Concept# Central tendency

Summary

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.
Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.
The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."
The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysis may judge whether data has a strong or a weak central tendency based on its dispersion.
Measures
The following may be applied to one-dimensional data. Depend

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As shown by Michel and Ramakrishnan (2007) and later generalized by Feigon and Whitehouse (2008), there are "stable" formulas for the average central L-value of the Rankin-Selberg convolutions of some holomorphic forms of fixed even weight and large level against a fixed imaginary quadratic theta series. We obtain exact finite formulas for twisted first moments of Rankin-Selberg L-values in much greater generality and prove analogous "stable" formulas when one considers either arbitrary modular twists of large prime power level or real dihedral twists of odd type associated to a Hecke character of mixed signature. (C) 2013 Elsevier Inc. All rights reserved.

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Brain-machine interfaces (BMIs) - devices that connect brain areas to external actuators - strive to restore limb mobility and sensation to patients suffering from paralysis or limb loss. Here we report a novel BMI that controls two virtual arms simultaneously. The development of BMIs for bimanual control is important because even the most basic daily movements such as opening a jar or buttoning a shirt require two arms. We for the first time have designed and implemented a bimanual BMI where activity of multiple cortical areas is translated in real-time into center-out reaching movements performed by two virtual arms. Eight multielectrode arrays, a total of 768 electrode channels, were implanted in the primary motor (M1), sensory (S1), supplementary motor (SMA), dorsal premotor (PMd), and posterior parietal (PP) cortices of both hemispheres of a rhesus monkey. Movement kinematics of each arm were extracted from the same ensemble of 400 neurons using a Wiener filter and an unscented Kalman filter (UKF). Typically, a single neuron contributed to the movements of both left and right arms. Movements were enacted by arms of a virtual rhesus monkey avatar on a computer screen presented in first-person to the monkey. On each trial, the virtual arms moved their central locations to peripheral targets presented simultaneously on the right and left sides of the computer screen. Peri-event time histograms and linear discriminant analysis revealed a highly distributed encoding scheme, with movement directions of both limbs represented by both ipsilateral and contralateral areas. Furthermore, movements were represented by multiple cortical regions, including both primary and non-primary motor areas which have been previously identified areas important for bimanual coordination. Over the course of several weeks of real-time BMI control, the monkey’s performance clearly improved both when the monkey continued to move the joystick and when the joystick was removed. These results support the feasibility of cortically-driven clinical neural prosthetics for bimanual operations.

2012We derive a Motohashi-type formula for the cubic moment of central values of -functions of level cusp forms twisted by quadratic characters of conductor , previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for -functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.