Standard Normal distribution

The Gaussian distribution and its related functions are widely used in statistics, engineering, and physics, including hypothesis testing, quality control, and digital communications. As the standard normal cumulative distribution function (CDF) lacks a closed-form, accurate approximations are essential for practical applications. This study presents new, highly accurate improvements to existing approximations of the standard normal distribution function while preserving simplicity. The proposed methodology refines current formulas by introducing additional parameters and systematically optimizing them using R-based iterative procedures guided by the Kolmogorov-Smirnov statistic to minimize maximum and mean absolute errors. Evaluations across a dense grid confirm the precision and robustness of the proposed models. Results demonstrate significant improvements, with one enhancement reducing the maximum error by a factor of 4.86 compared to previous models, while other formulas show notable accuracy gains. The improved models remain straightforward to implement, making them suitable for real-time engineering and applied statistical computations. A practical application is demonstrated by calculating the bit error rate (BER) in Binary Phase Shift Keying (BPSK) systems under Gaussian noise, where the proposed approximations closely replicate the exact BER curve, confirming their effectiveness in telecommunications system analysis. These improvements provide engineers and scientists with precise, efficient tools, supporting modeling, risk assessment, and system reliability in applications requiring accurate Gaussian evaluations.

This study analyses trends in rainfall, temperature, and humidity over a 15-year period (2006-2020) to investigate the effects of climate change on Kabul City. The Afghanistan Meteorological Department provided the data used to evaluate long-term trends and spatiotemporal variability. The findings show considerable climatic variability, with monthly rainfall generally rising but noticeably falling during the monsoon season. With the exception of the summer and monsoon seasons, when increases were noted, temperature study shows that monthly maximum temperatures are decreasing for the majority of the months. Trends in humidity indicate a steady yearly increase with noticeable seasonal variations. Two severe drought events (2007-2008 and 2016-2018) and one intermediate rainy spell (2011-2014) are highlighted in the precipitation record, highlighting the insufficiency of water resources to satisfy the demands of Kabul's rapidly expanding urban population. A negative water balance has been exacerbated by over-extraction and inappropriate water usage, which has weakened urban resilience, especially in agriculture, and increased the likelihood of water scarcity. In addition to offering insights to enable adaptation methods in Kabul's urban climate environment, the study highlights the urgency of evidence-based policies for sustainable water resource management.

The standard normal distribution is a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. Its probability density function is symmetric about zero and exhibits a bell-shaped curve, making it a fundamental tool in statistical analysis. Key properties include the use of standard scores (z-scores) to measure the relative position of data points and its role in simplifying computations for hypothesis testing, probability calculations, and confidence interval estimation. This distribution is essential for standardizing data and serves as a cornerstone in probability and inferential statistics.

In this paper, Tocher's approximation for standard normal distribution function is improved and three new approximations are proposed. The quality of the new approximations was computed based on two criteria; the maximum absolute error and the mean absolute error. We found that the maximum absolute errors of the proposed approximations fall between 7 7.62 10and 5 4.95 10and the mean absolute errors-based on 5001 values between 0 and 5-fall between 7

Due to the widespread applicability and use of the normal distribution, a need has arisen to approximate its cumulative distribution function (cdf). In this article, five new simple approximations to the standard normal cdf are developed. In order to assess the accuracy of the proposed approximations, both maximum absolute error and mean absolute error were used. The maximum absolute errors of the proposed approximations lie between 0.00095 and 0.00946, which is highly accurate if compared to the existing simple approximations and quite sufficient for many real-life applications. Even though simple approximations may not as accurate as complicated ones, they are, though, fairly good when judged vis-à-vis their simplicity. A numerical comparison study is conducted between several existing simple approximations and the proposed approximations to explore how well these approximations work.

The unnatural change in the globe under influence of devastating global warming has been quashing the overall functioning of ecosystem since industrial revolution. Thus, the human-induced disaster caused by proportional increase of greenhouse gases in the atmosphere has affected the normal functioning of hydrologic cycle. Under the undesirable condition, the amount of hydrologic variables began to diverge over time. Hydrologic variable should be homogeneous for the reliability of hydraulic structure while predicting necessary design criteria for its construction. Therefore, the test of whether this requirement is true should be performed in the context of any given hydrologic data‘s homogeneity before being passed to the implementation of statistical approaches to the data. The study carried out in Yesilirmak basin was realized on homogeneity of seasonal maximum streamflow data from eight gauging stations operated by The General Directorate of State Hydraulic Works (DSI). Yesilirmak...

The unnatural change in the globe under influence of devastating global warming has been quashing the overall functioning of ecosystem since industrial revolution. Thus, the human-induced disaster caused by proportional increase of greenhouse gases in the atmosphere has affected the normal functioning of hydrologic cycle. Under the undesirable condition, the amount of hydrologic variables began to diverge over time. Hydrologic variable should be homogeneous for the reliability of hydraulic structure while predicting necessary design criteria for its construction. Therefore, the test of whether this requirement is true should be performed in the context of any given hydrologic data’s homogeneity before being passed to the implementation of statistical approaches to the data. The study carried out in Yesilirmak basin was realized on homogeneity of seasonal maximum streamflow data from eight gauging stations operated by The General Directorate of State Hydraulic Works (DSI). Yesilirmak River basin area is approximately 5% of surface area of Turkey. Yesilirmak River is one of the major rivers of Turkey and its long is 519 kilometers. There are three main tributaries of the Yesilirmak River, named as Kelkit, Cekerek and Tersakan. Its water is mostly used for purposes as irrigation, drinking, fisheries and wildlife. The parametric and non-parametric procedures, called as standard normal homogeneity, Pettitt, Buishand range and von Neuman ratio were used for this reason. Statistically significant inhomogeneity with respect to the all of the statistic tests taken into account in the study was detected in the considered streamflow data sequences presented.

Of all statistical distributions, the standard normal is perhaps the most popular and widely used. Its use often involves computing the area under its probability curve. Unlike many other statistical distributions, there is no closed form theoretical expression for this area in case of the normal distribution. Consequently it has to be approximated. While there are a number of highly complex but accurate algorithms, some simple ones have also been proposed in literature. Even though the simple ones may not be very accurate, they are nevertheless useful as accuracy has to be gauged vis-à-vis simplicity. In this short paper, we present another simple approximation formula to the cumulative distribution function of standard normal distribution. This new formula is fairly good when judged vis-à-vis its simplicity.

F. Hosseinzadeh Lotfi et al economic state where inputs are overly invested. Evidence of congestion occurs whenever reducing some inputs can increase outputs. A congestion in stochastic DEA model is presented and then it is reformulated in the manner that the congestion in stochastic model incorporate future information. This research applies the approach to plan the restructure strategy of Iranian commercial banks.

Due to the widespread applicability and use of the normal distribution, a need has arisen to approximate its cumulative distribution function (cdf). In this article, five new simple approximations to the standard normal cdf are developed. In order to assess the accuracy of the proposed approximations, both maximum absolute error and mean absolute error were used.  The maximum absolute errors of the proposed approximations lie between 0.00095 and 0.00946, which is highly accurate if compared to the existing simple approximations and quite sufficient for many real-life applications. Even though simple approximations may not as accurate as complicated ones, they are, though, fairly good when judged vis-a-vis their simplicity.

In this article, we present a new approximation to the cumulative distribution function of standard normal distribution. The approximation is fairly accurate with minimum accuracy of seven decimal digits. To the best of our knowledge, this formula outperforms other such approximations available in literature.

Of all statistical distributions, the standard normal is perhaps the most popular and widely used. Its use often involves computing the area under its probability curve. Unlike many other statistical distributions, there is no closed form theoretical expression for this area in case of the normal distribution. Consequently it has to be approximated. While there are a number of highly complex but accurate algorithms, some simple ones have also been proposed in literature. Even though the simple ones may not be very accurate, they are nevertheless useful as accuracy has to be gauged vis-à-vis simplicity.

In this short paper, we present another simple approximation formula to the cumulative distribution function of standard normal distribution. This new formula is fairly good when judged vis-à-vis its simplicity.

Climatic changes is uncertain in nature and other
changes are not expected to be uniform, rather there may be
dramatic regional differences. Considerable effort should be given
to understand change in climate at the regional level. Here we had
conducted the study to establish the rainfall trends in the Tel river
of Kesinga and also to provide the evidence of climate change by
analyzing available rainfall record for 11years period from 2003 to
2013. Records of monthly rainfall were obtained from the CWC
station at Kesinga for analysis. Graphs were constructed to
illustrate the changing trends within the months of the study area.
From the graphs the highest and lowest amount of rainfall was
obtainedand the variation in rainfall was studied. The mean,
median, standard deviation, variance of rainfall data of each month
was obtained to know the variation in 11 years of data. Trend
analysis is parametric type, i.e. linear regression analysis revealed
positive and negative trend in some months.

Estimation of individual and population slopes using
Linear Minimum Variance Unbiased Estimation by Wu and
Bailey (1988) is applied for incomplete longitudinal studies. In
these studies the right censoring process is considered
informative, in which the number of observations made for each
subject is assumed to vary depending on the rate of change or
slope of the individual response variable. This process of
‘missing not at random’ poses problems in the estimation of the
slope parameters. The proposed method provides better estimates
under different parametric models for the censoring distribution.
The method’s performance is studied through a simulation study
under Standard Normal and Uniform censoring distributions.