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According to this definition, E[X] exists and is finite if and only if E[X +] and E[X −] are both finite. Due to the formula |X| = X + + X −, this is the case if and only if E|X| is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite ...
These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where: E = (a + 4m + b) / 6 SD = (b − a) / 6. E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. SD measures the variability or uncertainty in the estimate.
The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point.
A probability distribution is not uniquely determined by the moments E[X n] = e nμ + 1 / 2 n 2 σ 2 for n ≥ 1. That is, there exist other distributions with the same set of moments. [4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution. [citation needed]
For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical ...
For example, to calculate the 95% prediction interval for a normal distribution with a mean (μ) of 5 and a standard deviation (σ) of 1, then z is approximately 2. Therefore, the lower limit of the prediction interval is approximately 5 ‒ (2⋅1) = 3, and the upper limit is approximately 5 + (2⋅1) = 7, thus giving a prediction interval of ...
If X is a random variable from a normal distribution with mean μ and standard deviation σ, its Z-score may be calculated from X by subtracting μ and dividing by the standard deviation: Z = X − μ σ {\displaystyle Z={\frac {X-\mu }{\sigma }}}
We can then infer that the probability that it has between 600 and 1400 words (i.e. within = standard deviations of the mean) must be at least 75%, because there is no more than / = / chance to be outside that range, by Chebyshev's inequality.