# When To Use Pdf And Cdf Statistics

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Say you were to take a coin from your pocket and toss it into the air. While it flips through space, what could you possibly say about its future? Will it land heads up?

## Probability density functions

Typical Analysis Procedure. Enter search terms or a module, class or function name. While the whole population of a group has certain characteristics, we can typically never measure all of them. In many cases, the population distribution is described by an idealized, continuous distribution function.

In the analysis of measured data, in contrast, we have to confine ourselves to investigate a hopefully representative sample of this group, and estimate the properties of the population from this sample.

A continuous distribution function describes the distribution of a population, and can be represented in several equivalent ways:. In the mathematical fields of probability and statistics, a random variate x is a particular outcome of a random variable X : the random variates which are other outcomes of the same random variable might have different values.

Since the likelihood to find any given value cannot be less than zero, and since the variable has to have some value, the PDF has the following properties:. The integral over the PDF between a and b gives the likelihood of finding the value of x in that range. Together, this gives us. Probability Density Function left and Cumulative distribution function right of a normal distribution. The Figure Utility functions for continuous distributions, here for the normal distribution.

Probability density function PDF : note that to obtain the probability for the variable appearing in a certain interval, you have to integrate the PDF over that range. Cumulative distribution function CDF : gives the probability of obtaining a value smaller than the given value.

Survival function SF : 1-CDF: gives the probability of obtaining a value larger than the given value. Utility functions for continuous distributions, here for the normal distribution. When we have a datasample from a distribution, we can characterize the center of the distribution with different parameters:.

The median is that value that comes half-way when the data are ranked in order. In contrast to the mean, it is not affected by outlying data points. In some situations the geometric mean can be useful to describe the location of a distribution.

It is usually close to the median, and can be calculated via the arithmetic mean of the log of the values. This one is fairly easy: it is the difference between the highest and the lowest data value. The only thing that you have to watch out for: after you have acquired your data, you have to check for outliers , i.

Often, such points are caused by errors in the selection of the sample or in the measurement procedure. There are a number of tests to check for outliers. One of them is to check for data which lie more than 1. The Cumulative distribution function CDF tells you for each value which percentage of the data has a lower value Figure Utility functions for continuous distributions, here for the normal distribution.

The value below which a given percentage of the values occur is called centile or percentile , and corresponds to a value with a specified cumulative frequency. Also important are the quartiles , i. The difference between them is sometimes referred to as inter-quartile range IQR. The Figure below indicates why the sample standard deviation underestimates the standard deviation of the underlying distribution. Gaussian distributions fitted to selections of data from the underlying distribution: While the average mean of a number of samples converges to the real mean, the sample standard deviation underestimates the standard deviation from the distribution.

Since the t-distribution has longer tails than the normal distribution, it is much less sensitive to outliers see Figure above. While the standard deviation is a good measure for the distribution of your values, often you are more interested in the distribution of the mean value. For example, when you measure the response to a new medication, you might be interested in how well you can characterize this response, i. This measure is called the standard error of the mean , or sometimes just the standard error.

For the sample standard error of the mean , which is the one you will be working with most of the time, we have. The most informative parameter that you can give for a statistical variable is arguably its confidence interval.

Most of the time you want to determine the confidence interval for normally distributed data, which is given by. Note: If you want to know the confidence interval for the mean value, you have to replace the standard deviation by the standard error! In scipy. Should the definition of a distribution require more than two parameters, the following parameters are called shape parameters.

The exact meaning of each parameter can be found in the function definition. The scale parameter describes the width of a probability distribution. If s is large, then the distribution will be more spread out; if s is small then it will be more concentrated. If the probability density exists for all values of s , then the density as a function of the scale parameter only satisfies.

A shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter.

If location and shape already completely determine the distribution as is the case for e. It follows that the skewness and kurtosis of these distribution are constants. Distributions are skewed if they depart from symmetry. For example, if you have a measurement that cannot be negative, which is usually the case, then we can infer that the data have a skewed distribution if the standard deviation is more than half the mean.

Such an asymmetry is referred to as positive skewness. The opposite, negative skewness, is rare. Right The leptokurtic Laplace distribution has an excess kurtosis of 3, and the platykurtic Wigner semicircle distribution an excess kurtosis of Distributions with negative or positive excess kurtosis are called platykurtic distributions or leptokurtic distributions respectively.

The variable for a standardized distribution function is often called statistic. The Normal distribution or Gaussian distribution is by far the most important of all the distribution functions. This is due to the fact that the mean values of all distribution functions approximate a normal distribution for large enough sample numbers.

For smaller sample numbers, the sample distribution can show quite a bit of variability. For example, look at 25 distributions generated by sampling numbers from a normal distribution:. The central limit theorem states that for identically distributed independent random variables also referred to as random variates , the mean of a sufficiently large number of these variables will be approximately normally distributed.

The figure below shows that averaging over 10 uniformly distributed data already produces a smooth, almost Gaussian distribution. Center Histogram of average over two datapoints. Right Histogram of average over 10 datapoints. To illustrate the ideas behind the use of distribution functions, let us go step-by-step through the analysis of the following problem:. The average weight of a newborn child in the US is 3.

If we want to check all children that are significantly different from the typical baby, what should we do with a child that is born with a weight of 2. The chance that a healthy baby weighs 2. The chance that the difference from the mean is that much is twice that much, as the lighter blue area must be added.

In the following, we will describe these distributions in more detail. Other distributions you should have heard about will be mentioned briefly:. The reason is that the sample mean does not coincide exactly with the population mean. This modified distribution is the t-distribution , and converges for larger values towards the normal distribution.

A very frequent application of the t-distribution is in the calculation of Confidence intervals :. For comparison, I also calculate the corresponding value from the normal distribution. Since the t-distribution has longer tails than the normal distribution, it is much less sensitive to outliers see Figure below. Since the Chi-square distribution describes the distribution of the summed squares of random variates from a standard normal distribution , we have to normalize our data before we calculate the corresponding CDF-value:.

In other words, the batch matches the expected standard deviation. The cutoff values in an F table are found using three variables:. ANOVA compares the size of the variance between two different samples.

This is done by dividing the larger variance over the smaller variance. The formula for the resulting F statistic is:. If you want to investigate whether two groups have the same variance, you have to calculate the ratio of the sample standard deviations squared:.

Take for example the case that you want to compare two methods to measure eye movements. The two methods can have different accuracy and different precision. With your test you want to determine if the precision of the two methods is equivalent, or if one method is more precise than the other.

Accuracy and precision of a measurement are two different characteristics! When you look 20 deg to the right, you get the following results: Method 1: [ The code sample below shows that the F statistic is close to the center of the distribution, so we cannot reject the hypothesis that the two methods have the same precision.

In some circumstances a set of data with a positively skewed distribution can be transformed into a symmetric distribution by taking logarithms. Taking logs of data with a skewed distribution will often give a distribution that is near to normal see Figure below.

It has two parameters, which allow it to handle increasing, decreasing or constant failure-rates see Figure below. It is defined as. Its complementary cumulative distribution function is a stretched exponential function. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:. In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus.

For a stochastic variable X with an exponential distribution , the probability distribution function is:. This is a simple one: an even probability for all data values see Figure below. Not very common for real data.

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This tutorial provides a simple explanation of the difference between a PDF probability density function and a CDF cumulative distribution function in statistics. There are two types of random variables: discrete and continuous. Some examples of discrete random variables include:. Some examples of continuous random variables include:. For example, the height of a person could be

Cumulative distribution functions are also used to specify the distribution of multivariate random variables. The proper use of tables of the binomial and Poisson distributions depends upon this convention. The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating [3] using the Fundamental Theorem of Calculus ; i. Every function with these four properties is a CDF, i. Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function ccdf or simply the tail distribution or exceedance , and is defined as. This has applications in statistical hypothesis testing , for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed.

Cumulative Distribution Functions (CDF); Probability Density Function (PDF); Interactive the future states of a system in some useful way, we use random variables. Also, interactive plots of many other CDFs important to the field of statistics.

## Cumulative distribution function

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The cumulative distribution function CDF calculates the cumulative probability for a given x-value.

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