probabilityand statistics, 1.96 is the approximate value of the 97.5 percentilepoint of the normal distribution. 95% of the area under a normal curvelies 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),
and as the normal distribution is symmetric,
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 functionof the normal distribution, the exact value of "z".025 is determined by
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=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|page=47] 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 placeswas 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.
*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
Margin of error
Standard error (statistics)
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