मुफ्त मानक सामान्य कैलकुलेटर
मानक सामान्य वितरण (Z-तालिका) से प्रायिकता और क्षेत्र निकालें।
PDF phi(z)
0.05844094
PDF phi(z) vs Z-Value
सूत्र
Standard Normal Distribution
PDF Formula
phi(z) = (1 / sqrt(2*pi)) * e^(-z^2/2)
The standard normal distribution has mean 0 and standard deviation 1. It is the reference distribution for z-scores and hypothesis testing. The PDF is symmetric around z=0, meaning phi(z) = phi(-z). Common critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%).
हल किया गया उदाहरण
Find the PDF value at z = 1.96 (the 95% critical value).
- 01phi(1.96) = (1 / sqrt(2*pi)) * e^(-1.96^2 / 2)
- 02= 0.39894 * e^(-1.9208)
- 03= 0.39894 * 0.14634
- 04= 0.05844
अक्सर पूछे जाने वाले प्रश्न
Why is z = 1.96 special?
The area between -1.96 and 1.96 under the standard normal curve is approximately 0.95 (95%). This makes it the critical value for 95% confidence intervals and two-sided tests at alpha = 0.05.
What is the maximum value of the standard normal PDF?
The maximum occurs at z = 0, where phi(0) = 1/sqrt(2*pi) ≈ 0.3989. The curve decreases symmetrically from this peak.
How is the standard normal related to other normal distributions?
Any normal distribution N(mu, sigma) can be converted to the standard normal N(0,1) by the transformation Z = (X - mu) / sigma. This is why z-tables work for all normal distributions.
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