Psychology
Normal Distribution Psychology
The normal distribution in psychology refers to a bell-shaped curve that represents the distribution of many psychological traits and behaviors in the population. It is characterized by a symmetrical shape with the majority of scores falling near the mean and fewer scores at the extremes. This distribution is important for understanding and analyzing various psychological phenomena and measurements.
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10 Key excerpts on "Normal Distribution Psychology"
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Normal distribution 3 A theoretical mathematical distribution that specifies the relative frequency of a set of scores in a population. mean ( ) and the standard deviation ( ) of the distribution. A normal distribution with and is illustrated in Figure 6.1. The distribution shown is a mathemati- cal distribution that does not represent any actually measured population of scores. No real population of scores is distributed so precisely that it will be exactly normal, but it will be useful at times to assume that some populations of scores are approximately nor- mally distributed. In this chapter, however, we focus on the normal distribution as a the- oretical mathematical distribution. The Normal Distribution and the Behavioral Sciences The normal distribution developed from the work of Jakob Bernoulli (1654–1705), Abraham de Moivre (1667–1754), Pierre Remond de Montmort (1678–1719), Karl Friedrich Gauss (1777–1855), and others. Their interests were in developing mathematical approximations for probabilities encountered in various games of chance or in the distribution of errors to be expected in observations, such as in astronomy or physics. The normal distribution soon became an important distribution to statisticians because many problems in statistics can be solved only if a normal distribution is assumed. = 10 = 70 CHAPTER 6 The Normal Distribution, Probability, and Standard Scores 107 6.1 TABLE Notation for samples and populations Sample Population Descriptive characteristics Statistics Parameters Mean Standard deviation s Variance s 2 2 X .40 .30 .20 .10 Relative frequency 100 + 3 90 + 2 80 + 1 70 60 – 1 50 – 2 40 – 3 Scores .00 Figure 6.1 A normal distribution with and . = 10 = 70 The normal distribution also gained importance for behavioral scientists because it is reasonable to assume that any behavioral measures determined by a large number of inde- pendent variables will be approximately normally distributed for a population.
- 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.
eBook - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
6 | THE NORMAL DISTRIBUTION Figure 6.1 If you ask enough people about their shoe size, you will find that your graphed data is shaped like a bell curve and can be described as normally distributed. (credit: Ömer Ünlϋ) Introduction Chapter Objectives By the end of this chapter, the student should be able to: • Recognize the normal probability distribution and apply it appropriately. • Recognize the standard normal probability distribution and apply it appropriately. • Compare normal probabilities by converting to the standard normal distribution. The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some of your instructors may use the normal distribution to Chapter 6 | The Normal Distribution 361 help determine your grade. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is extremely important, but it cannot be applied to everything in the real world. In this chapter, you will study the normal distribution, the standard normal distribution, and applications associated with them. The normal distribution has two parameters (two numerical descriptive measures), the mean (μ) and the standard deviation (σ). If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ), we designate this by writing Figure 6.2 The probability density function is a rather complicated function. Do not memorize it. It is not necessary. f(x) = 1 σ ⋅ 2 ⋅ π ⋅ e − 1 2 ⋅ ⎛ ⎝ x − µ σ ⎞ ⎠ 2 The cumulative distribution function is P(X < x). It is calculated either by a calculator or a computer, or it is looked up in a table.
- 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.
- Martin Lee Abbott(Author)
- 2016(Publication Date)
- Wiley(Publisher)
Martin Lee Abbott. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc. 78 THE NORMAL DISTRIBUTION –3 2.15% 13.59% 34.13% 34.13% 13.59% 2.15% –2 –1 0 +1 +2 +3 x y Figure 4.1 The normal curve with known properties. smooth line you see but rather a more jagged line. However, as a database increases its size, the histogram approximates the smooth normal curve in variables that are normally distributed; the jagged line becomes filled in as more cases are added. When we speak of the normal distribution and how our sample data set is normally distributed, we actually speak about our data approximating a normal distribution. We refer to the perfect normal distribution as an ideal so that we have a model distribution for comparison to our actual data. Thus, the normal curve is a kind of perfect ruler with known features and dimensions. In fact, we can mathematically chart the perfect normal curve and derive a picture of how the areas under the curve are distributed. Because of these features, we refer to the perfect normal distribution as a standard normal distribution. Look at Figure 2.12, which is reproduced here as Figure 4.1. As you can see, the perfect normal curve is represented as having known proportions of the total area between the mean and given standard deviation (SD) units. A standard normal curve (also known as a z distribution) has a mean of 0 and an SD of 1.0. This is always a bit puzzling until you consider how the mean and SD are calculated. Since a perfect distribution has equal numbers of scores lying to the left and to the right of the mean, calculating the mean is akin to adding positive and negative values resulting in 0. Dividing 0 by N, of whatever size, will always equal to 0. Therefore, the mean of a perfect standard normal distribution is equal to 0. The standard normal distribution has an SD equal to 1 unit. This is simply an easy way to designate the known areas under the curve.
eBook - PDFComparative Analysis Of Nations
Quantitative Approaches
- Robert Perry(Author)
- 2019(Publication Date)
- Routledge(Publisher)
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. Without a minimal appreciation of the basic lan-guage and logic of the normal curve and areas under the normal curve, the student faces the daunting task of trying to make sense out of the most basic tools of hypothe-sis testing within cross-national analysis. THE LOGICAL PLACE OF THE NORMAL DISTRIBUTION WITHIN CROSS-NATIONAL ANALYSIS At the center of hypothesis testing is the statistical concept of the normal curve, first introduced to you in Chapter 3 (Figure 3.2). In this chapter, we turn our attention to two questions. First, how does the concept of the normal curve fit meaningfully into cross-national analysis? Second, how are the mean and standard deviation crucial to the application of the normal distribution? In this section, we summarize the five basic steps by which we apply our sample and the logic of the normal curve to quantitative cross-national analysis. This five-step process is characteristic of all quantitative research. Figure 4.1 graphically illustrates the five logical steps by which we incorporate the logic of the normal sampling distri-bution into cross-national analysis, and subsequently apply it to the refinement of our generalized understanding of political behavior and institutional authority.
eBook - PDF- M.H. Quenouille(Author)
- 2014(Publication Date)
- Pergamon(Publisher)
It can be shown mathematically that whenever a measurement is the sum total of a large number of small independent effects, no one of which predominates, the distribution of observations will take the same form. This form is called the normal distribution and a knowledge of the origin and scale determines it completely i.e. once the mean and spread are known all else is determined. This is not generally true of distributions, since the mean and spread convey no information, for example, on the asymmetry of the distribution, but the normal distribution is symmetrical and all of its properties can be found from its mean and spread. Figure 7 gives, in fact, the frequency curve of the normal distribution, and the same curve is shown in Figure 8 with different spreads. That this curve is not 24 NORMAL DISTRIBUTION a mere mathematical abstraction has been demonstrated by numerous practical applications, some of which are given in examples 8 and 9 and Figures 5, 6, 9, 10 and 11. Figure 8. Normal curves with relative spreads of 3:4:6:8 It is interesting to postulate some of the many causes which have contributed jointly to the measurements in each of these applications. For instance, the yields of wheat and the girths of trees will have resulted partly from variations in soil fertility, humidity, temperature, activity of pests, competition, and other environmental factors and partly from a multitude of genetical factors. Likewise, most of the other measurements result from a complex of environmental and genetical factors. The exceptions, here, are the distribution of observations of the right ascension of the Pole Star which is caused by a series of small errors in measurement, and the distribution of the total score on six dice in which the individual scores on the dice act as six independent effects contributing to the final measurement. The normal distribution cannot r be applied to every measurement and the reasons why this is so are worth noting.
eBook - ePubResearching Education
Perspectives and Techniques
- Kanka Mallick, Gajendra Verma(Authors)
- 2005(Publication Date)
- Routledge(Publisher)
It can be seen that the tops of the columns lie approximately on a curve and that the area under the curve is equivalent to that of the columns. However, to obtain the probability of 8 or more heads by using the curve we need to obtain the area under the tail of the curve from 7.5 to 10.5. Since tables of areas under the normal curve are obtainable in books, they are often used to obtain probabilities in large distributions.Figure 8.2: Normal distribution and the 10-coin testAlthough laborious, it would be possible to work out the probabilities if 100 coins were tossed together. If the result were represented in a column graph, the tops of the columns would form an almost perfect curve known as the ‘normal curve’.Many distributions of measurements of human beings such as heights, weights, sizes of shoes, gloves or hats, the distances individuals of the same age and sex can throw a ball, and so on, fit very well to the normal curve. So do many educational and psychological test scores, although in the case of standardized tests, this may be partly because educationists and psychologists designed them to do so. Before we can make use of the normal curve in order to say how exceptional a pupil’s score is as compared with other children of the same age, we need to turn his or her score in a test into a standard score. To do this we must first obtain the mean and standard deviation for the distribution of scores.Taking a simplified example initially: in six tests, a boy obtains marks of 36, 49, 52, 60, 65, 74. The mean (or average) score is defined as the total of the scores for all the tests divided by the number of tests, i.e. the boy’s mean score is .Deviations of the scores above and below the mean are as follows:Score 52 49 65 36 60 74 Deviation from 56 -4 -7 9 -20 4 18 Total=0 Having obtained the variance by squaring the deviation (which gave us 147.67 in this example) the ‘standard deviation’ is arrived at by finding the square root of 147.67 which is approximately 12.2. It is now possible to arrive at the ‘standard score’ in each subject by using the formula
eBook - PDFStatistics
Principles and Methods
- Richard A. Johnson, Gouri K. Bhattacharyya(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
It can be symmetric about the mean of X or it can be skewed, meaning that it has a long tail to either the left or the right. The probability that X lies in an interval from a to b is determined by the area under the probability density curve between a and b. The total area under the curve is 1, and the curve is never negative. The population 100 p-th percentile is an x value that has probability p to its left and probability 1 − p to its right. When X has mean and standard deviation , the standardized variable Z = X − has mean 0 and standard deviation 1. The normal distribution has a symmetric bell-shaped curve centered at the mean. The intervals extending one, two, and three standard deviations around the mean contain the probabilities .683, .954, and .997, respectively. If X is normally distributed with mean and standard deviation , then Z = X − has the standard normal distribution. When the number of trials n is large and the success probability p is not too near 0 or 1, the binomial distribution is well approximated by a normal distribution with mean n p and sd = √ n p ( 1 − p ). Specifically, the probabilities for a binomial variable X can be approximately cal- culated by treating Z = X − n p √ n p ( 1 − p ) as standard normal. For a moderate number of trials n, the approximation is improved by appro- priately adjusting by 1 2 called a continuity correction. 206 CHAPTER 6/THE NORMAL DISTRIBUTION The normal scores are an ideal sample from a standard normal distribution. Plotting each ordered observation versus the corresponding normal score creates a normal-scores plot, which provides a visual check for possible departures from a normal distribution. Transformation of the measurement scale often helps to convert a long-tailed distribution to one that resembles a normal distribution.
eBook - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
The z-score allows us to compare data that are normally distributed but scaled differently. CHAPTER REVIEW 6.1 The Standard Normal Distribution A z-score is a standardized value. Its distribution is the standard normal, Z ~ N(0, 1). The mean of the z-scores is zero and the standard deviation is one. If z is the z-score for a value x from the normal distribution N(µ, σ), then z tells you how many standard deviations x is above—greater than—or below—less than—µ. 6.2 Using the Normal Distribution The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell- shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ. A special normal distribution, called the standard normal distribution, is the distribution of z-scores. Its mean is zero, and its standard deviation is one. FORMULA REVIEW 6.0 Introduction X ∼ N(μ, σ) μ = the mean, σ = the standard deviation 6.1 The Standard Normal Distribution Z ~ N(0, 1) z = a standardized value (z-score) mean = 0, standard deviation = 1 To find the k th percentile of X when the z-score is known, k = μ + (z)σ z-score: z = x – μ σ Z = the random variable for z-scores 6.2 Using the Normal Distribution Normal Distribution: X ~ N(µ, σ), where µ is the mean and σ is the standard deviation Standard Normal Distribution: Z ~ N(0, 1). Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation) Calculator function for the k th percentile: k = invNorm (area to the left of k, mean, standard deviation) PRACTICE 6.1 The Standard Normal Distribution 1. A bottle of water contains 12.05 fluid ounces with a standard deviation of 0.01 ounces. Define the random variable X in words. X = ____________. 2. A normal distribution has a mean of 61 and a standard deviation of 15.
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