Standard normal distribution (English: standard normal distribution, German: Standard normal distribution) is a very important distribution in mathematics, physics, engineering and other fieldsprobabilityDistributionstatisticsIt has great influence in many aspects.expected valueμ=0，That is, the symmetry axis of the curve image is the Y axis,standard deviationσ=1Normal distribution, marked as N (0, 1).
The standard normal distribution is also calledU distribution, is 0Mean, take 1 asstandard deviationThe normal distribution of is N (0,1).
Standard normal distribution curvelowerArea distribution ruleYes: the area under the curve is 0.9500 in the range of -1.96~+1.96, and 0.9900 in the range of -2.58~+2.58.The statisticians also developed a statistical table（freedomby∞With the help of this table, the area under the curve within the range of some special u1 and u2 values can be estimated.
Attached Table Standard Normal Distribution Table
Normally distributedprobability density functionThe curve is bell shaped, so people often call itBell curve。What we usually sayStandard normal distributionIs a positional parameterMean0, scale parameter:standard deviationIs the normal distribution of 1 (see the green curve in Figure 1).
Figure 1 Standard Normal Distribution
characteristic
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Symmetry of density function with respect to mean value
Average value, its statistical mode andmedian(medium) The same value.
The area of 68.268949% under the function curve is about the averagestandard deviationWithin the range.
95.449974% of the area is within two standard deviations of the average.
99.730020% of the area is within three standard deviations of the average.
99.993666% of the area is within four standard deviations of the average.
CurvilinearInflection point(inflection point) is the position one standard deviation from the average.
standard deviation
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standard deviation
The dark blue area is the range of values less than one standard deviation from the average value.stayNormal distributionIn, the ratio of this range is all values68%, according to the normal distribution, the ratio within the two standard deviations is95%；The ratio within the three standard deviations is99%。
In practical application, it is often considered that a group of data with approximateNormal distributionThe probability distribution of.If its assumption is correct, then approximately68.3%The value distribution is within 1 standard deviation from the average value, about95.4%The values are distributed within 2 standard deviations from the mean, and about99.7%The values are distributed within 3 standard deviations from the mean.Is called“68-95-99.7 rule”Or“Rule of thumb”[1]。
