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I write a small program which calculates a confidence interval for a given norm value. The output is then visualized by plotting the confidence interval and the corresponding hypothetical norm value distribution. The software is supposed to be used for psychodiagnostics when a test provides you with norm values but does not report confidence intervals for them.

I use norm.pdffrom scipy.stats to calculate the y-values of the normal distribution (see this link). norm.pdf takes an array of x-values, a mean and a standard deviation as arguments. The normal distribution curve is then plotted using matplotlib. The software works fine but I am not sure if I use norm.pdf properly according to what I want to achieve.

Let's say I have Person which has a norm value of 60 in Test A (T-values, M = 50, SD = 10) and a norm value of 1 in Test B (z-values, M = 0, SD = 1). On both scales the person's test value is exactly one standard deviation above average. Intuitively I would expect that the y-values (aka. probability densities) would be the same for both test values. Instead the y-values seem to change in relation to the given standard deviation and therefore the y-axis labels. Is this behavior statistically correct or do I make false assumptions about the behavior of the normal distribution?

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norm value = 1, M = 1, SD = 1

enter image description here

norm value = 60, M = 50, SD = 10

Johannes Wiesner
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  • Have you read https://stats.stackexchange.com/questions/4220? – whuber Feb 25 '19 at 13:35
  • No I haven't and I don't know how this relates to my post or solves my question. Could you explain why this link is useful to answer my question? Please excuse my poor statistical understanding on this issue. – Johannes Wiesner Feb 25 '19 at 13:52
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    In short: when talking about a PDF, the **area** under it is equal to 1, regardless of it's location and width. So that a very narrow, _localized_, distribution will have a very high peak. On the other hand, a very broad distribution will be very low. Picture it as rectangles: one is $0.1\times 10$ the other is $10\times 0.1$. The PDF values will be different for your measurements. But what is meaningful, is, e.g., the probability that the measurement will be higher than $\mu+\sigma$, which in both cases is $\approx 17\%$. – corey979 Feb 25 '19 at 13:56
  • Thank you all for your help. I think I get it now. Correct me If I am wrong: since the PDF **doesn't output probabilities** but **probability densities** probabilities are defined as integrals between two points. This means that I will only get probabilities if I calculate the integral between two norm values. As far as the program and user interface is concerned: one could argue if it makes sense to present the y-axis labels. If I don't get any information from it unless I calculate an integral would you recommend to just leave out the y-axis labels? – Johannes Wiesner Feb 26 '19 at 10:08

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