probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the 97.5th percentile point of the standard

normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...

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 deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

s of the

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

. Because of the

central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...

, 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. This convention seems particularly common in medical statistics, but is also common in other areas of application, such as earth sciences, social sciences and business research. There is no single accepted name for this number; it is also commonly referred to as the "standard normal deviate", "

normal score Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

" 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), :

\mathrm(X > 1.96) \approx 0.025, \,

\mathrm(X < 1.96) \approx 0.975, \,

and as the normal distribution is symmetric, :

\mathrm(-1.96 < X < 1.96) \approx 0.95. \,

One notation for this number is ''z''_.975. From the

probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...

of the standard normal distribution, the exact value of ''z''_.975 is determined by :

\frac\int_^ e^ \, \mathrmx = 0.95.

Its square, about 3.84146, is the 95th percentile point of a

chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...

with 1 degree of freedom, often used for testing 2×2

contingency table In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the multivariate frequency distribution of the variables. They are heavily used in survey research, business int ...

History

The use of this number in applied statistics can be traced to the influence of

Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...

's classic textbook,

Statistical Methods for Research Workers ''Statistical Methods for Research Workers'' is a classic book on statistics, written by the statistician R. A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods, together with his '' T ...

, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... 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. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. This is not recommended but is occasionally seen.

Software functions

The inverse of the standard normal

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

(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:

History

Software functions

See also

References

Further reading