You could draw a Probability Plot: "The probability plot ... is a graphical technique for assessing whether or not a data set follows a given distribution such as the normal or Weibull. "The data are plotted against a theoretical distribution in such a way that the points should form approximately a straight line. Departures from this straight line indicate departures from the specified distribution." Source: Online Engineering Statistics Handbook http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm
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Normal Distribution is a key to Statistics. It is a limiting case of Binomial and Poisson distribution also. Central limit theorem converts random variable to normal random variable. Also central limit theorem tells us whether data items from a sample space lies in an interval at 1%, 5%, 10% siginificane level.
You integrate the probability distribution function to get the cumulative distribution function (cdf). Then find the value of the random variable for which cdf = 0.5.
I will give first the non-mathematical definition as given by Triola in Elementary Statistics: A random variable is a variable typicaly represented by x that has a a single numerical value, determined by chance for each outcome of a procedure. A probability distribution is a graph, table or formula that gives the probabability for each value of the random variable. A mathematical definition given by DeGroot in "Probability and Statistics" A real valued function that is defined in space S is called a random variable. For each random variable X and each set A of real numbers, we could calculate the probabilities. The collection of all of these probabilities is the distribution of X. Triola gets accross the idea of a collection as a table, graph or formula. Further to the definition is the types of distributions- discrete or continuous. Some well know distribution are the normal distribution, exponential, binomial, uniform, triangular and Poisson.
It will be the same as the distribution of the random variable itself.
The value of the distribution for any value of the random variable must be in the range [0, 1]. The sum (or integral) of the probability distribution function over all possible values of the random variable must be 1.