Chi square distribution is a continuous probability distribution primarily used in hypothesis testing, contingency analysis, and construction of confidence limits in inferential statistics but not necessarily in the modeling of real-life phenomena. The closed-form expression for the quantile function (QF) of Chi square is not available because the cumulative distribution function cannot be transformed to yield the QF and consequently places limitations on the use of the QF. Researchers have over the years proposed approximations that improve over time in terms of speed, computational efficiency, and precision, and so on. However, most of the available closed-form expressions (quantile approximation) fail at the extreme tails of the distribution. This paper used the Quantile mechanics approach to obtain second-order nonlinear ordinary differential equations whose solutions using the power series method yielded initial approximates in form of series for different values of the degrees of freedom. The initial approximate varies with the exact (R software) values which serve as the reference and the error between them was minimized by the natural cubic spline interpolation. The final approximates are correct up to an average of 8 decimal places, have small error, and is closer to the exact when compared with some other results from other researchers. The upper tail of the distribution was considered and excellent results were obtained which is a major improvement over the existing results in the literature. The approach used in this work is a hybrid of series expansions and numerical algorithms. Computer codes can be written for the application.

卡方分布是一种连续的概率分布，主要用于假设检验，偶发性分析和推论统计中的置信极限的构建，但不一定用于现实现象的建模。卡方平方分位数函数（QF）的封闭形式表达式不可用，因为无法将累积分布函数转换为QF，因此限制了QF的使用。多年来，研究人员提出了一些近似值，这些近似值会随着时间的流逝在速度，计算效率和精度等方面不断提高。但是，大多数可用的闭式表达式（分位数逼近）在分布的最末端出现故障。本文采用分位数力学方法获得了二阶非线性常微分方程，使用幂级数方法的解对于不同的自由度值产生了一系列级数的初始近似值。初始近似值随作为参考的精确（R软件）值而变化，并且通过自然三次样条插值最小化了它们之间的误差。与其他研究人员的其他一些结果相比，最终的近似值是正确的，平均小数点后8位都是正确的，误差很小，并且更接近精确值。考虑了分布的上尾部，并获得了出色的结果，这是对文献中现有结果的重大改进。在这项工作中使用的方法是级数展开和数值算法的混合。可以为该应用编写计算机代码。