Mathematics
Normal Distribution Percentile
The normal distribution percentile refers to the percentage of data points that fall below a particular value in a normal distribution curve. It is a measure of relative standing within the distribution, with the 50th percentile representing the median. This concept is widely used in statistics and probability to understand the distribution of data.
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7 Key excerpts on "Normal Distribution Percentile"
eBook - PDFPharmaceutical Statistics
Practical and Clinical Applications, Fifth Edition
- Sanford Bolton, Charles Bon(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
To compute percentiles, the data are ranked in order of magnitude, from smallest to largest. The n th percentile denotes a value below which n % of the data are found, and above which (100 − n ) % of the data are found. The 10th, 25th, and 75th percentiles represent values below which 10%, 25%, and 75%, respectively, of the data occur. For the tablet potencies shown in Table 1.5, the 10th percentile is 95.5 mg; 10% of the tablets contain less than 95.5 mg and 90% of the tablets contain more than 95.5 mg of drug. The 25th, 50th, and 75th percentiles are also known as the first, second, and third quartiles, respectively. The mode is less often used as the central, or typical, value of a distribution. The mode is the value that occurs with the greatest frequency. For a symmetrical distribution that peaks in the center, such as the normal distribution (see chap. 3), the mode, median, and mean are identical. For data skewed to the right (e.g., incomes), which contain a relatively few very large values, the mean is larger than the median, which is larger than the mode (Fig. 10.1). 1.5 MEASUREMENT OF THE SPREAD OF DATA The mean (or median) alone gives no insight or information about the spread or range of values that comprise a data set. For example, a mean of five values equal to 10 may comprise the numbers 0, 5, 10, 15, and 20 or 5, 10, 10, 10, and 15. The mean, coupled with the standard deviation or range , is a succinct and minimal descrip-tion of a group of experimental observations or a data distribution. The standard deviation and the range are measures of the spread of the data; the larger the magnitude of the standard deviation or range, the more spread out the data are. A standard deviation of 10 implies a wider range of values than a standard deviation of 3, for example. 1.5.1 Range The range , denoted as R , is the difference between the smallest and the largest values in the data set. For the data in Table 1.1, the range is 152, from − 97 to + 55 mg%.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
However, whether this assumption is valid or not in practice is debatable. A famous remark of Lippmann says: “Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics.” • In standardized testing, results can be made to have a normal distribution. This is done by either selecting the number and difficulty of questions (as in the IQ test), or by transforming the raw test scores into “output” scores by fitting them to the normal distribution. For example, the SAT’s traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100. • Many scores are derived from the normal distribution, including percentile ranks ( “percentiles” or “quantiles”), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, a number of behavioral statistical procedures are based on the assumption that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores. • In hydrology the distribution of long duration river discharge or rainfall (e.g monthly and yearly totals, consisting of the sum of 30 respectively 360 daily values) is often thought to be practically normal according to the central limit theorem. The blue picture illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis. ________________________ WORLD TECHNOLOGIES ________________________ Generating values from normal distribution The bean machine, a device invented by sir Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins.
No longer available |Learn more- K. Paul Nesselroade, Jr., Laurence G. Grimm, K. Paul Nesselroade, Jr(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
From this point in the text through Chapter 16, the normal curve will be heavily relied upon and extensively used, so much so that data sets will be assumed to be normally distributed unless there is specific information to the contrary. The Importance of Normal Distributions Normal distributions are of fundamental importance in the field of statistics for two reasons. First, it is a very common distribution shape. Measurements of many naturally occurring phenomena, including psychological concepts like intelligence, anxiety, mood, and so on, are normally distributed. Second, if we were to take a sample of scores from any shaped population, calculate M , then replace the scores and take another sample of the same size, calcu-late M , replace them, and so on until we had an exceedingly large number of sample means; those means would be normally distributed. This second observation forms an important theoretical basis for most statistical ana-lyses used to test hypotheses. This point will be developed extensively in Chapter 7. Characteristics of Normal Distributions For a distribution to be called normal, it must conform to a certain mathemat-ical model: y = 1 2 πσ 2 e -X -μ 2 2 σ 2 where y = the ordinate on the graph, that is, the height of the curve for a given X X = any given score μ = population mean σ 2 = population variance π = the value of pi: 3.1416 (rounded) e = 2.718 (rounded), the base of the system of natural logarithms Do not experience “ formula shock ” ; we will likely never use this equation. However, this formula can be used to make some valid points about normal dis-tributions. The formula for a normal curve is a general formula; it is not tied to a specific set of scores. All the values in the equation are fixed, except for X , μ , and 5.2 The Normal Distributions 133 σ 2 , which will vary from distribution to distribution. To draw any curve, we need to know, for each X , how far up on the graph to go to plot a point.
No longer available |Learn more- Anthony Hayter(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
For example, the 80th percentile satisfies ( x ) = 0 . 8 and can be found by using the table “backward” by searching for the value 0.8 in the body of the table. It is found that ( 0 . 84 ) = 0 . 7995 and that ( 0 . 85 ) = 0 . 8023, so that the 80th percentile point is somewhere between 0.84 and 0.85. (If further accuracy is required, interpolation may be attempted or a computer software package may be utilized.) If the value x is required for which P ( | Z | ≤ x ) = 0 . 7 as illustrated in Figure 5.10, notice that the symmetry of the standard normal distribution implies that ( − x ) = 0 . 15 Table I then indicates that the required value of x lies between 1.03 and 1.04. The percentiles of the standard normal distribution are used so frequently that they have their own notation. For α < 0 . 5, the ( 1 − α) × 100th percentile of the distribution is denoted by z α , so that ( z α ) = 1 − α as illustrated in Figure 5.11, and some of these percentile points are given in Table I. The percentiles z α are often referred to as the “critical points” of the standard normal distribution. Copyright 2011 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.
- David Howell(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
The normal distribution is a very common distribution in statistics, and it is often taken as a good description of how observations on a dependent variable are distributed. We very often assumed that the data in our sample came from a normally distributed population. This chapter began by looking at a pie chart representing people under correc-tional supervision. We saw that the area of a section of the pie is directly related to the probability that an individual would fall in that category. We then moved from the pie chart to a bar graph, which is a better way of presenting the data, and then moved to a histogram of data that have a roughly normal distribution. The purpose of those transitions was to highlight the fact that area under a curve can be linked to probability. The normal distribution is a symmetric distribution with its mode at the cen-ter. In fact, the mode, median, and mean will be the same for a variable that is nor-mally distributed. We saw that we can convert raw scores on a normal distribution to z scores by simply dividing the deviation of the raw score from the population mean ( m ) by the standard deviation of the population ( s ). The z score is an important statistic because it allows us to use tables of the standard normal distribution (often denoted N ( m , s 2 )). Once we convert a raw score to a z score we can immediately use the tables of the standard normal distribution to compute the probability that any observation will fall within a given interval. We also saw that there are a number of measures that are directly related to z . For example, data are often reported as coming from a population with a mean of 50 and a standard deviation of 10.
- William DeCoursey(Author)
- 2003(Publication Date)
- Newnes(Publisher)
157 CHAPTER 7 The Normal Distribution This chapter requires a good knowledge of the material covered in sections 2.1, 2.2, 3.1, 3.2, and 4.4. Chapter 6 is also helpful as background. The normal distribution is the most important of all probability distributions. It is applied directly to many practical problems, and several very useful distributions are based on it. We will encounter these other distributions later in this book. 7.1 Characteristics Many empirical frequency distributions have the following characteristics: 1. They are approximately symmetrical, and the mode is close to the centre of the distribution. 2. The mean, median, and mode are close together. 3. The shape of the distribution can be approximated by a bell: nearly flat on top, then decreasing more quickly, then decreasing more slowly toward the tails of the distribution. This implies that values close to the mean are relatively frequent, and values farther from the mean tend to occur less frequently. Remember that we are dealing with a random variable, so a frequency distribution will not fit this pattern exactly. There will be random variations from this general pattern. Remember also that many frequency distributions do not conform to this pattern. We have already seen a variety of frequency distributions in Chapter 4, and many other types of distribution occur in practice. Example 4.2 showed data on the thickness of a particular metal part of an optical instrument as items came off a production line. A histogram for 121 items is shown in Figure 4.4, reproduced here. Figure 4.4: Histogram of Thickness of Metal Part 0 10 20 30 40 50 Class Frequency per Class Width of 0.05 mm 3.220 3.270 3.320 3.370 3.420 3.470 3.520 3.570 Thickness, mm Thickness of Part Relative Class Frequency 0 0.083 0.165 0.248 0.331 0.413 Chapter 7 158 We can see that the characteristics stated above are present, at least approxi-mately, in Figure 4.4.
eBook - PDFComparative Analysis Of Nations
Quantitative Approaches
- Robert Perry(Author)
- 2019(Publication Date)
- Routledge(Publisher)
Unfortunately, as any student of cross-national analysis is all too painfully aware, data availability and the sheer limitation of the number of countries within the world make 94 The Role if the Normal Distribution in Cross-National Research 95 samples that perfectly conform to assumptions of the normal distribution a rarity (if not an impossibility). This, however, should not preclude students from becoming fa-miliar with the uncomfortable, and often challenging, task of thinking beyond the properties of the literal sample to considering the more abstract, and practical, impli-cations that are inherent within the statistical analysis of their samples. In this chapter we will introduce you to a more detailed discussion of the logic of the normal distribution, with particular reference to the standard areas under the nor-mal curve. Once you have grasped the logic of the normal distribution, we can then show how to use the statistical output from univariate analysis to formulate assess-ments of the population's general properties and to go from there to a consideration of country-specific comparisons. The workhorse statistics in this chapter are once again the mean and standard deviation. Our goal, in this chapter and the next, is to apply the mean and standard deviation, not merely interpret their meaning. In applying these two statistics we often rely upon the construction of a critical statistic called the z-score. We will show how one com-putes and interprets a z-score, and how it plays an indispensable role in cross-national analysis. While the beginning student often dismisses discussions of the normal distribution as boring and overly detailed, the truth is that the logic of the normal distribution of-fers the student of cross-national analysis important insight that is invaluable in the broader enterprise of comparison.
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