Kurtosis. In probability theory and statistics, kurtosis (from the Greek word κυρτός, kyrtos or kurtos, meaning curved, arching) is any measure of the "peakedness" of the probability distribution of a real-valued random variable.[1] In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population.
There are various interpretations of kurtosis, and of how particular measures should be interpreted; these are primarily peakedness (width of peak), tail weight, and lack of shoulders (distribution primarily peak and tails, not in between). The "Darkness" data is platykurtic (−0.194), while "Far Red Light" shows leptokurtosis (0.055) Pearson moments[edit]
Skewness. Example of experimental data with non-zero (positive) skewness (gravitropic response of wheatcoleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
The skewness value can be positive or negative, or even undefined. Variance. In probability theory and statistics, variance measures how far a set of numbers is spread out.
(A variance of zero indicates that all the values are identical.) Variance is always non-negative: A small variance indicates that the data points tend to be very close to the mean (expected value) and hence to each other, while a high variance indicates that the data points are very spread out from the mean and from each other. The square root of variance is called the standard deviation. The variance is a parameter that describes, in part, either the actual probability distribution of an observed population of numbers, or the theoretical probability distribution of a sample (a not-fully-observed population) of numbers. In the latter case, a sample of data from such a distribution can be used to construct an estimate of its variance: in the simplest cases this estimate can be the sample variance.
Definition[edit] , or simply σ2 (pronounced "sigma squared"). Continuous random variable[edit] where. Mode (statistics) The mode is the value that appears most often in a set of data.
The mode of a discrete probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. The mode of a continuous probability distribution is the value x at which its probability density function has its maximum value, so, informally speaking, the mode is at the peak. The above definition tells us that only global maxima are modes. Slightly confusingly, when a probability density function has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. In symmetric unimodal distributions, such as the normal (or Gaussian) distribution (the distribution whose density function, when graphed, gives the famous "bell curve"), the mean (if defined), median and mode all coincide. Expected value.
In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) refers, intuitively, to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained.
More formally, the expected value is a weighted average of all possible values. In other words, each possible value that the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value. The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution.[3] For random variables such as these, the long-tails of the distribution prevent the sum/integral from converging. The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. Definition[edit] Example 2. Lemma. Proof.