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dnorm() and dgamma() are used to return the density of a given distribution function. How we interpret their value? For example, if the mark of stat students is normally distributed with mean 2 and standard deviation 0.5. Then dnorm(3,2,0.5) will give 0.1078919. So what does it mean? Can I say that the probability density that a student got 3 in the stat exam is 0.10?

Kindly note that I am not asking about the difference between probability density and probability. I am asking what does the value of dnorm means at a single value? How to explain the return value?

Dave
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Alice
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  • I think I found the answer. dnorm(3, 2, 0.5) is how often a stusents get 3 in stat exam which is about 10%. – Alice Jul 11 '22 at 16:46
  • That is incorrect. If you take the distribution to be normal (which student scores won't be exactly), then the probability of scoring a $3$ is zero! – Dave Jul 11 '22 at 16:48
  • @Dave I am not asking about probability itself. Of course it will be zero as normal is continuous. But I am asking about dnorm. Which is the probability density which is different than the probability. – Alice Jul 11 '22 at 16:50
  • Then how do students score $3$ about $10%$ of the time? – Dave Jul 11 '22 at 16:51
  • @Dave I mean relative frequency. Is the dnorm at a single value return the height of the density at this value? The hight is the frequency/ relative frequency. – Alice Jul 11 '22 at 16:53
  • @Dave if there is 100 students. Then 10% of them will got 3. Not the probability that a students will got 3 in the exam. – Alice Jul 11 '22 at 16:56
  • @Dave my question is not a duplicate as I am asking about the meaning of a return value or interpretation and not the difference between probability and probability density as the other question. – Alice Jul 11 '22 at 17:04
  • What happens when the dnorm function returns a value in excess of $1$, such as dnorm(0, 0, 1/5)? – Dave Jul 11 '22 at 17:07
  • @Dave dnorm can exceed 1. It is not a probability so it can exceed one. It is the f(x). – Alice Jul 11 '22 at 17:09
  • So then over $190%$ of the students can earn $3$ if the mean if $3$ and the standard deviation is $1/5?$ dnorm(3, 3, 1/5) = 1.99 – Dave Jul 11 '22 at 17:11
  • @Dave So, what do you think 1.99 mean? – Alice Jul 11 '22 at 17:18
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    That's answered in the duplicate thread, which specifically explains how to interpret the density: I encourage you to read it. – whuber Jul 11 '22 at 17:36
  • @whuber I have read it. But I am still confused. The density is a smooth curve of histogram. Then, the higher of any bar in histogram is the frequency of that bar. I think the same is for density. That is my point. – Alice Jul 11 '22 at 17:58
  • The height of the pdf represents density. The comparison with a histogram is a good one when the histogram is correctly understood as representing frequency by the areas of the bars, not their heights. – whuber Jul 11 '22 at 18:34

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