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मानक प्रसामान्य चर (Standard Normal Variable) का तात्पर्य उस सामान्य वितरण (Normal Distribution) से है जिसका माध्य (Mean, μ) शून्य (0) होता है और मानक विचलन (Standard Deviation, σ) एक (1) होता है। इसे Z-वितरण (Z-distribution) भी कहा जाता है और इसे Z से निरूपित किया जाता है। यदि कोई सामान्य चर X हो जिसका माध्य μ और मानक विचलन σ हो, तो उसे मानकीकरण (Standardization) के माध्यम से Z-स्कोर में बदला जा सकता है, जो इस प्रकार है: Z = (X – μ) / σ यह प्रक्रिया किसी भी सामान्य वितरण को मानक प्रसामान्य वितरण में बदलने के लिए की जाती है, जिससे विभिन्न वितरणों की तुलनात्मक व्याख्या की जा सके। मानक प्रसामान्य वितरण सांख्यिकीय विश्लेषण में एक अत्यंत महत्वपूर्ण भूमिका निभाता है क्योंकि इसका उपयोग Z-स्कोर तालिका (Z-table), परिकल्पना परीक्षण (hypothesis testing), आत्मविश्वास अंतराल (confidence intervals), और सांख्यिकीय अनुमान (statistical inference) में किया जाता है। अब हम मानक प्रसामान्य चर के महत्वपूर्ण परिगुणों (Important Properties/Characteristics of Standard Normal Variable) का विस्तृत वर्णन करते हैं: --- ### 1. माध्य (Mean = 0) मानक प्रसामान्य वितरण का केंद्र बिंदु उसका माध्य होता है, जो शून्य होता है। इसका अर्थ है कि वितरण पूरी तरह से अपने केंद्र (Z = 0) पर संतुलित होता है। Z-स्कोर = 0 का अर्थ है कि कोई भी अवलोकन अपने वितरण के औसत के ठीक बराबर है। यह मानककरण प्रक्रिया का एक मुख्य उद्देश्य होता है — सभी _____ __________ _________ _____ ___ _______.
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