Mean

In mathematics, mean has several different definitions depending on the context.

In probability and statistics, mean and expected value are used synonymously to refer to one 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; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving . An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value is for n = 1, 2, 3, ....

For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted ) to distinguish it from the population mean (denoted or ).

Taylor Swift

Taylor Alison Swift (born December 13, 1989) is an American singer-songwriter. Raised in Wyomissing, Pennsylvania, she moved to Nashville, Tennessee, at the age of 14 to pursue a career in country music. She signed with the independent label Big Machine Records and became the youngest songwriter ever signed by the Sony/ATV Music publishing house. The release of Swift's eponymous debut album in 2006 marked the start of her career as a country music singer. Her third single, "Our Song", made her the youngest person to single-handedly write and perform a number-one song on the Hot Country Songs chart.

Swift's second album, Fearless, was released in 2008. Buoyed by the pop crossover success of the singles "Love Story" and "You Belong with Me", Fearless became the best-selling album of 2009 in the United States. The album won four Grammy Awards, making Swift the youngest ever Album of the Year winner. Swift's third and fourth albums, 2010's Speak Now and 2012's Red, both sold more than one million copies within the first week of their U.S release. Speak Now's "Mean" won two Grammy Awards, while Red's singles "We Are Never Ever Getting Back Together" and "I Knew You Were Trouble" were successful worldwide. Swift's fifth album, the pop-focused 1989, was released in 2014 and sold more copies in its opening week than any album in the previous 12 years, making Swift the first and only act to have three albums sell more than one million copies in the opening release week. Its singles "Shake It Off", "Blank Space", and "Bad Blood" reached number one on the Billboard Hot 100. The album would go on to win three awards at the 2016 Grammys including Album of the Year making Swift the first and thus far only female artist to receive the award twice.

Justin Bieber

Justin Drew Bieber (/ˈbiːbər/; born March 1, 1994) is a Canadian singer, songwriter, and record producer. Bieber released his debut EP, My World, in late 2009. It was certified platinum in the U.S. He became the first artist to have seven songs from a debut record chart on the Billboard Hot 100. Bieber released his first full-length studio album, My World 2.0, in 2010. It debuted at or near number one in several countries and was certified triple platinum in the U.S. It was preceded by his most successful single to date, "Baby".

Following his debut album, he had his first headlining tour, the My World Tour, released the remix albums My Worlds Acoustic and Never Say Never – The Remixes -- and the 3D biopic-concert film Justin Bieber: Never Say Never. He released his second studio album, Under the Mistletoe, in November 2011, when it debuted at number one on the Billboard 200. Bieber released his third studio album, Believe, in 2012. His fourth studio album Purpose was released in November 2015. His U.S. album and singles sales total 44.7 million.