In statistics, mean has two related meanings:

• the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
• the expected value of a random variable, which is also called the population mean.

There are other statistical measures that should not be confused with averages - including 'median' and 'mode'. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example.

For a data set, the mean is the sum of the values divided by the number of values. The mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} , pronounced "x bar". This mean is a type of arithmetic mean. If the data set were based on a series of observations obtained by sampling a statistical population, this mean is termed the "sample mean" (Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} ) to distinguish it from the "population mean" (Failed to parse (Missing texvc executable; please see math/README to configure.): \mu or Failed to parse (Missing texvc executable; please see math/README to configure.): \mu _x). The mean is often quoted along with the standard deviation: the mean describes the central location of the data, and the standard deviation describes the spread. An alternative measure of dispersion is the mean deviation, equivalent to the average absolute deviation from the mean. It is less sensitive to outliers, but less mathematically tractable.

If a series of observations is sampled from a larger population (measuring the heights of a sample of adults drawn from the entire world population, for example), or from a probability distribution which gives the probabilities of each possible result, then the larger population or probability distribution can be used to construct a "population mean", which is also the expected value for a sample drawn from this population or probability distribution. For a finite population, this would simply be the arithmetic mean of the given property for every member of the population. For a probability distribution, this would be a sum or integral over every possible value weighted by the probability of that value. It is a universal convention to represent the population mean by the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): \mu .^[1] In the case of a discrete probability distribution, the mean of a discrete random variable x is given by taking the product of each possible value of x and its probability P(x), and then adding all these products together, giving Failed to parse (Missing texvc executable; please see math/README to configure.): \mu = \sum x P(x) .^[2]

The sample mean may differ from the population mean, especially for small samples, but the law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.^[3]

As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.

Examples of means[link]

Arithmetic mean (AM)[link]

The arithmetic mean is the "standard" average, often simply called the "mean".

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = \frac{1}{n}\cdot \sum_{i=1}^n{x_i}

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{4 + 36 + 45 + 50 + 75}{5} = \frac{210}{5} = 42.

Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions.

The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.

Nevertheless, many skewed distributions are best described by their mean – such as the exponential and Poisson distributions.

Geometric mean (GM)[link]

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^\tfrac1n

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

Failed to parse (Missing texvc executable; please see math/README to configure.): (4 \times 36 \times 45 \times 50 \times 75)^{^1/_5} = \sqrt[5]{24\;300\;000} = 30.

Harmonic mean (HM)[link]

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15.

Relationship between AM, GM, and HM[link]

AM, GM, and HM satisfy these inequalities:

Failed to parse (Missing texvc executable; please see math/README to configure.): AM \ge GM \ge HM \,

Equality holds only when all the elements of the given sample are equal.

Generalized means[link]

Power mean[link]

The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined for a set of n positive numbers x_i by

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^\tfrac1m

By choosing different values for the parameter m, the following types of means are obtained:

Failed to parse (Missing texvc executable; please see math/README to configure.): m\rightarrow\infty	maximum
Failed to parse (Missing texvc executable; please see math/README to configure.): m=2	quadratic mean
Failed to parse (Missing texvc executable; please see math/README to configure.): m=1	arithmetic mean
Failed to parse (Missing texvc executable; please see math/README to configure.): m\rightarrow0	geometric mean
Failed to parse (Missing texvc executable; please see math/README to configure.): m=-1	harmonic mean
Failed to parse (Missing texvc executable; please see math/README to configure.): m\rightarrow-\infty	minimum

[edit] ƒ-mean

This can be generalized further as the generalized f-mean

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right)

and again a suitable choice of an invertible ƒ will give

Weighted arithmetic mean[link]

The weighted arithmetic mean (or weighted average) is used, if one wants to combine average values from samples of the same population with different sample sizes:

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}.

The weights Failed to parse (Missing texvc executable; please see math/README to configure.): w_i

represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

Truncated mean[link]

Sometimes a set of numbers might contain outliers, i.e. a datum which is much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. In this case one can use a truncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values.

Interquartile mean[link]

The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i}

assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.

Mean of a function[link]

In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx.

Recall that a defining property of the average value Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{y}

of finitely many numbers Failed to parse (Missing texvc executable; please see math/README to configure.): y_1, y_2, \dots, y_n

is that Failed to parse (Missing texvc executable; please see math/README to configure.): n\bar{y} = y_1 + y_2 + \cdots + y_n . In other words, Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{y}

is the constant value which when

added to itself Failed to parse (Missing texvc executable; please see math/README to configure.): n

times equals the result of adding the Failed to parse (Missing texvc executable; please see math/README to configure.): n
terms of Failed to parse (Missing texvc executable; please see math/README to configure.): y_i

. By analogy, a defining property of the average value Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f}

of a function over the interval Failed to parse (Missing texvc executable; please see math/README to configure.): [a,b]
is that

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx

In other words, Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f}

is the constant value which when integrated over Failed to parse (Missing texvc executable; please see math/README to configure.): [a,b]
equals the result of

integrating Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

over Failed to parse (Missing texvc executable; please see math/README to configure.): [a,b]

. But by the second fundamental theorem of calculus, the integral of a constant Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f}

is just

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_a^b\bar{f}\,dx = \bar{f}x\bigr|_a^b = \bar{f}b - \bar{f}a = (b - a)\bar{f}

See also the first mean value theorem for integration, which guarantees that if Failed to parse (Missing texvc executable; please see math/README to configure.): f

is continuous then there exists a point Failed to parse (Missing texvc executable; please see math/README to configure.): c \in (a, b)
such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_a^bf(x)\,dx = f(c)(b - a)

The point Failed to parse (Missing texvc executable; please see math/README to configure.): f(c)

is called the mean value of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
on Failed to parse (Missing texvc executable; please see math/README to configure.): [a,b]

. So we write Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f} = f(c)

and rearrange the preceding equation to get the above definition.

In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.

This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be

Failed to parse (Missing texvc executable; please see math/README to configure.): \exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).

More generally, in measure theory and probability theory either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.

Mean of a probability distribution[link]

See expected value.

Mean of angles[link]

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities.

Fréchet mean[link]

The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher).

Other means[link]

Properties[link]

All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

Weighted mean[link]

A weighted mean M is a function which maps tuples of positive numbers to a positive number

Failed to parse (Missing texvc executable; please see math/README to configure.): M : (0,\infty)^n \to (0,\infty)

such that the following properties hold:

• "Fixed point": M(1,1,...,1) = 1
• Homogeneity: M(λ x₁, ..., λ x_n) = λ M(x₁, ..., x_n) for all λ and x_i. In vector notation: M(λ x) = λ Mx for all n-vectors x.
• Monotonicity: If x_i ≤ y_i for each i, then Mx ≤ My

It follows

• Boundedness: min x ≤ Mx ≤ max x
• Continuity: Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x\to y} M x = M y

• There are means which are not differentiable. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a median), but is not differentiable.
• All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
  • If f is bijective, then the generalized f-mean satisfies the fixed point property.
  • If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
  • In general a generalized f-mean will miss homogeneity.

The above properties imply techniques to construct more complex means:

If C, M₁, ..., M_m are weighted means and p is a positive real number, then A and B defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): A x = C(M_1 x, \dots, M_m x) ,

Failed to parse (Missing texvc executable; please see math/README to configure.): B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)} ,

are also weighted means.

Unweighted mean[link]

Intuitively spoken, an unweighted mean is a weighted mean with equal weights. Since our definition of weighted mean above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.

Thus we define M to be an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same.

Symmetry: Mx = M(πx) for all n-tuples π and permutations π on n-tuples.

Analogously to the weighted means, if C is a weighted mean and M₁, ..., M_m are unweighted means and p is a positive real number, then A and B defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): A x = C(M_1 x, \dots, M_m x) ,

Failed to parse (Missing texvc executable; please see math/README to configure.): B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)} ,

are also unweighted means.

Convert unweighted mean to weighted mean[link]

An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean M and weight the numbers by natural numbers Failed to parse (Missing texvc executable; please see math/README to configure.): a_1,\dots,a_n . (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by

Failed to parse (Missing texvc executable; please see math/README to configure.): A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).

Means of tuples of different sizes[link]

If a mean M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely

• Given an arbitrary tuple x, which is partitioned into y₁, ..., y_k, then

Failed to parse (Missing texvc executable; please see math/README to configure.): M x \in \mathrm{convexhull}(M y_1, \dots, M y_k).

(See Convex hull.)

Population and sample means[link]

The mean of a population has an expected value of μ, known as the population mean. The sample mean makes a good estimator of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a random variable, not a constant, and consequently it will have its own distribution. For a random sample of n observations from a normally distributed population, the sample mean distribution is

Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.

Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with n − 1 degrees of freedom.

See also[link]

Statistics portal

• Algorithms for calculating mean and variance
• Average, same as central tendency
• Descriptive statistics
• For an independent identical distribution from the reals, the mean of a sample is an unbiased estimator for the mean of the population.
• Kurtosis
• Law of averages
• Median
• Mode (statistics)
• Spherical mean
• Summary statistics
• Taylor's law

References[link]

External links[link]

• Weisstein, Eric W., "Mean" from MathWorld.
• Weisstein, Eric W., "Arithmetic Mean" from MathWorld.
• Comparison between arithmetic and geometric mean of two numbers
• Some relationships involving means

Taylor Swift
Swift performing during the Australian leg of her Speak Now Tour in 2012
Background information
Birth name	Taylor Alison Swift
Born	(1989-12-13) December 13, 1989 (age 22) Reading, Pennsylvania, United States
Genres	Country-pop, pop, country, alternative rock
Occupations	Singer-songwriter, actress
Instruments	Vocals, guitar, banjo, ukulele, piano
Years active	2006–present
Labels	Big Machine
Website	TaylorSwift.com

Taylor Alison Swift (born December 13, 1989) is an American singer-songwriter and occasional actress. Raised in Wyomissing, Pennsylvania, Swift moved to Nashville, Tennessee at the age of fourteen to pursue a career in country music. She signed to the independent label Big Machine Records and became the youngest songwriter ever hired by the Sony/ATV Music publishing house. The release of Swift's self-titled debut album in 2006 established her as a country music star. "Our Song", her third single, made her the youngest sole writer and singer of a number one song on the country chart. She received a Best New Artist nomination at the 50th Grammy Awards.

Swift's second album, Fearless, was released in late 2008. Buoyed by the chart success of the singles "Love Story" and "You Belong with Me", Fearless attracted a crossover audience and became the top-selling album of 2009. The record won four Grammy Awards, with Swift becoming the youngest ever Album of the Year winner. Fearless also received Album of the Year plaudits at the American Music Awards, Academy of Country Music Awards and Country Music Association Awards, making it the most awarded album in country music history. In 2010, Swift released her third album, Speak Now, which sold over one million copies in its first week. She then embarked on the 111-date Speak Now World Tour, which was attended by over 1.6 million fans and has become one of the highest-grossing concert tours of all time. The album's third single, "Mean", won two Grammy Awards for Best Country Song and Best Country Solo Performance. Swift is currently recording her fourth studio album, due for release in the fall of 2012.

Swift's work has earned her numerous accolades, including six Grammy Awards, ten AMAs, seven CMAs, six ACMs and 13 BMI Awards. She has sold over 22 million albums and 50 million song downloads worldwide, making her one of the best-selling music artists of all time. Swift has endorsement deals with Target, CoverGirl, Sony, Elizabeth Arden, Walmart and American Greetings. As a philanthropist, Swift has placed particular emphasis on arts education, children's literacy and natural disaster relief funds. In addition to her music career, Swift has appeared as an actress in the crime drama CSI: Crime Scene Investigation (2009), the romantic comedy Valentine's Day (2010) and the animated film The Lorax (2012).

Early life[link]

Taylor Alison Swift was born on December 13, 1989 in Reading, Pennsylvania.^[1] She is the daughter of Scott Swift, a Merrill Lynch financial adviser,^[2][3] and Andrea (née Gardner), a homemaker who previously worked as a mutual fund marketing executive.^[1][4] Swift was named after singer James Taylor; her mother believed a gender-neutral name would help her forge a successful business career.^[5][6] She has a younger brother, Austin, who attends Vanderbilt University.^[7] She spent the early years of her life on an eleven-acre Christmas tree farm in Montgomery County, Pennsylvania and was educated at the fee-paying Wyndcroft School.^[8] When Swift was nine years old, the family moved to Wyomissing, Pennsylvania,^[9][10] where she attended Wyomissing Area Junior/Senior High School.^[11] Swift spent her summers at her parent's vacation home in Stone Harbor, New Jersey^[12] and has described it as the place "where most of my childhood memories were formed".^[10]

Swift's first hobby was English horse riding. Her mother Andrea put her in a saddle when Swift was nine months old and she later competed in horse shows.^[13][14] Her family owned several Quarter horses and a Shetland pony.^[14] At the age of nine, Swift turned her attention to musical theatre and performed in Berks Youth Theatre Academy productions of Grease, Annie, Bye Bye Birdie and The Sound of Music.^[15][8] She travelled regularly to Broadway, New York for vocal and acting lessons. However, "after a few years of auditioning in New York and not getting anything”, Swift became interested in country music.^[16] She was inspired by LeAnn Rimes's Blue^[17] and her maternal grandmother, Marjorie Finlay, an opera singer.^[8] When she was in fourth grade, Swift won a national poetry contest with a three-page poem, "Monster in My Closet".^[18] At the age of eleven, after many attempts,^[19] Swift won a local talent competition by singing a rendition of LeAnn Rimes’s “Big Deal”, and was given the opportunity to appear as the opening act for Charlie Daniels at a Strausstown amphitheater.^[15] She spent her weekends performing at local festivals, fairs, coffeehouses, karaoke contests, garden clubs and Boy Scout meetings.^[20][4] This interest in country music isolated Swift from her middle school peers.^[21][22]

After watching a Behind the Music episode about Faith Hill,^[23] Swift recorded a demo of karaoke covers, and travelled with her mother to Nashville, Tennessee for spring break to leave a copy of the demo with record labels along Music Row. She received label rejections and realized that "everyone in that town wanted to do what I wanted to do. So, I kept thinking to myself, I need to figure out a way to be different".^[23] Swift began performing the "The Star Spangled Banner" at many sporting events, hoping she would be offered a recording contract. On one occasion, an eleven-year-old Swift high-fived Jay-Z after singing the national anthem at a 76ers game in Philadelphia.^[24] At the age of twelve, Swift was shown by a computer repairman how to play three chords on a guitar, inspiring her to write her first song, "Lucky You".^[25][26] She then recorded a second demo of original songs.^[15] In 2003, Swift and her parents began working with music manager Dan Dymtrow, after he spotted her singing at the US Open.^[27] Swift's second demo then caught the attention of RCA Records, who offered the eight-grader an artist development deal. In 2004, Swift modelled for Abercrombie and Fitch as part of their "Rising Stars" campaign and had an original song included in a Maybelline Cosmetics compilation CD.^[28]

When Swift was fourteen, her father transferred to the Nashville office of Merrill Lynch and the family relocated to a lake-shore house in Hendersonville, Tennessee.^[1] Swift has said her parents "presented it as a move to a nice community" rather than as her chance to become a star.^[29] Her mother has said, "We've always told her that this is not about putting food on our table or making our dreams come true."^[30] In Tennessee, Swift attended Hendersonville High School for her freshman and sophomore years.^[31] Later, to accommodate her touring schedule, Swift transferred to the Aaron Academy, a private Christian school which offered homeschooling services, and earned her high school diploma in 2008.^[32]

Music career[link]

[edit] 2004–08: Nashville songwriting sessions and Taylor Swift release

Swift moved to Nashville at the age of fourteen, having secured an artist development deal with RCA Records. Her musical influences included the Dixie Chicks, Shania Twain, LeAnn Rimes, Faith Hill,^[29] Tom Petty,^[33] Sheryl Crow,^[34] Melissa Etheridge, Pat Benatar,^[35] Reba McEntire, Garth Brooks,^[36] Patsy Cline, Loretta Lynn and Tammy Wynette.^[37] Swift had writing sessions with experienced songwriters such as Troy Verges, Brett Beavers, Brett James, Mac McAnally and the Warren Brothers,^[38][39] but eventually formed a lasting working relationship with Liz Rose.^[40] Swift saw Rose performing at an RCA songwriter event and suggested that they write together.^[41] They began meeting for two-hour writing sessions every Tuesday afternoon after school.^[42] Rose has said that the sessions were "some of the easiest I've ever done. Basically, I was just her editor. She'd write about what happened in school that day. She had such a clear vision of what she was trying to say. And she'd come in with the most incredible hooks".^[43][42] Swift also began recording demos with producer Nathan Chapman.^[41] After performing at a BMI Songwriter's Circle showcase at The Bitter End, New York,^[39] Swift became the youngest songwriter ever hired by the Sony/ATV Tree publishing house.^[44]

Swift performing at the Maverick Saloon & Grill in Santa Maria, California in 2006

Swift left RCA Records when she was fifteeen; the company wanted her to record the work of other songwriters and wait until she was eighteen to release an album, but she felt ready to launch her career with her own material.^[45][19] She also parted ways with manager Dan Dymtrow, who later took legal action against Swift and her parents.^[28] At an industry showcase at Nashville's The Bluebird Café in 2005, Swift caught the attention of Scott Borchetta, a Dreamworks Records executive who was preparing to form his own independent record label, Big Machine Records.^[40] Swift was one of the new label's first signings.^[40] As an introduction to the country music business, Borchetta arranged for Swift to intern at the CMA Music Festival; she acted as the artist escort for Darryl Worley.^[46][47]

Swift began working on her self-titled debut album shortly after signing her record deal. After experimenting with veteran Nashville producers, Swift persuaded Big Machine to hire her demo producer Nathan Chapman.^[19] It was his first time to record a studio album but Swift felt they had the right "chemistry".^[19] In the end, Chapman produced all but one of the tracks on Taylor Swift.^[19] She has described the album as the "diary" of her early teens^[48] and most songs were written during her freshman year of high school.^[19] As a result, the songs describe coming of age experiences such as insecurity, young love, and teenage angst.^[19] She has said that, although "it sounds like I've had 500 boyfriends", a lot of the songs are observational.^[19] Swift wrote three of the album's songs alone, including two singles, and co-wrote the remaining eight with writers such as Liz Rose, Robert Ellis Orrall and Angelo Petraglia.^[49] Musically, the album has been described as "a mix of trad-country instruments and spry rock guitars".^[50]

Taylor Swift was released in October 2006 and received generally positive reviews from music critics.^[51] PopMatters hoped Swift would be "able to find an accomodation between the country tradition and her very obvious pop sensibilities, because Taylor Swift suggests she has much to offer".^[52] The New Yorker's Sasha Frere-Jones described Swift as a "prodigy". She noted that "Our Song" "stop[ed] me in my tracks" and praised the lyrics: “He’s got a one-hand feel on the steering wheel, the other on my heart".^[53] Country Weekly felt Swift demonstrated "an honesty, intelligence and idealism with which listeners of any age will be able to connect" and noted that "the more thoughtful material suggests a talent poised to last well past high school".^[54] AllMusic stated that, while listening to the album, it was "never in doubt" that "Swift is a talent to be reckoned with".^[55] Rolling Stone described Swift as "bright-eyed but remarkably seasoned", and admired "Our Song"'s "insanely hooky sing-song melody that's as Britney as it is Patsy".^[50]

Swift performing at Yahoo HQ in Sunnyvale, California in 2007

Big Machine Records was still in its infancy upon the release of the lead single "Tim McGraw" in June 2006, and Swift and her mother helped "stuff the CD singles into envelopes to send to radio".^[30] She spent much of 2006 promoting Taylor Swift in a radio tour and later commented, "Radio tours for most artists last six weeks. Mine lasted six months."^[19] Swift painted canvases (inspired by Jackson Pollock) to gift to radio station managers who played her music.^[56] She took part in "GAC Short Cuts", a part-documentary, part-music-video series designed to introduce her to country music fans.^[57][58] She also made television appearances on Good Morning America,^[59] The Megan Mullally Show,^[59] America's Got Talent,^[60] and TRL.^[61] Swift, a self-described "kid of the internet",^[62] used MySpace to build a fanbase. She wrote her own blog posts, left comments on her fans' accounts and personally respond to the messages that were sent to her.^[63] This was, at the time, "revolutionary in country music".^[64][57] Borchetta has said that his decision to sign a sixteen year old singer-songwriter initially raised eyebrows among his record industry peers^[57] but Swift tapped into a previously unknown market: teenage girls who listen to country music.^[1][57]

Following "Tim McGraw", four further singles were released throughout 2007 and 2008: "Teardrops on My Guitar", "Our Song", "Picture to Burn" and "Should've Said No". All were highly successful on the Billboard Hot Country Songs chart, with "Our Song" and "Should've Said No" both reaching number one. "Our Song" made Swift the youngest sole writer and singer of a number one country song.^[65] "Teardrops on My Guitar" became a minor pop hit; it reached number thirteen on the Billboard Hot 100.^[66][67] The album sold 39,000 copies during its first week^[68] and, as of March 2011, has sold over 5.5 million copies worldwide.^[69] Swift also released a holiday album, Sounds of the Season: The Taylor Swift Holiday Collection, in October 2007 and an EP, Beautiful Eyes, in July 2008.^[70][71]

Swift toured extensively in support of Taylor Swift. In addition to festival and theater dates, Swift performed as an opening act for several country artists' concert tours. She opened for Rascal Flatts on several dates in late 2006.^[59][72] In 2007, she served as the opening act on twenty dates for George Strait's 2007 United States tour,^[73] several dates on Kenny Chesney's tour^[74] and selected dates on Brad Paisley's Bonfires & Amplifiers Tour.^[75][76] Also in 2007, Swift appeared as the opening act on several dates for Tim McGraw and Faith Hill's joint Soul2Soul II Tour.^[77] Swift again opened for Flatts for their Still Feels Good Tour in 2008.^[78] In addition to performing her own material, Swift played covers of songs by Beyoncé and Rihanna.^[79] She conducted meet-and-greet sessions with fans before and after her concerts; these lasted for up to four hours.^[80][81]

In 2007, Swift and Alan Jackson were jointly named the Nashville Songwriters Association's "Songwriter/Artists of the Year". Swift was the youngest person ever to be honored with the title.^[82] She also won the Country Music Association’s Horizon Award for Best New Artist.^[83] In 2008, she won Top New Female Vocalist at the Academy of Country Music Awards^[84] and Favorite Country Female Artist at the American Music Awards.^[85] She won seven BMI Awards for songs featured on Taylor Swift.^[86] Swift was also nominated for a 2008 Grammy Award in the category of Best New Artist, but lost to Amy Winehouse.^[87]

[edit] 2008–10: Fearless release, MTV VMA incident and media backlash

Swift's second studio album, Fearless, was released in November 2008.^[88] She wrote seven of the album's songs alone, including two singles, and co-wrote the remaining six with songwriters Liz Rose, John Rich, Colbie Caillat and Hillary Lindsey.^[49] While Swift wrote many of the songs on tour, she made a conscious effort not to include "road songs": "As a 16-year-old girl, I could never relate to those songs about playing a different city every night. I want to write about feelings and love and the lack of love".^[89] She has said that the album title "means you’re afraid of a lot of things, but you jump anyway".^[90] She co-produced the album with Nathan Chapman.^[49] Musically, the record is characterized by "loud, lean guitars and rousing choruses", with the occasional "bit of fiddle and banjo tucked into the mix".^[91]

Swift performing at the 2010 Cavendish Beach Music Festival in Canada

Fearless received generally positive reviews from music critics.^[92] The New York Times described Swift as "one of pop's finest songwriters, country’s foremost pragmatist and more in touch with her inner life than most adults".^[93] The Village Voice felt she displayed "preternatural wisdom and inclusiveness", "masterfully avoiding the typical diarist's pitfalls of trite banality and pseudo-profound bullshit".^[94] Rolling Stone described her as "a songwriting savant with an intuitive gift for verse-chorus-bridge architecture" whose "squirmingly intimate and true" songs seemed to be "literally ripped from a suburban girl's diary".^[95] USA Today found it "a pleasure to hear a gifted teenager who sounds like a gifted teenager, rather than a mouthpiece for a bunch of older pros' collective notion of adolescent yearning."^[96] The New Yorker described it as an album "without a bad track", adding that "the album’s finest effort, "Fifteen", will feature in yearbook quotes for years".^[97] Entertainment Weekly noted that the album would appeal mainly to young girls – "she sounds like a real teen, not some manufactured vixen-Lolita" – but predicted it would be "exciting to watch her precocious talent grow".^[98] Music critic Robert Christgau described Swift as "an uncommonly-to-impossibly strong and gifted teenage girl".^[99]

Swift promoted Fearless heavily upon its release. An episode of The Ellen DeGeneres Show was dedicated to the album launch^[57] and Swift appeared on many other chat shows.^[100] She communicated with fans using social media platforms such as twitter and personal video blogs.^[57] In October 2008, she appeared at the CMT Giants: Alan Jackson event, performing a cover of Jackson's "Drive (For Daddy Gene)".^[101] In November 2008, Swift took part in a joint, televised concert with rock band Def Leppard in Nashville.^[102][103] She performed her song "Fifteen" with Miley Cyrus at the 51st Grammy Awards in February 2009.^[104] In April 2009, she performed a cover of George Strait's "Run" at a televised ACM event honoring Strait as Artist of the Decade.^[105] Swift hosted and appeared as the musical guest on Saturday Night Live in November 2009.^[106] The lead single from the album, "Love Story", was released in September 2008 and became the second best-selling country single of all time, peaking at number four on the Billboard Hot 100 chart.^[67][107] Four more singles were released throughout 2008 and 2009: "White Horse", "You Belong with Me", "Fifteen" and "Fearless". "You Belong with Me" was the album's highest-charting single, peaking at number two on the Billboard Hot 100.^[108] The album debuted at number one on the Billboard 200 Album Chart with sales of 592,304 and has since sold over 8.6 million copies worldwide.^[109][110] It was the top-selling album of 2009 and brought Swift much crossover success.^[111]

Swift carried out her first headlining tour from April 2009 to June 2010. As part of the 105-date Fearless Tour, Swift played 90 dates in North America, six dates in Europe, eight dates in Australia and one date in Asia.^[112] The stage show included multiple costume changes, dancers and a fairy-tale castle set.^[113] She sang a cover of Justin Timberlake's "What Goes Around... Comes Around" nightly, intertwined with her own "You're Not Sorry".^[114] Swift invited John Mayer, Faith Hill and Katy Perry to perform one-off duets with her at various dates during the North American tour.^{[115][116][117]} Justin Bieber, Kelly Pickler and Gloriana were the support acts.^[118] The tour was attended by more than 1.1 million fans and has grossed over $63 million.^[119] Taylor Swift: Journey to Fearless, a concert film, was aired on television and later released on DVD and Blu-ray.^[120] Also in 2009, Swift performed as a supporting act for Keith Urban.^[121]

In September 2009, Swift became the first country music artist to win an MTV Video Music Award when "You Belong with Me" was named Best Female Video.^[122] Her acceptance speech was interrupted by rapper Kanye West, who had been involved in a number of other award show incidents.^[123] West declared Beyoncé's video for "Single Ladies (Put a Ring on It)", nominated in the same category, to be "one of the best videos of all time." Many audience members booed West,^[124][125] prompting him to flip off the crowd.^[126][124] He then handed the microphone back to a speechless Swift.^[124][127] Backstage, Swift was seen "hysterically crying".^[124] According to Rolling Stone, when Swift's mother confronted West, he gave "a half-hearted apology in which he added he still thought Beyoncé's video was superior".^[128] West was removed from the event.^{[124][126][129]} When Beyoncé later won the award for Video of the Year, she invited Swift onstage to finish her speech.^[124][130] In the event's press room, Swift, who in 2008 had expressed a desire to sing a hook on a Kanye West rap song,^[131] was asked if she had "any hard feelings" towards West: "I don’t know him, and I’ve never met him, so... I don’t want to start anything because I had a great night tonight."^[128][132]

The incident received much media attention and inspired many Internet memes.^[133] President Barack Obama, in an "off the record" comment, called West a "jackass".^[134] Former US President Jimmy Carter said West's interruption was "completely uncalled for".^[135] West's behavior was criticized by celebrities, including Eminem, Snoop Dogg and Lady Gaga.^{[126][136][137][128]} The following day, West apologized for his verbal outburst both in a blog entry and during an appearance on The Jay Leno Show.^[124][125] He maintained that, while Swift was "very talented", "Beyoncé's video was the best of this decade!!!! ... I gave my awards to Outkast when they deserved it over me ... I'm not crazy y'all, I'm just real."^[138] Two days after the VMAs, Swift told an interviewer that West had not spoken to her since the ceremony.^[139] West then contacted her to offer a personal apology, which Swift accepted: "Kanye did call me, and he was very sincere in his apology."^[122][139] It has been said that the incident and subsequent media attention turned Swift into "a bona-fide mainstream celebrity".^[140]

Swift released a cover of Tom Petty's "American Girl", exclusively through Rhapsody in June 2009^[141] and continues to make her stage entrance to Petty's recording of the song.^[142] Swift contributed backing vocals to John Mayer's "Half of My Heart", featured on his fourth album, in November 2009.^[143] Mayer wrote the song as a tribute to Tom Petty and Fleetwood Mac: "I thought, 'Well, if this is going to be my love letter to that style of music, who's going to be the Stevie Nicks in this equation?' And I thought, 'This Taylor Swift girl is going to be around for a long time."^[144] The song received positive reviews from music critics.^{[145][146][147]} Swift and Mayer performed the song live at Madison Square Garden, New York in December 2009.^[148] It was released as the album's third single in June 2010 and peaked at number 25 on the Billboard Hot 100.^[149] Swift collaborated with a number of other artists in 2008 and 2009. She co-wrote and recorded "Best Days of Your Life" with Kelly Pickler.^[150] She co-wrote two songs for the Hannah Montana: The Movie soundtrack – "You'll Always Find Your Way Back Home" and "Crazier" – with Martin Johnson and Robert Ellis Orrall, respectively.^[151] Swift also provided vocals for Boys Like Girls's "Two Is Better Than One", written by Martin Johnson.^{[152][153][154][155]} In January 2010, Swift contributed two songs – including "Today Was a Fairytale" – to the Valentine's Day soundtrack^[156] and recorded a cover of Better Than Ezra's "Breathless" for the Hope for Haiti Now album.^[157]