standard deviation of the data. The y-intercept
indicates the mean of the data. For
sets of < 30 observations, only substantial departure from linearity should be
interpreted as conclusive evidence of non normality. How the plot differs from a
straight line can give you information about how the data distribution differs from a
If the graph is (X(i)
, Zi), A light-tailed distribution (relative to the Normal) will
S-shaped plot with the left end of the plot curving upward. A heavy-tailed distribution
(relative to the Normal)
will give an S-shaped plot with the left end of the plot curving
downward. A right-skewed distribution will give a plot having middle points falling above
the line and end points falling below the line. Note: Some statistical packages plot (X(i) , Zi)
instead of (Zi, X(i)). The interpretation of the shape of the plot as an indication of how the
distribution differs from normality will then be reversed.
Probability plots can be used to check distributional assumptions for distributions other than
Normal. For example, to check whether data comes from an exponential distribution, compare
percentiles of data to percentiles of a standard Exponential (with rate λ = 1). Additionally,
to check whether two sets of data come from the same underlying distribution, plot the
percentiles of the first data set versus the percentiles of the second data set. The plot
should be approximately linear.
The exponential probability density function is widely used in engineering to describe the
distribution of many types of variables, most often, the distribution of waiting times
between occurrences of successive events. A random variable X has an exponential
distribution with parameter λ (λ >0) if its pdf is f(x) = λ exp(-λx), x >0,.
mean of an exponential r.v. with parameter λ is 1/λ . The standard deviation of an
exponential r.v. with parameter λ is also 1/λ . The StatConcepts Lab “How are
Populations Distributed” allows you to visualize exponential pdfs with different parameters λ.
Suppose the number of occurrences of an event in a time interval of length t follows a
Poisson process with rate αt and the number of occurrences in nonoverlapping intervals
are independent. It can be shown that the waiting time until the first occurrence of the
event follows an exponential distribution with parameter α.
The exponential distribution has the memoryless property, meaning that the probability we
wait at least b additional minutes for an event to occur, given that we’ve already waited at
least a minutes (a<b) is the same as the probability that we have to wait b minutes from
the start. In other words, the distribution of additional waiting time is exactly the same as
the distribution of original waiting time, or distribution of additional waiting time is
independent of how long you’ve already waited. (The distribution of the number of
occurrences until the first success, given the events are independent with constant
probability of success, p, also has this property.)