Gaussian distribution confidence interval

Figure 1 Normal (Gaussian) distribution, confidence limits, and confidence intervals.

If the observed data follow a bell-shaped Gaussian (normal) distribution, the confidence interval Cl of mean is defined by... 

N number of data xi P probability of significant deviation of a value from data within interval confidence t - borders of the (1 - P)th part of the area of the Gaussian distribution for a given F. 

A control chart is a visual representation of confidence intervals for a Gaussian distribution. The chart warns us when a monitored property strays dangerously far from an intended target value. 

For independent measurements with a gaussian distribution the confidence interval of a mean may be written... 

For regression analysis a nonUnear relation often can be transformed into a linear one by plotting a simple function such as the logarithm, square root, or reciprocal of one or both of the variables. Nonlinear transformations should be used with caution because the transformation will convert a distribution from gaussian to nongaussian. Calculations of confidence intervals usually are based on data having a gaussian distribution. 

The parametric method for the determination of percentiles and their confidence intervals assumes a certain type of distribution, and it is based on estimates of population parameters, such as the mean and the standard deviation. We are, for example, using a parametric method if we believe that the true distribution is Gaussian and determine the reference limits (percentiles) as the values located 2 standard... 

General estimates for the 100a and 100(1 - a) percentiles and their 0.90 confidence intervals can be determined by the following method, provided that data (original or transformed) fit the Gaussian distribution ... 

In presenting the analytical results, the confidence interval should necessarily be mentioned, i.e. the interval in which the average concentration C exists at a given confidence level. For a Gaussian distribution of n parallel measurements (n > 20) and probability P = 0.95, the confidence interval is given by the expression... 

Figure 22.2 Gaussian curves. When the number of measurements increases and if the interval determining the group is narrow, then the graphical shape will take (measurement/frequency) the form of a Gaussian curve (Normal distrihution law). Below, two series of results centred on two different true means /t. If the numher of measurements is very small, it is not possible to estimate the average distribution, thus the true mean. At the bottom right, a reduced form of the Gaussian distribution is shown. The reliabUity of the mean is given by the 95 per cent confidence interval.

For example, we can build B = 400 parametric bootstrap samples, each by adding noise to the model for predicting y from the last example. We assume the noise, denoted s has a Gaussian distribution with mean zero and variance = YH=i( yi y<9 In, written as a,- A1(0, ri). Then we can write y = a -1-/3/fc > 0. 5) -b Si, where a and /3 are the maximum-UkeUhood estimates of the intercept and slope, respectively. As B increases, the parametric bootstrap samples capture the variability of the estimates for a and a -b /3. For example, as with the nonparametric bootstrap, with B = 400, we can extract a 95% confidence interval for the estimates by selecting the 0.025 and 0.975 quantiles of the bootstrap estimates for each z-value. 

Confidence intervals are somewhat arbitrary but the generally accepted criterion requires a 95% probability of the true value falling within the confidence interval. For very large samples, (n > 30), the confidence interval is calculated assuming the distribution is Gaussian. For sample sizes less than 30, the value of s fluctuates substantially from sample to sample and thus the distribution is no longer a standard normal distribution. For this case, we represent the distribution with a statistic that is known as the Student s r-statistic. 

The uncertainty. A, is defined as the product of the standard error of the mean and a confidence interval (arbitrarily defined by the experimenter) and for a Gaussian distribution it is calculated according to the following relation ... 

Values of the Student s t-statistic are summarized in Table 2.2. For a sample size of six (five degrees of freedom, n — 1) and a 95% confidence interval a = 95%), the value of the Student s t is 2.571 ( 2.6). It is approximately equal to 2 for a Gaussian distribution. Table 2.3 compares the Student s t with the Gaussian distribution for several common combinations of sample numbers... 

Uncertainty can be expressed as a standard uncertainty u(y). The standard uncertainty can be multiplied by a number k such that it gives a confidence interval for the measurand Y. Then the probability of T to be in the region y — k u y) y + k u y) can be expressed as a percentage, and the region is called the confidence interval. With some approximations and the assumption of a Gaussian distribution, it can be stated that taking k = 2 will generate a confidence interval of approximately 95 %. 

In order to be able to establish confidence intervals, it also assumed that Ck is Gaussian distributed. The ARX model (1) is characterized by 3 numbers Ua, the auto-regressive order ny, the exogeneous order and rik, the pure time delay between input and output. A regression model is an ARXOlO model (with [ua, ny, n-k] = [0,1,0]). 

When running the NN, the user is provided with overtopping rates based on the CLASH database and the NN prediction. Together with these results the user will also obtain the uncertainties of the prediction through the 5% and 95% confidence intervals. Assuming a normal distribution will allow the standard deviation of overtopping to be estimated from those confidence intervals, and (if required) the whole Gaussian distribution. 

A tabulation or histogram of how often the various values of x occur in replication is approximated by this distribution. From the Gaussian distribution the statistical confidence of the individual values is obtained. From the cumulative values of this distribution, 68.3% of all measurement values are within the interval x + a, 95.4% are within x + 2cr, and 99.7% are within x + 3statistical concepts can be found in Refs. 5 and 8. 

FIGURE 2.18 Synopsis of the probability distribution relevant values and intervals for a normal (Gaussian) distribution of working data repartition with t-Student confidence intervals. 

FIGURE 2.19 The symmetrical confidence interval information/formation for a normal (Gaussian) t-Student distribution. 

When all the standard uncertainties of Type A and Type B have been determined in this way, they should be combined to produce the combined standard uncertainty (suggested symbol u), which may be regarded as the estimated standard deviation of the measurement result. This process, often called the law of propagation of uncertainty or root-sum-of-squares, involves taking the square root of the sum of the squares of all the u.. In many practical measurement situations, the probability distribution characterized by the measurement result y and its combined standard uncertainty ufy) is approximately normal (Gaussian). When this is the case, ufy) defines an interval y - ufy) to y + ufy) about the measurement result y within which the value of the measurand Yestimated by y is believed to lie with a level of confidence of approximately 68 percent. That is, it is believed with an approximate level of confidence of 68 percent that y - ufy) < Y < y + ufy), which is commonly written as T = y ufy). 

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