Probability bell shaped curve

The normal, or Gaussian, distribution occupies a central place in statistics and measurement. Its familiar bell-shaped curve (the probability density function or pdf, figure 2.1) allows one to calculate the probability of finding a result in a particular range. The x-axis is the value of the variable under consideration, and the y-axis is the value of the pdf. 

If normal distributions are followed, the probability function curves for the concentration of pseudoephedrine hydrochloride from the previous example should follow the familiar bell-shaped curve as shown in Figure 3.1, where p specifies the population mean concentration for a species and x represents an individual concentration value for that species. The probability function for the normal distribution is given by the function,... 

Usefulness of the normal distribution curve lies in the fact that from two parameters, the true mean p. and the true standard deviation true mean determines the value on which the bell-shaped curve is centered, and most probability concentrated on values near the mean. It is impossible to find the exact value of the true mean from information provided by a sample. But an interval within which the true mean most likely lies can be found with a definite probability, for example, 0.95 or 0.99. The 95 percent confidence level indicates that while the true mean may or may not lie within the specified interval, the odds are 19 to 1 that it does.f Assuming a normal distribution, the 95 percent limits are x 1.96 where a is the true standard deviation of the sample mean. Thus, if a process gave results that were known to fit a normal distribution curve having a mean of 11.0 and a standard deviation of 0.1, it would be clear firm Fig. 17-1 that there is only a 5 percent chance of a result falling outside the range of 10.804 and 11.196. 

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the curve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical bell-shaped curve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

Since we are not able to find the true value for any parameter, we often make do with the average of all of the experimental data measured for that parameter, and consider this as the most probable value." Measured values, which necessarily contain experimental errors, should lie in a random manner on either side of this most probable value as expressed by the normal or Gaussian distribution. This distribution i. a bell-shaped curve that represents the number of measurements N that have a specific value x (which deviates from the mean or most probable value Xq by an amount x - Xo, representative of the error). Obviously the smaller the value of x - Xo, the higher the probability that the quantity being measured lies near the most likely value xq, which is at the top of the peak. A plot of N against x, shown in Figure 10.1, is called a Gaussian distribution or error curve, expressed mathematically as ... 

The observed frequencies with 3, 5, 7 and 14 different and equally probable random error sources are shown in Fig. 3,6c-f. With many different sources of experimental error, it is seen that the frequency of the experimental response data can be approximately described by the bell-shaped curve in Fig. 3.6g. 

The probability distribution of r is a bell-shaped curve, similar to the normal distribution, see Fig. 3.11. However, the shape of the curve depends on the number of the degrees of freedom used to estimate the standard error. When this number increases, the t distribution approaches the normal distribution. 

Probability density function (pdf) The mathematical function that describes a distribution in terms of the probability of finding a result. For the normal distribution the pdf is the bell-shaped curve. (Section 1.8.2 equation 1.1)... 

Note that the pdf is a function of. v—the values that can be taken by the data. A probability density function is defined in terms of its area the probability of finding a result between two values of. v (say x and x2) is the area under the pdf between x and. v2. The shape of this pdf is the familiar bell-shaped curve shown overlaying the histogram of figure 1.3. 

The dose of a drug is an individual consideration with many factions contributing to the size and effectiveness of that given. The correct drug dose would be the smallest effective amount. This would probably vary among individuals. It could also vary in that same individual on different occasions. A normal distribution or bell-shaped curve would be indicative of these scenarios, and would produce an average effect in the majority of individuals. 

Fig. 2.3 shows the famous bell-shaped curve that is the plot of the probability density of the standard normal distribution,... 

By far the most widely assumed probability distribution applicable to biological data is the Normal Probability Distribution, or Gaussian distribution. When plotted, this distribution forms the familiar bell-shaped curve that is symmetrical about the mean. The math atical expression describing this distribution is... 

The graph of the probability density function is a bell-shaped curve that is symmetrical to the value of p and whose form is determined by the parameter o (Fig. A.9). The maximum of the curve is found at p and has a height given by the... 

There have been many attempts to explain the bell-shaped curve of enzyme activity versus Wo. It is likely that several factors contribute and that the relative importance of different parameters varies with the type of enzyme studied [40,41]. However, it seems probable that diffusion effects play a major role, and a diffusion model applicable to a hydrophilic enzyme located in the core of the water droplet and hydrophilic substrates also situated in the droplets was worked out by Walde and coworkers [42,43]. Before the enzyme-catalyzed reaction can take place, two different diffusion processes must occur. In the first of these, an interdroplet diffusion step, drops containing the substrate and drops containing the enzyme must collide. In the second process, an intradroplet diffusion step, the substrate reaches the enzyme s active site. Whereas the rate of the first process increases with droplet radius, the reverse is true for the second process. These two counteracting dependencies of reaction rate on droplet size (and thus on Wo at constant surfactant concentration) may lead to a bell-shaped activity versus Wo curve. 

Based on data provided, we have calculated the probabilities corresponding to the time intervals that people took to assemble the parts. The probability distribution for Example 19.4 is shown in Table 19.10 and Figure 19.5. Again, note that the sum of probabilities is equal to 1. Also note that if we were to connect the midpoints of time results (as shown in Figure 19.5), we would have a curve that approximates a bell shape. As the number of data points increases and the intervals decrease, the probability-distribution curve becomes smoother. A probability distribution that has a bell-shaped curve is called a normal distribution. The probability distribution for many engineering experiments is approximated by a normal distribution. 

Figure 4-2 is typical of many laboratory measurements The most probable response is at the center, and the probability of observing other responses decreases as the distance from the center increases. The smooth, bell-shaped curve superimposed on the data is called a Gaussian distribution. The more measurements made on any physical system, the closer the bar chart comes to the smooth curve. 

The Gaussian distribution and the normal law of error are both often expressed as the same relationship. The Gaussian distribution law is the theoretical frequency distribution for a set of data of any normal, repetitive function, due to chance, represented by a bell-shaped curve symmetrical about a mean. The relationship of the number of events occuring and frequency when the events occur are due to chance only. The probability for distributions that occur due to chance is ... 

Bell-shaped curves are obtained for both cases with maximal activity at 75% neutrality (a = 0.75) with poly(vinylimidazole). These results are best explained if we assume that enough cationic sites (25%) have to be present on the polymer for substrate binding but also a large portion of the polymer residues must be neutral (75%). It is these neutral imidazole rings that are probably responsible for the hydrolysis of the ester substrates. 

Gaussian and Poisson distributions are related in that they are extreme forms of the Binomial distribution. The binomial distribution describes the probability distribution for any number of discrete trials. A Gaussian distribution is therefore used when the probability of an event is large (this results in more symmetric bell-shaped curves), whereas a Poisson distribution is used when the probability is small (this results in asymmetric curves). The Lorentzian distribution represents... 

This probability distribution is called the binomial probability distribution. Eq. (15.16) is the version for the case that p = 1/2. Figure 15.1 shows the binomial probability distribution for m heads outcomes out of 10 throws with an unbiased coin. This distribution has a single peak and resembles the familiar bell-shaped curve. The curve for larger values of n is even more nearly bell-shaped. 

---