what is the centeral tendency "mean, mode & median " for continuous data?

Answer 1

a_math_guy got the formula for the median wrong. For a continuous variable, the median is a value m (not necessarily unique) such that P(X≥m) = P(X≤m). If the random variable has a probability density function f, this equation can be stated as [m, ∞]∫f(x) dx = [-∞, m]∫f(x) dx (not x f(x) dx, as a_math_guy stated).

The formula for the mean for a random variable with pdf f is, as a_math_guy stated, [-∞, ∞]∫x f(x) dx.

Despite a_math_guy's claim to the contrary, the mode is sometimes defined for continuous data - specifically, it is the value(s) at which the pdf attains it's maximum (if the pdf does in fact attain a maximum).

Edit: don't worry about it a_math_guy, I've made some dandy screwups myself.

About the mode: if the pdf is taken to be the derivative of the cumulative distribution function, then we can say that if f(a) > f(b), then a is more probable than b in the sense that ∃δ>0 such that ∀h<δ, P(a-h≤X≤a+h) > P(b-h≤X≤b+h) (that is, for sufficiently small δ, it will always be more likely that an event in a δ-neighborhood of a will happen than an event in a δ-neighbourhood of b).

That said, you do bring up an interesting point - if we admit more general pdfs which are not the derivatives of a cumulative distribution function (say, because they differ from said derivative on a set of Lebesgue measure zero), these pdfs will result in an identical distribution function, but may have different modes, and it's more difficult to say whether that has any philosophical meaning. I suppose that you might try to classify the relative probabilities of P=0 events using infinitesimal hyperreal numbers, but I don't have enough experience with the theory of distributions to tell whether such an interpretation could be made wholly consistent. Thus, it would seem here that the formal definition of the mode (which is the maximum of the pdf) may, in these cases, differ from the intuitive concept of the most frequently occurring event.

Answer 2

There are easier ways to explain this conceptually.

For continuous data, you should have a distribution function. Imagine that this function is made of solid material sitting atop the number line. In this case, the mean (arithmetic average) is the balance point for this distribution. Means will be pulled by extreme values into the tails of the distribution--for example, income is right-skewed, meaning that there are many lower incomes and fewer higher incomes. Because of this, the mean will be pulled toward the tail with the higher incomes.

The median divides the ordered data in half. It is also the "equal areas" point--50% of the area is to its left and 50% of the area is to its right.

For continuous data, it does not make sense to talk about a mode unless we are looking at a range of data values. If we are, then the mode will be the data range that is the "tallest."

It's also good to consider the relationship between the mean and the median. When the data are (roughly) symmetric, the mean and median will be about the same. When the data are right-skewed (like in the income example), the mean is greater than the median. When the data are left-skewed, the mean is less than the median. (An example of left-skewed data would be an easy test. Many people would earn high scores and fewer people would earn low scores.)

Hope this helps!

Answer 3

I don't know the formulas off the top of my head, but the function is y = f(x), the mean is Integral ( f(x) dx)/f(x)

The mode is the maximum of the function
The median is the value of p for which integral (f(x) dx) with limits -infinity to p is half the value of the integral from -infinity to infinity.

Answer 4

You need calculus for these concepts.

The median is that value M such that integral of x*P(X=x)dx over xM.

The mean is the integral of x*P(X=x) over all x.

There is no mode for a continuous variable, the expected value of any one occurence is 0.

Whoops as Pascal corrected me, I got expected value integral of x*P(X=x) and probability integral of P(X=x) confused, sorry. I am going to have to think about the mode tho'. Sure the max of the probability function is where you would look for the mode, but in what sense is the most frequent when P(X=a)=0 for any single value a?