# 1.96

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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%). [Cite book| last=Rees | first=DG |title=Foundations of Statistics| page=p.246 |publisher=CRC Press |isbn=0412285606| quote=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.] [] [cite book| title=Real-Life Math: Statistics |first=Eric T | last=Olson | coauthors=Tammy Perry Olson|page=p.66| publisher=Walch Publishing | year=2000| isbn=0825138639 | quote=While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.] This convention seems particularly common in medical statistics, [ Citation |title=Why 95% confidence limits? | first=Steve | last=Simon | url=http://www.childrens-mercy.org/stats/ask/why95.asp | year=2002| access-date=2008-02-01] [] [] but is also common in other areas of application, such as earth sciences [cite book|quote=For simplicity, we adopt the common earth sciences convention of a 95% confidence interval.| page=p.79| title=Statistics of Earth Science Data|first=Graham J. |last=Borradaile |publisher=Springer| year=2003| isbn=3540436030] and business research. [cite book|title=Measuring Customer Service Effectiveness|first=Sarah |last= Cook| page=p.24|quote=Most researchers use a 95 per cent confidence interval | publisher=Gower Publishing| year=2004|isbn=0566085380]

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),

:$mathrm\left\{P\right\}\left(X > 1.96\right) = 0.025, ,$

:$mathrm\left\{P\right\}\left(X < 1.96\right) = 0.975, ,$

and as the normal distribution is symmetric,

:$mathrm\left\{P\right\}\left(-1.96 < X < 1.96\right) = 0.95. ,$

One notation for this number is "z".025. [Citation| first=J.| last=Gosling | title=Introductory Statistics| publisher=Pascal Press |year=1995 | isbn=1864410159|pages=78-9] From the probability density function of the normal distribution, the exact value of "z".025 is determined by

:$frac\left\{1\right\}\left\{sqrt\left\{2piint_\left\{z_\left\{.025^infty e^\left\{-x^2/2\right\} , mathrm\left\{d\right\}x = 0.025.$

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." [Citation| first=Ronald | last=Fisher |author-link=Ronald Fisher |title=Statistical Methods for Research Workers |year=1925 |isbn=0-05-002170-2|page=47]
In Table 1 of the same work, he gave the more precise value 1.959964. [Citation| first=Ronald | last=Fisher |author-link=Ronald Fisher |title=Statistical Methods for Research Workers |year=1925 |isbn=0-05-002170-2, [http://psychclassics.yorku.ca/Fisher/Methods/tabI-II.gifTable 1] ] .In 1970, the value truncated to 20 decimal places was calculated to be:1.95996 39845 40054 23552... [Citation | title=Tables of Normal Percentile Points | first=John S. | last=White | journal =Journal of the American Statistical Association | volume=65 | number=330 | month=June | year=1970 | pages=635-638 |url=http://links.jstor.org/sici?sici=0162-1459%28197006%2965%3A330%3C635%3ATONPP%3E2.0.CO%3B2-F]

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.

Notes

*Citation |editor-last=Gardner |editor-first=Martin J |editor2-last=Altman| editor2-first=Douglas G | editor2-link=Doug Altman |title= Statistics with confidence | publisher= BMJ Books|year= 1989 | isbn=978-0727902221

ee also

*Margin of error
*Probit
*Reference range
*Standard error (statistics)
*68-95-99.7 rule

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