quirS- we were doing inferential statistics and in our last lecture we did a sampling distribution or sampling distribution Ko samples were not possible to study the entire population because it is a very big population, what we do is we draw a sample from a population and we estimate the population parameter of the rate of calculate of theuinely calculated अगर हम किसी भी देफाँईट पापूलेशन में से, अगर हम किसी भी एन साइस के all possible samples draw करें, और फिर उन सब का मीन निकालें, और वो मीन पापूलेशन मीन से different होगा, लिकिन अगर हम उन सारे मीनस का again mean calculate करें, तो फिर वो population mean के exactly equal होगा, you remember in the last example we did a very small population of 3, अखर हम नहीं उस 2 2 के samples निकालें और फिर उन 2 2 के samples में हर samples का मीन निकालां, और फिर उन सारे मीनस को एक अद्ता करें, उन सारे मीनस का एक मीन निकालां, और अगर फिर शामपल मीन जाए, मीन of all sample means would be equal to population mean. Imagine, अगर तहीं अगर after us करें, अगर हम नहीं लास इसका होगा के n factorial और n minus r factorial और r factorial और अगर हम इसको n हमारा 25 हैं और हमारा जो r है वो 5 हैं if we put the values in the formula अगर हम इसको solve करें तो you remember के factorial का मतलब यह होता है के हम उसको descending order में multiply करते जाते तो हम करेंगे 25 multiplied by 24 multiplied by 23 multiplied by 22 multiplied by 21 multiplied by 20 n minus r i.e. 25 में से हम 5 minus करेंगे तो it will be 20 factorial तो 20 factorial और 20 factorial क्रोस हो जाएगा और r factorial को आपने फिर दबारा कोलने है 5 multiplied by 4 multiplied by 3 multiplied by 2 और अगर हम इसको solve करेंगे तो possibly में खयाल से इसके आपके 53136 all possible samples बनेंगे तो 53136 samples के आप 5-5 के निकालने है आप हर सामपल की मीं निकालेंगे और फिर उन सारी 53,000 means को एकटा जब हम कंबाइन करेंगे तो हम ये कहेंगे को उन सारे मींस का जो total mean होगा, that will be equal to the population mean. तो this is a very golden rule in statistics क्यो? क्यो के वी know के अगर हमारे पास all possible samples निकालेंगे और हम उनको plot करेंगे तो हमारी जो distribution की शेप होगी वो normal होगी and mean of all sample means would be equal to population mean. तो this is actually the central limit theorem central limit theorem क्यो के अगर अगर किसी भी population में से अगर अगर अप एक सामपल निकालें उसका mean निकालेंगे और उसको plot करेंगे distribution की अपर अप again a sample drop करेंगे उसी population से उसका mean निकालेंगे और उसको plot करेंगे तो आप का जो distribution of means होगा that will approximate normal distribution. उसकी वो shape होगी उसकी normal distribution की तरां बनेगी और जा वो normal distribution की तरां बनेगी तो mean of all sample means would be equal to population. फिर आप को 53,000, 54,000 सामपल लेके हरे की mean निकालेंगे अगर निकालेंगे उसरत निकालेंगे अगर अगर निकालेंगे तो हमारे लिए possible भी अजीली हम पापलेशन में से एक सामपल निकालते है और उसी सामपल से हमने population parameter को अस्टिमेट करनाउते है लेकिन once we know the rule once we know the law the central element theorem तो हमारे लिए चीसे बड़या सान हो जाएंगी किं कि अगर आगर चलके मैंगाई अपको तो हमने normal distribution पे हमें पता है कि हम किसी भी चीस को जब normal distribution भी लेजाते है और उसका mean standard deviation पाईंडोड कर लिते है तो हम उसका z पाईंडोड कर सकते है और उस z से हम पिर उसका area find out कर सकते है and we remember that area is equal to the probability of any event occurring in that particular region. So what is the central limit theorem? Just I will read it for any population with the mu and standard deviation the distribution of sample means for sample size n will have a mean of population mean and the standard deviation of that sampling distribution will be equal to standard deviation under... So mean of all sample means will be equal to mu and standard deviation of the sampling distribution of means will be equal to sigma over n under root of population standard deviation or n under root So this is our sampling distribution Just like normal distribution in which we found out the mean of standard deviation we have a sampling distribution where instead of x values we have the means or standard deviations and then we take out the mean or standard deviation. So central limit theorem mean of all sample means and the standard deviation of sampling distribution of mean would be equal to sigma over n under root Central limit theorem describes the distribution of sample means by identifying the three basic characteristics that describe any distribution meaning shape of the distribution central tendency and variability Shape of the distribution the distribution of sample means tends to be normal distribution in fact this distribution is almost perfectly normal just like our n size increase means if you have taken out 2 sample size or 5 sample size but we increase n if n is greater than 30 then our distribution would be exactly normal distribution and its shape would be perfectly normal the population from which this sample was taken out is already normally distributed so when we take out different samples and plot them then the distribution would be approximately normal but if you increase n size means the size of every sample would be approximately normal but if you increase n size means the size of every sample would be approximately normal and if you increase the size of every sample and take out samples and take out the mean of every sample and plot the means then the distribution would be perfectly normal because its n size is bigger and last time we did two examples if you remember in sampling distribution we took out 2 samples and then we took out 2 samples from the 4 population but when we increase sample size then our distribution when we plot the means then it is perfectly normal distribution the mean of distribution of sample means is the expected value of mean and the mean of distribution of sample mean is equal to the population mean if our n size is bigger then the mean of sampling distribution would be equal to the population mean and remember whenever we are making sampling distribution of means or of proportions or standard deviations we will always remember that all possible samples means would be of the population mean and this is the central limit theorem and it is also called expected value of m and the standard deviation of our sampling distribution would be standard deviation of the population divided by n under root as soon as we make sampling distribution of means similarly we make sampling distribution of proportions and we make standard deviations and our central limit theorem also says that if we have the sampling distribution of proportions i.e we have calculated instead of means proportions which I will talk shortly about we will do this that our mean of sampling distribution of proportions will be equal to population proportion and the standard deviation of sampling distribution of proportions will be equal to we will put a formula which is p 1 minus p and we will square it divided by n we will do it shortly sampling distribution of proportions