Using counts of a continuous random variable to establish an unknown probability density function

Question

The precision of a given measuring device defines a window over the probability density function of whatever is being measured. For instance, if a measuring tape is precise to 0.1 inches, then the probability density function for population height is split into bins of 0.1 inches, from 0 to $\infty$.

When we measure some people in a population, individual measures using this tape are single counts that we collect, and there is no known probability associated with them.

Though each count’s specific real value by itself does not have any probability to it (ie, the integral of the probability density function at the point is 0), the likelihood of the value occurring is given by the area of the probability bin at that value. This is reflected by a greater number of counts for events that fall into an area of greater probability--say 6 feet, as opposed to 9 feet.

Even though we have no known probability density function, from these counts, we can sum and divide by the number of counts to establish averages, and then get deviations and other descriptions for the probability density function.

My question is, if this is how we handle continuous data for which a probability density function is unknown, then is it not true that the averaging of our measurements (and other calculations) in essence has us treating these values as discrete counts? If so, then how does this play into the theoretical approaches for managing continuous data, where given a probability density function, we use integrals to establish means, deviations, and other details?

In essence, if the density function is known then we use calculus to establish descriptions of this function (quartiles, means, deviations, correlations, etc.), but if the function is not known, then even though the data is continuous, it seems we must use discrete approaches to establish a description of the density function. Is this not true?

Answer 1

If your measurements are precise to 0.1 inches, but the range of human heights is 0 feet to 9 feet, then there are 1080 discrete values your measurement could take. Although this distribution is discrete, its properties are well-approximated* by a continuous distribution.

Plus, at the points where the approximation fails, usually it's the continuous version that has the desirable property. For instance, continuous distributions are differentiable, which helps when one is doing numerical optimization. Combined with the fact that integrals are often easier to compute analytically than discrete sums, this is why continuous distributions are often used instead of very fine discrete distributions in practice.

*Some of the discrete distribution's properties at extremely fine scales are not well-approximated by a continuous distribution. For instance, if $H_c$ is a continuous height variable and $H_d$ is quantized to 0.1 inch, then $E(\cos 20\pi \cdot H_c) = 0$ but $E(\cos 20\pi\cdot H_d) = 1$. But at larger scales, say 1 inch and up, the two distributions are almost indistinguishable.