 # How to interpret the standard normal distribution curve (standardized value vs probability density)?

In a standard normal distribution curve, if x-axis is the standardized value of a specific feature and y-axis is the probability density of that feature then how are we plotting the curve? Precisely to ask what is probability density and how do we arrive at the values of the probability density for plotting standard normal distribution curve and how is it different from probability distribution?

I think probably you are looking for a more deeper insight , but I will try with one possible way to look at it.

In normal distribution, the Random Variable (RV) is continuous, lets take temperature as the R.V. Since the RV is continuous we can only talk about its probability for a range of values i.e., temperature lying between t1 and t2 etc.
If we want to calculate temperature probability for a specific value, that would make temperature a discrete variable (taking specific values, so no longer continuous). This is what I think needs most convincing and thinking.

Next, if we agree that probability can only be calculated for a specific range, we can decide a range interval (probability very tiny range intervals), collect data, and calculate those probabilities from the data, plot it as a histogram and approximate with a curve.

For continuous r.v., note that Probability density itself doesn’t equals probability. Integrating `Probability Density Function` over a interval gives the probability for taking a range of values.

For discrete r.v., we use probability mass function, which basically can be used to directly calculate the probability (for a specific value or a range of discrete values, dice showing 6 or dice showing a number between 2-4).

You can look for `Probability Mass Function vs Probability Density Function`, you might find it useful.

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For continuous r.v., note that Probability density itself doesn’t equals probability. Integrating `Probability Density Function` over a interval gives the probability for taking a range of values.

For discrete r.v., we use probability mass function, which basically can be used to directly calculate the probability (for a specific value or a range of discrete values, dice showing 6 or dice showing a number between 2-4).

You can look for `Probability Mass Function vs Probability Density Function`, you might find it useful.

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