There are several kinds of means in various branches of mathematics. For a data set, the arithmetic mean called the mathematical expectation or average, is the central value of a discrete set of numbers: the sum of the values divided by the number of values; the arithmetic mean of a set of numbers x1, x2... xn is denoted by x ¯. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean to distinguish it from the mean of the underlying distribution, the population mean. In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value. An analogous formula applies to the case of a continuous probability distribution.

Not every probability distribution has a defined mean. Moreover, for some distributions the mean is infinite. For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals; the sample mean may differ from the population mean for small samples. The law of large numbers dictates that the larger the size of the sample, the more it is that the sample mean will be close to the population mean. Outside probability and statistics, a wide range of other notions of "mean" are used in geometry and analysis; the arithmetic mean of a sample x 1, x 2, …, x n denoted by x ¯, is the sum of the sampled values divided by the number of items in the sample x ¯ = 1 n = x 1 + x 2 + ⋯ + x n n For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: 4 + 36 + 45 + 50 + 75 5 = 210 5 = 42.

The geometric mean is an average, useful for sets of positive numbers that are interpreted according to their product and not their sum. X ¯ = 1 n = 1 n For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: 1 5 = 24 300 000 5 = 30; the harmonic mean is an average, useful for sets of numbers which are defined in relation to some unit, for example speed. X ¯ = n − 1 For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is 5 1 4 + 1 36 + 1 45 + 1 50 + 1 75 = 5 1 3 = 15. {\displaystyle {\frac {++{\tfrac

List of honors and awards received by Barack Obama

The following is a list of accolades and honors conferred upon the 44th U. S. President, Barack Obama. 2005: Obama received NAACP Image Award – Chairman's Award upon his election to the U. S. Senate 2006: Best Spoken Word Album Grammy Awards for abridged audiobook version of Dreams from My Father 2007: He received the NAACP Image Award for Outstanding Literary Work, Nonfiction for Dreams from My Father. 2008: Best Spoken Word Album Grammy Awards for abridged audiobook version of The Audacity of Hope 2008: Harold Washington Award from the Congressional Black Caucus as well as the 2008 Phoenix Award from the CBC Foundation, Inc. 2008: His concession speech after the New Hampshire primary was set to music by independent artists as the music video "Yes We Can", viewed 10 million times on YouTube in its first month and received a Daytime Emmy Award. 2009: Nobel Peace Prize 2011: Transparency Award jointly offered by OMB Watch, the National Security Archive, the Project on Government Oversight, the Reporters' Committee for Freedom of the Press and

2013: NME Award for Hero of the Year 2013: Freedom of the City of Cape Town. The freedom scroll was accepted on their behalf by Acting U. S. Ambassador, Virginia Palmer. 2013: Israeli President's Medal of Distinction 2014: "Ambassador of Humanity" Award from the USC Shoah Foundation Institute for Visual History and Education 2017: Department of Defense Medal for Distinguished Public Service. 7 May 2017: John F. Kennedy Profile in Courage Award. 2 November 2017: For Inspiration and Recognition of Science and Technology's "Make It Loud" award for contributions as President supporting FIRST and STEM education 9 February 2020: Best Documentary Feature Academy Award for "American Factory," made by the Obamas' production company, Higher Ground

Churra, Spain

Churra is a hamlet belonging to the municipality of Murcia, with a municipal board. It is a semi-agricultural town, although in recent years it has opened its economy to industry, the service sector, with two hypermarkets, it had an area of 5.6 square kilometers. The distance to Murcia city center is 3.5 kilometers, served with a tram, lies at an altitude of 57 meters above sea level. It was founded during the Islamic domination around the ninth century and since its history has always been linked to that of Murcia, becoming in this hamlet of the eighteenth century. There are vestiges of its history in the village, such as the New and Old Ditches and Torre Alcayna House. Churra. Merino Álvarez, Abelardo. Geografía histórica de la provincial de Murcia. Consejo Superior de Investigaciones Científicas. Calvo García-Tornel, Francisco. Geografía humana de Murcia. Olkos-Tau Ediciones