducibility in living material. On the other hand, the variability in chemical and instrumental analyses of drugs is usually relatively small. Thus it is not unusual to find a CV of less than 1% for some analytical procedures. 1.5.4 Standard Deviation of the Mean (Standard Error of the Mean) The s.d. is a measure of the spread of a group of individual observations, a measure of their variability.

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  Mastering Corporate Finance Essentials

  The Critical Quantitative Methods and Tools in Finance

  • Stuart A. McCrary(Author)
  • 2010(Publication Date)
  • Wiley

    (Publisher)

  The median is an alternative to the mean for establishing the general magnitude of data. The median is the point in a population or sample for which half of the remaining data is larger than the median and half of the remaining data is smaller than the median. In many cases, the mean and the median are approximately equal.

  Most applications of statistics in finance use the mean rather than the median. The examples in this book use the mean in all cases.

  STANDARD DEVIATION MEASURES THE NOISE

  The mean simplifies a collection of observations, but much of the detail is lost. Without seeing the individual observations, it is impossible to tell whether most observations are approximately equal to the mean or whether they vary tremendously. The most common measures of dispersion are variance and the closely related statistic called standard deviation. The observations in Figure 2.1 spread out over a mean of 9.5 percent. In fact, observations near 9.5 percent (including values slightly above and slightly below 9.5 percent) are the most commonly occurring observations. Observations below 8.5 percent or above 10.5 percent occur less frequently. Observations below 7.5 percent or above 11.5 percent are rare.

  FIGURE 2.1 Yields with 9.50 percent Mean and 1 percent Standard Deviation

  Figure 2.2 shows a similar set of observations. These observations are also centered around a mean of 9.5 percent. However, observations 1 or 2 percent above or below the mean are common.

  Table 2.1 is a histogram of the 1,000 returns used to create Figure 2.1 and Figure 2.2 . The first line contains a return of 2.5 percent followed by a count of zero instances of a return of 2.5 percent or lower in both Figures 2.1 and 2.2 . The following line contains a count of the number of returns between 2.5 percent and 3.5 percent. Figure 2.1 contains no returns between 2.5 percent and 3.5 percent, but Figure 2.2 contains four returns in that range.

  Notice that the number of returns close to 9.5 percent is higher for the returns in Figure 2.1 than for those in Figure 2.2 .

  Figure 2.3 displays the count of returns in Table 2.1 visually. As with Table 2.1 , it is clear that the returns in Figure 2.2 span a wider range. In other words, although the returns average about 9.5 percent in both cases, the chance that a return differs from the mean is much higher for the data in Figure 2.2 than for the data in Figure 2.1

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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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  Modern Engineering Statistics

  • Thomas P. Ryan(Author)
  • 2007(Publication Date)
  • Wiley-Interscience

    (Publisher)

  In one case (the normal distribution), µ and σ do appear explicitly in the function that represents the distri- bution, and these are the mean and standard deviation, respectively. Generally, this is not the case, however, with the mean and variance being functions of the parameters that appear explicitly in the probability function. In those cases, the objective is the same: to estimate the unknown parameters and to estimate the mean, variance, and/or standard deviation. Although simple statistics such as x are the ones most frequently used, it is important to recognize that robust statistics such as the trimmed mean, trimmed variance, and trimmed standard deviation will generally be superior to their untrimmed counterparts in the presence of extreme observations. Estimation is discussed in more detail in Chapter 4. We should also keep in mind that in this age of extremely large data sets and data mining, there are challenges in data summarization and data visualization that of course were not envisioned when simple statistics were used extensively and were deemed adequate. Clearly, we would not want to compute the average of 10 million observations, even if we have the capability of doing so as those 10 million observations would almost certainly come from a mixture of distributions with different means. Therefore, we should keep in mind the computing environment at the time and the objectives that exist in particular situations regarding parameter estimation. REFERENCE Rocke, D. M. (1989). Robust control charts. Technometrics, 31, 173–184. EXERCISES 2.1. The following are six measurements on a check standard, with the check standard measurements being resistivities at the center of a 100 ohm · cm wafer: 96.920, 97.118, 97.034, 97.047, 97.127, and 96.995. EXERCISES 59 (a) Compute the average and the standard deviation of the measurements.

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  Introductory Statistics

  • Prem S. Mann(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  17. The mean, as a measure of center, has the disadvantage of being influenced by extreme values. Illustrate this point with an example. 18. The range, as a measure of spread, has the disadvantage of being influenced by extreme values. Illustrate this point with an example. 19. When is the value of the standard deviation for a data set zero? Give one example of such a data set. Calculate the standard deviation for that data set to show that it is zero. 20. The following table gives the frequency distribution of the num- bers of computers sold during each of the past 25 weeks at an elec- tronics store. Computers Sold Frequency 4 to 9 2 10 to 15 4 16 to 21 10 22 to 27 6 28 to 33 3 a. What does the frequency column in the table represent? b. Calculate the mean, variance, and standard deviation. 21. The members of a very large health club were observed on a ran- domly selected day. The distribution of times they spent that day at the health club was found to have a mean of 91.8 minutes and a stand- ard deviation of 16.2 minutes. Suppose these values of the mean and standard deviation hold true for all members of this club. a. Using Chebyshev’s theorem, find the minimum percentage of this health club’s members who spend time at this health club between i. 59.4 and 124.2 minutes ii. 51.3 and 132.3 minutes *b. Using Chebyshev’s theorem, find the interval that contains the times spent at this health club by at least 89% of members. 22. The ages of cars owned by all people living in a city have a bell- shaped distribution with a mean of 7.3 years and a standard deviation of 2.2 years. a. Using the empirical rule, find the (approximate) percentage of cars in this city that are i. 5.1 to 9.5 years old ii. .7 to 13.9 years old *b. Using the empirical rule, find the interval that contains the ages of (approximate) 95% of the cars owned by all people in this city.

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