| dbpprop:abstract
|
- 1.96 is the approximate value of the 97.5 percentile point of the normal distribution used in probability and statistics. 95% of the area under a normal curve lies within roughly 1.96 standard deviations of the mean, and due to the central limit theorem, this number is therefore used in the construction of approximate 95% confidence intervals. Its ubiquity is due to the arbitrary but common convention of using confidence intervals with 95% coverage rather than other coverages (such as 90% or 99%). Olson, Eric T; Tammy Perry Olson (2000). Real-Life Math: Statistics. Walch Publishing. p. p.66. ISBN 0825138639. "While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians. " </ref> This convention seems particularly common in medical statistics, Simon, Steve (2002), Why 95% confidence limits?, http://www. childrens-mercy. org/stats/ask/why95. asp, retrieved 2008-02-01 </ref> but is also common in other areas of application, such as earth sciences Borradaile, Graham J. (2003). Statistics of Earth Science Data. Springer. p. p.79. ISBN 3540436030. "For simplicity, we adopt the common earth sciences convention of a 95% confidence interval. " </ref> and business research. Cook, Sarah (2004). Measuring Customer Service Effectiveness. Gower Publishing. p. p.24. ISBN 0566085380. "Most researchers use a 95 per cent confidence interval" </ref> There is no single accepted name for this number; it is also commonly referred to as the "standard normal deviate", "normal score" or "Z score" for the 97.5 percentile point, or .975 point. If X has a standard normal distribution, i.e. X ~ N(0,1), <math> \mathrm{P}(X > 1.96) = 0.025, \,</math> <math> \mathrm{P}(X < 1.96) = 0.975, \,</math> and as the normal distribution is symmetric, <math> \mathrm{P}(-1.96 < X < 1.96) = 0.95. \,</math> One notation for this number is z.025. From the probability density function of the normal distribution, the exact value of z.025 is determined by <math> \frac{1}{\sqrt{2\pi}}\int_{z_{.025}}^\infty e^{-x^2/2} \, \mathrm{d}x = 0.025. </math> The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: "The value for which P = .05, or 1 in 20, is 1.96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. " In Table 1 of the same work, he gave the more precise value 1.959964. Fisher, Ronald (1925), Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, ISBN 0-05-002170-2, Table 1</ref>. In 1970, the value truncated to 20 decimal places was calculated to be 1.95996 39845 40054 23552... The commonly-used approximate value of 1.96 is therefore accurate to better than one part in 50 000, which is more than adequate for applied work.
|
![[RDF Data]](/statics/sw-cube.png)
