1.96
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%).[1][2][3] This convention seems particularly common in medical statistics,[4][5][6] but is also common in other areas of application, such as earth sciences[7] and business research.[8]
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.[9] From the probability density function of the normal distribution, the exact value of z.025 is determined by
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[edit] History
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."[10]
In Table 1 of the same work, he gave the more precise value 1.959964.[11] In 1970, the value truncated to 20 decimal places was calculated to be
- 1.95996 39845 40054 23552...[12]
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.
[edit] Software functions
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 | NORMSINV(0.975) |
| Mathematica | InverseCDFNormalDistribution[0,1],0.975] |
| MATLAB | max(norminv([0.025, 0.975])) |
| R | qnorm(0.975) |
[edit] Notes
- ^ Rees, DG (1987), Foundations of Statistics, CRC Press, p. 246, ISBN 0412285606, "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. http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm. 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; Tammy Perry Olson (2000), Real-Life Math: Statistics, Walch Publishing, p. 66, ISBN 0825138639, "While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians."
- ^ Simon, Steve (2002), Why 95% confidence limits?, http://www.childrens-mercy.org/stats/ask/why95.asp, 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, http://www.consort-statement.org/index.aspx?o=1320
- ^ "Resources for Authors: Research". BMJ Publishing Group Ltd. http://resources.bmj.com/bmj/authors/types-of-article/research. 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 3540436030, "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 0566085380, "Most researchers use a 95 per cent confidence interval"
- ^ Gosling, J. (1995), Introductory Statistics, Pascal Press, pp. 78–9, ISBN 1864410159
- ^ 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
[edit] Further reading
- Gardner, Martin J; Altman, Douglas G, eds. (1989), Statistics with confidence, BMJ Books, ISBN 978-0727902221
