probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...

and statistics, the 97.5th percentile point of the standard

normal distribution In 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 e^ The parameter \mu ...

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 deviations of the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...

. Because of the

central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...

, 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 The term normal score is used with two different meanings in statistics. One of them relates to creating a single value which can be treated as if it had arisen from a standard normal distribution (zero mean, unit variance). The second one relates t ...

" or "

Z score In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...

" 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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...

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

\frac\int_^ e^ \, \mathrmx = 0.975.

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 ...

's classic textbook, Statistical Methods for Research Workers, 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 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