# 1.96

In probability and statistics, **1.96** is the approximate value of the 97.5 percentile point of the normal distribution. 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%).^{[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, or .975 point.

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

and as the normal distribution is symmetric,

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

## History[edit]

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."

^{[11]}

In Table 1 of the same work, he gave the more precise value 1.959964.^{[12]} In 1970, the value truncated to 20 decimal places was calculated to be

- 1.95996 39845 40054 23552...
^{[13]}

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.

## Software functions[edit]

The inverse of the standard normal CDF can be used to compute the value, the following is a table of function calls that return 1.96 in some commonly used applications:

Application | Function call |
---|---|

Excel | NORM.S.INV(0.975) |

MATLAB | norminv(0.975) |

R | qnorm(0.975) |

scipy | scipy.stats.norm.ppf(0.975) |

SAS | probit(0.025); |

SPSS | x = COMPUTE IDF.NORMAL(0.975,0,1). |

Stata | invnormal(0.975) |

Wolfram Language (Mathematica) | InverseCDF[NormalDistribution[0, 1], 0.975]^{[14]}^{[15]} |

## See also[edit]

## References[edit]

**^**Rees, DG (1987),*Foundations of Statistics*, CRC Press, p. 246, ISBN 0-412-28560-6,Why 95% confidence? Why not some other

*confidence level*? The use of 95% is partly convention, but levels such as 90%, 98% and sometimes 99.9% are also used.**^**"Engineering Statistics Handbook: Confidence Limits for the Mean". National Institute of Standards and Technology. Archived from the original on 5 February 2008. Retrieved 2008-02-04.Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used.

**^**Olson, Eric T; Olson, Tammy Perry (2000),*Real-Life Math: Statistics*, Walch Publishing, p. 66, ISBN 0-8251-3863-9,While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.

**^**Swift, MB. "Comparison of Confidence Intervals for a Poisson Mean - Further Considerations".*Communications in Statistics - Theory and Methods*.**38**(5). pp. 748–759. doi:10.1080/03610920802255856.In modern applied practice, almost all confidence intervals are stated at the 95% level.

**^**Simon, Steve (2002),*Why 95% confidence limits?*, archived from the original on 28 January 2008, retrieved 2008-02-01**^**Moher, D; Schulz, KF; Altman, DG (2001), "The CONSORT statement: revised recommendations for improving the quality of reports of parallel-group randomised trials.",*Lancet*,**357**(9263): 1191–1194, doi:10.1016/S0140-6736(00)04337-3, PMID 11323066, retrieved 4 February 2008**^**"Resources for Authors: Research". BMJ Publishing Group Ltd. Archived from the original on 18 July 2009. Retrieved 2008-02-04.For standard original research articles please provide the following headings and information: [...] results - main results with (for quantitative studies) 95% confidence intervals and, where appropriate, the exact level of statistical significance and the number need to treat/harm

**^**Borradaile, Graham J. (2003),*Statistics of Earth Science Data*, Springer, p. 79, ISBN 3-540-43603-0,For simplicity, we adopt the common earth sciences convention of a 95% confidence interval.

**^**Cook, Sarah (2004),*Measuring Customer Service Effectiveness*, Gower Publishing, p. 24, ISBN 0-566-08538-0,Most researchers use a 95 per cent confidence interval

**^**Gosling, J. (1995),*Introductory Statistics*, Pascal Press, pp. 78–9, ISBN 1-86441-015-9**^**Fisher, Ronald (1925),*Statistical Methods for Research Workers*, Edinburgh: Oliver and Boyd, p. 47, ISBN 0-05-002170-2**^**Fisher, Ronald (1925),*Statistical Methods for Research Workers*, Edinburgh: Oliver and Boyd, ISBN 0-05-002170-2, Table 1**^**White, John S. (June 1970), "Tables of Normal Percentile Points",*Journal of the American Statistical Association*, American Statistical Association,**65**(330): 635–638, doi:10.2307/2284575, JSTOR 2284575**^**InverseCDF, Wolfram Language Documentation Center.**^**NormalDistribution, Wolfram Language Documentation Center.

## Further reading[edit]

- Gardner, Martin J; Altman, Douglas G, eds. (1989),
*Statistics with confidence*, BMJ Books, ISBN 978-0-7279-0222-1