Given a histogram and the probability mass function values for each observation, when plotting the histogram and the curve (this is bell curve since the data is assumed to be normal) on the same figure, the two will not be of the same scaling. How does one actually scale the probability mass/density function so that it is 'equivalent' to the histogram?


1 Answer 1


This is pretty simple. Let's assume that $F_X(x)$ is cummulative density function, such that $$F_X(x) = \int_{-\infty}^{x} f_X(x) dx \quad \cdots \cdots (1)$$ where $f_X(x)$ is density function. Differentiating on the both side (ignoring subscript X), we get $$\frac{d}{dx}\, F(x) = f(x) $$ It can be written as; $$dF(x)=f(x) \,d(x)$$ $$F(x+h) - F(x) = f(x)dx$$ so, more precisely your density $f(x)$ is: $$f(x) = \frac{F(x+h)-F(x)}{dx}\quad \cdots \cdots (2)$$

Now, you can use identity 2, to scale your histogram into density. But your degree of accuracy depend upon the width of class interval. To scale your histogram into density:

  1. Convert your frequency for each class into probability by dividing total number of observations{ensure that your class interval is sufficiently small}. This represent your $dF(x)$.

  2. Now, divide your probability from $dx$ is size of class interval and you will get $f(x)$.

You can reverse the procedure to get the frequency from the density and vice-versa.


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