97.5th percentile point

In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Because of the central limit theorem, this number is 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% probability in science and frequentist statistics, though other probabilities (90%, 99%, etc.) are sometimes used.^[1][2][3][4] This convention seems particularly common in medical statistics,^[5][6][7] but is also common in other areas of application, such as earth sciences,^[8] social sciences and business research.^[9]

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, the .975 point, or just its approximate value, 1.96.

If X has a standard normal distribution, i.e. X ~ N(0,1),

${\displaystyle \mathrm {P} (X>1.96)\approx 0.025,\,}$

${\displaystyle \mathrm {P} (X<1.96)\approx 0.975,\,}$

and as the normal distribution is symmetric,

${\displaystyle \mathrm {P} (-1.96<X<1.96)\approx 0.95.\,}$

One notation for this number is z_.975.^[10] From the probability density function of the standard normal distribution, the exact value of z_.975 is determined by

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z_{.975}}e^{-x^{2}/2}\,\mathrm {d} x=0.975.}$