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With a Little Help from My Friends
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This article is about the song. For the albums, see With a Little Help from My Friends (Joe Cocker album) and With a Little Help from My Friends (Neal Morse album). For the album by The Flaming Lips, see With a Little Help from My Fwends.
"With a Little Help from My Friends"
Song by the Beatles
from the album Sgt. Pepper's Lonely Hearts Club Band
Released
26 May 1967[1]
Recorded
EMI Studios
29–30 March 1967
Genre
Pop rock, psychedelic pop
Length
2:44
Label
Parlophone PMC 7027 (mono), PCS 7027 (stereo)
Songwriter(s)
Lennon–McCartney
Producer(s)
George Martin
"Sgt. Pepper's Lonely Hearts Club Band"/"With a Little Help from My Friends"
Single by the Beatles
B-side
"A Day in the Life"
Released
14 August 1978 (US)
30 September 1978 (UK)
Format
7"
Recorded
February–March 1967
Length
4:46
Label
Capitol 4612 (US)
Parlophone R6022 (UK)
Songwriter(s)
Lennon–McCartney
The Beatles UK singles chronology
"Back in the U.S.S.R."
(1976) "Sgt. Pepper's Lonely Hearts Club Band" / "With a Little Help from My Friends"
(1978) "The Beatles Movie Medley"
(1982)
"Back in the U.S.S.R."
(1976) "Sgt. Pepper's Lonely Hearts Club Band" / "With a Little Help from My Friends"
(1978) "The Beatles Movie Medley"
(1982)
The Beatles US singles chronology
"Ob-La-Di, Ob-La-Da"
(1976) Ob-La-Di, Ob-La-Da1976 "Sgt. Pepper's Lonely Hearts Club Band" / "With a Little Help from My Friends"
(1978) Sgt. Pepper's Lonely Hearts Club BandWith a Little Help from My Friends1978 "The Beatles Movie Medley"
(1982) The Beatles Movie Medley1982
"With a Little Help from My Friends" is a song by the English rock band the Beatles from their 1967 album Sgt. Pepper's Lonely Hearts Club Band. It was written by John Lennon and Paul McCartney, and intended as the album's featured vocal for drummer Ringo Starr. The group recorded the song towards the end of the sessions for Sgt. Pepper, with Starr singing as the character "Billy Shears".
A subsequent recording of the track by Joe Cocker became a hit single in 1968 and an anthem for the Woodstock era.[2] In 1978, the Beatles' recording, paired with "Sgt. Pepper's Lonely Hearts Club Band", was reissued as a single, and peaked at number 63 in Britain and number 71 in the United States. Starr has regularly performed the song in concert as a solo artist. "With a Little Help from My Friends" was ranked number 311 on Rolling Stone's list of the 500 Greatest Songs of All Time.
Contents
[hide]
1 Origins
2 Recording
3 Personnel
4 Live performances
5 Cover versions
6 Joe Cocker version
7 In popular culture
8 Notes
9 References
10 External links
Origins[edit]
Lennon and McCartney finished writing this song in mid-March 1967,[3] written specifically as Starr's track for the album. McCartney said: "It was pretty much co-written, John and I doing a work song for Ringo, a little craft job." In 1970 Lennon stated: "Paul had the line about 'a little help from my friends.' He had some kind of structure for it, and we wrote it pretty well fifty-fifty from his original idea.", but in 1980 Lennon said: "This is Paul, with a little help from me. 'What do you see when you turn out the light/ I can't tell you, but I know it's mine...' is mine."[4] It was briefly called "Bad Finger Boogie" (later the inspiration for the band name Badfinger),[5] supposedly because Lennon composed the melody on a piano using his middle finger after having hurt his forefinger.
The song begins with the applause from the end of Sgt. Pepper's Lonely Hearts Club Band.
Lennon and McCartney deliberately wrote a tune with a limited range – except for the last note, which McCartney worked closely with Starr to achieve. Speaking in the Anthology, Starr explained that he insisted on changing the first line — which originally was "What would you think if I sang out of tune? Would you throw ripe tomatoes at me?" — so that fans would not throw tomatoes at him should he perform it live. (In the early days, after George Harrison made a passing comment that he liked jelly babies, the group was showered with them at all of their live performances.)[6]
The song's composition is unusually well documented, as Hunter Davies was present and described the writing process in the Beatles' official biography.
The song is partly in the form of a conversation in which the other three Beatles sing a question — e.g. "Would you believe in a love at first sight?" where Starr answers, "Yes, I'm certain that it happens all the time."
Recording[edit]
The Beatles began recording the song on 29 March 1967, the day before they posed for the Sgt. Pepper album cover. They recorded 10 takes of the song, wrapping up sessions at 5:45 in the morning.[7] The backing track consisted of Starr on drums, McCartney playing piano, Harrison playing lead guitar and Lennon beating a cowbell. At dawn, Starr trudged up the stairs to head home – but the other Beatles cajoled him into doing his lead vocal then and there, standing around the microphone for moral support.[4] The following day they added tambourine, backing vocals, bass and more electric guitar.[7]
Personnel[edit]
According to Ian MacDonald, except where noted:[8]
Ringo Starr – lead vocal, drums
John Lennon – backing vocal, cowbell
Paul McCartney – backing vocal, piano, bass guitar
George Harrison – backing vocal, rhythm[9] and lead guitars
George Martin – Hammond organ
Live performances[edit]
To date, Starr has closed every concert performed by each version of his All-Starr Band with this song. After he is done singing, Starr tells the audience "Peace and love...peace and love is the only way...and good night", then walks off the stage. Since 2008, the band segued right into "Give Peace a Chance", during which Starr comes back onstage, then walks off again.
Starr performed the song with George Harrison, Eric Clapton, Bruce Springsteen, George Michael, Phil Collins, Elton John, and many others at the 1987 Prince's Trust Concert at Wembley Arena, London.[10]
McCartney and Starr performed this song for the first time together at the David Lynch Foundation Benefit Concert in the Radio City Music Hall, New York on 4 April 2009. McCartney and Starr also performed the song together on "The Night That Changed America: A Grammy Salute to The Beatles", a commemorative show on 27 January 2014, that marked 50 years since the band's first appearance on The Ed Sullivan Show.[11], then again in 2015 at Ringo Starr's induction in the Rock'n Roll Hall of Fame.[12]
Cover versions[edit]
There have been at least 50 cover versions of the song[13] and it has achieved the number one position on the British singles charts three times: by Joe Cocker in 1968,[14] Wet Wet Wet in 1988,[15] and by Sam & Mark in 2004. Chris Clark, the first white female recording artist to have an album released on Motown, recorded the song for her second album, 1969's CC Rides Again.
The song was also covered by Mumford & Sons and Dawes on the Tour of Two Halves in 2012.[16] Mumford & Sons also covered the song to close out their headline set at the 2013 Glastonbury Festival with Vampire Weekend, The Vaccines, First Aid Kit, and The Staves on 30 June 2013.[17] For the 1981 compilation Rarities, the Beach Boys released their version recorded in 1967,[18] while in 2015, Brian Wilson named it one of his favorite songs,[19] and in that year German rock band Bonfire, with new frontman David Reece, recorded their version for the album Glörious.
Widespread Panic has performed the song twice, both times during their special 3-Set New Year's Eve shows.[20]
In March 2017, Broadway and Hollywood artists including Liz Callaway, FORTE, Annie Golden, Telly Leung and Chris Mann performed a gospel-flavored version of the song for a single and video to benefit Americans for the Arts.[21]
"With a Little Help from My Friends"
Single by Joe Cocker
from the album With a Little Help from My Friends
B-side
"Something's Coming On"
Released
October 1968 (UK)
Format
7"
Recorded
1968
Genre
Blues rock, hard rock, soul
Length
5:11
Label
Regal Zonophone
Songwriter(s)
Lennon–McCartney
Joe Cocker singles chronology
"Marjorine"
(1968) Marjorine1968 "With a Little Help from My Friends"
(1968) With a Little Help from My Friends1968 "Delta Lady"
(1969) Delta Lady1969
Audio sample
file
help
Joe Cocker version[edit]
English singer Joe Cocker's version of "With a Little Help from My Friends" was a radical re-arrangement of the original, in a slower, 6
8 meter, using different chords in the middle eight, and a lengthy instrumental introduction (featuring drums by Procol Harum's B.J. Wilson, guitar lines from Jimmy Page, and organ by Tommy Eyre). Cocker performed the song at Woodstock in 1969 and that performance was included in the documentary film, Woodstock. This version gained even more fame when it was used as the opening theme song for the television series The Wonder Years.[22] Cocker's cover was ranked number two in UpVenue's top 10 best music covers of all time in 2009.[23] In 2014, a BBC poll saw it voted the seventh best cover version ever.[24] The version heard in the film Across the Universe segues from the original to Cocker's arrangement at the end of the song. In 2001, Cocker's version of the song was inducted into the Grammy Hall of Fame.[25]
Personnel
Joe Cocker – lead vocals
Jimmy Page – guitar
Chris Stainton – bass
Tommy Eyre – organ
B.J. Wilson – drums
Madeline Bell – backing vocals
Rosetta Hightower – backing vocals
Patrice Holloway – backing vocals
Sunny Wheetman – backing vocals
"With a Little Help from My Friends"
Single by Wet Wet Wet
from the album Sgt. Pepper Knew My Father
A-side
"She's Leaving Home" (performed by Billy Bragg)
Released
9 May 1988 (UK)
Format
7"
Recorded
1988
Genre
Pop
Label
PolyGram
Songwriter(s)
Lennon–McCartney
Wet Wet Wet singles chronology
"Temptation"
(1988) "With a Little Help from My Friends" / "String Module Error: Match not found"
(1988) "Sweet Surrender"
(1989)
"Temptation"
(1988) "With a Little Help from My Friends"/"She's Leaving Home" (Billy Bragg and Cara Tivey)
(1988) "Sweet Surrender"
(1989)
In popular culture[edit]
Janice sang this song as an opening number to an episode of The Muppet Show where other members of Dr. Teeth and The Electric Mayhem were pretending to rescue her from a Hindu human sacrifice.
Sesame Street spoofed this song as "With a Little Yelp from My Friends", sung by a dog named Moe Cocker, who spoofs Joe Cocker, and the arrangement is clearly based on Cocker's version.[26]
The Animaniacs sketch Woodstock Slappy features Joe Cocker singing the song.
The song was featured in the EastEnders 2007 Children in Need sketch where the cast sang to famous Beatles songs, the song was the last to be sung by the cast with the characters of Ronnie Mitchell, Roxy Mitchell, Chelsea Fox, Carly Wicks, Jane Beale, Lucy Beale, Peter Beale, Ben Mitchell, Zainab Masood, Masood Ahmed, Honey Mitchell, Dawn Swann and Mickey Miller singing the majority of the song.
The Joe Cocker version was used as the title music for the 1988–1993 television series The Wonder Years.
The Cocker version appears in a 2017 TV commercial for Volkswagen.
"Lend me your ears" is a Shakespearean reference, taken from the "Friends, Romans, countrymen" speech in Julius Caesar and used at the time as a popular way of asking people to listen.
"With a Little Help from My Friends" was played as wake-up music on Space Shuttle Mission STS-61.[27]
On Saturday Night Live, Season-1 Ep.3: John Belushi performed a full-length impression of Cocker singing this song.[28]
The cult PBS film The Lathe of Heaven (from 1980) uses the original recording of the song. The main character (George Orr), who can manipulate reality with his dreams, comes upon a 45 of the song at a novelty shop run by an alien. The alien hands George the 45 saying "Help is available." The song plays in the soundtrack and morphs into a synthesizer version. The film was out of circulation for over 20 years. When it was finally re-aired on PBS and released on DVD in 2001, many fans were upset that the original Beatles recording was replaced by a singer with an acoustic guitar. This was due to changes in publishing rights that have occurred since 1980 involving the dissolution of The Beatles' original Northern Songs and the acquisition by Sony/ATV (partly owned by the Jackson family).[citation needed]
Notes[edit]
Jump up ^ Everett 1999, p. 123. "In the United Kingdom Sgt. Pepper's Lonely Hearts Club Band...was rush-released six days ahead of its official date, June 1."
Jump up ^ MacDonald 2005, p. 247.
Jump up ^ Dowlding 1989, p. 165.
^ Jump up to: a b "100 Greatest Beatles Songs. No. 61 – 'With a Little Help From My Friends'". Rolling Stone. Retrieved 18 June 2012.
Jump up ^ Matovina 2000.
Jump up ^ The Beatles 2000, p. 242.
^ Jump up to: a b Lewisohn 1988, p. 106.
Jump up ^ MacDonald 2005, p. 246.
Jump up ^ Everett 1999, p. 102.
Jump up ^ "With a Little Help From My Friends". Retrieved 11 November 2017
Jump up ^ "Paul McCartney and Friends: Change Begins Within". Radio City Music Hall. New York, NY: Madison Square Garden. Archived from the original on 21 January 2010. Retrieved 27 March 2010.
Jump up ^ https://www.youtube.com/watch?v=TEGwjAKK5z4
Jump up ^ "Song: With a Little Help From My Friends – John Lennon, Paul McCartney". Second Hand Songs. Retrieved 21 August 2011.
Jump up ^ "Number 1 Singles of the 1960s". everyHit.com. Retrieved 21 August 2011.
Jump up ^ "Number 1 Singles of the 1980s". everyHit.com. Retrieved 21 August 2011.
Jump up ^ "Number 1 Singles of the 2000s". everyHit.com. 16 March 2000. Retrieved 21 August 2011.
Jump up ^ "Mumford & Sons Get a Little Help From Their Friends for Glastonbury Finale". 30 June 2013. Retrieved 1 July 2013.
Jump up ^ bradelliott.com 1983.
Jump up ^ Wilson, Brian (May 8, 2015). "Brian Wilson: The Music That Made Me". Rolling Stone.
Jump up ^ "Everyday Companion".
Jump up ^ Kaufman, Gil (March 23, 2017). "Broadway Stars Come Together For All-Star Single 'With a Little Help from My Friends' to Support the Arts". Billboard. Retrieved March 24, 2017.
Jump up ^ Parrott, Billy (9 August 2013). "The Wonder Years: Music and References from Season One". The New York Public Library. Archived from the original on 2 September 2013.
Jump up ^ UpVenue.com 2010.
Jump up ^ "Pet Shop Boys' Always On My Mind tops cover version vote". BBC News. 27 October 2014. Retrieved 28 October 2014.
Jump up ^ "Grammy Hall of Fame Award. Grammy.org Retrieved 21 December 2012
Jump up ^ Sesame Street – With a Little YELP from my Friends on YouTube
Jump up ^ Fries 2009.
Jump up ^ "SNL Season 01 Episode 03 - Rob Reiner, John Belushi as Joe Cocker - NBC.com". NBC. Retrieved 2017-03-05.
References[edit]
The Beatles (2000). The Beatles Anthology. San Francisco: Chronicle Books. ISBN 0-8118-2684-8.
Chianello, Joanne (2 October 2009). "Harper gets on stage with a little help from his wife". Ottawa Citizen. Archived from the original on 6 October 2009. Retrieved 2 October 2009.
Dowlding, William J. (1989). Beatlesongs. New York: Simon & Schuster. ISBN 0-671-68229-6.
Everett, Walter (1999). The Beatles as Musicians: Revolver Through the Anthology. Oxford University Press. ISBN 978-0-19-512941-0.
Fries, Colin, ed. (30 November 2009). "Chronology of Wakeup Calls" (PDF). National Aeronautics and Space Administration.
Lewisohn, Mark (1988). The Beatles Recording Sessions. New York: Harmony Books. ISBN 0-517-57066-1.
MacDonald, Ian (2005). Revolution in the Head: The Beatles' Records and the Sixties (Second Revised ed.). London: Pimlico (Rand). ISBN 1-84413-828-3.
Matovina, Dan (2000). Without You: The Tragic Story of Badfinger. Frances Glover Books. ISBN 0-9657122-2-2. Apple's Neil Aspinall remembers, "(...) Badfinger just popped in my head. It was from an old Lennon thing. He was playing the piano and he had a bad finger so he called the piece he was playing 'Bad Finger Boogie' (which evolved into 'With A Little Help From My Friends')
"Original liner notes for Capitol's Beach Boys Rarities album". bradelliott.com. 1983.
"Ringo Starr – With a Little Help from My Friends". The Daily Show with Jon Stewart. 13 January 2010.
Kilpatrick, Sean (4 October 2009). "Stephen Harper rocks out". thestar.com. Toronto. Retrieved 22 October 2009.
"UpVenue Top 10 Best Music Covers". UpVenue.com. 2010.
External links[edit]
Quotations related to Sgt. Pepper's Lonely Hearts Club Band at Wikiquote
How B.J. Wilson Rescued a Classic Joe Cocker Track (page about B.J. Wilson and Joe Cocker's recording of the song)
Lyrics of this song at MetroLyrics
show
v
t
e
Sgt. Pepper's Lonely Hearts Club Band
Songs
Side one
"Sgt. Pepper's Lonely Hearts Club Band"
"With a Little Help from My Friends"
"Lucy in the Sky with Diamonds"
"Getting Better"
"Fixing a Hole"
"She's Leaving Home"
"Being for the Benefit of Mr. Kite!"
Side two
"Within You Without You"
"When I'm Sixty-Four"
"Lovely Rita"
"Good Morning Good Morning"
"Sgt. Pepper's Lonely Hearts Club Band (Reprise)"
"A Day in the Life"
Non-album single
"Strawberry Fields Forever"
"Penny Lane"
Outtakes
"Only a Northern Song"
"Carnival of Light"
Tribute albums
Easy Star's Lonely Hearts Dub Band
Sgt. Pepper Knew My Father
Sgt. Pepper Live
Sgt. Petsound's Lonely Hearts Club Band
With a Little Help from My Fwends
Related articles
The Beatles discography
Sgt. Pepper's cover
Sgt. Pepper's Lonely Hearts Club Band (film)
Sgt. Pepper's Lonely Hearts Club Band (soundtrack)
It Was Twenty Years Ago Today
Sgt. Pepper's Lonely Hearts Club Band on the Road
Yellow Submarine (film)
"Tina in the Sky with Diamonds"
Please Please Me
With the Beatles
A Hard Day's Night
Beatles for Sale
Help!
Rubber Soul
Revolver
Sgt. Pepper's Lonely Hearts Club Band
Magical Mystery Tour
The Beatles (White Album)
Yellow Submarine
Abbey Road
Let It Be
Past Masters
show
v
t
e
The Beatles singles discography
UK and US
(all labels)
1963
"Please Please Me" / "Ask Me Why"
"From Me to You" / "Thank You Girl"
"She Loves You" / "I'll Get You"
1964
"Can't Buy Me Love" / "You Can't Do That"
"I Feel Fine" / "She's a Woman"
1965
"Ticket to Ride" / "Yes It Is"
"Help!" / "I'm Down"
"We Can Work It Out" / "Day Tripper"
1966
"Paperback Writer" / "Rain"
"Yellow Submarine" / "Eleanor Rigby"
1967
"Penny Lane" / "Strawberry Fields Forever"
"All You Need Is Love" / "Baby, You're a Rich Man"
"Hello, Goodbye" / "I Am the Walrus"
1968
"Lady Madonna" / "The Inner Light"
"Hey Jude" / "Revolution"
1969
"Get Back" / "Don't Let Me Down"
"The Ballad of John and Yoko" / "Old Brown Shoe"
"Something" / "Come Together"
1970
"Let It Be" / "You Know My Name (Look Up the Number)"
1978
"Sgt. Pepper's Lonely Hearts Club Band/With a Little Help from My Friends" / "A Day in the Life"
1982
"The Beatles Movie Medley" / "I'm Happy Just to Dance with You"
1995
"Free as a Bird" / "Christmas Time (Is Here Again)"
1996
"Real Love" / "Baby's in Black"
UK only
(Parlophone,
Apple)
1962
"My Bonnie" / "The Saints"
"Love Me Do" / "P.S. I Love You"
1963
"I Want to Hold Your Hand" / "This Boy"
1964
"Ain't She Sweet" / "If You Love Me, Baby"
"A Hard Day's Night" / "Things We Said Today"
1976
"Yesterday" / "I Should Have Known Better"
"Back in the U.S.S.R." / "Twist and Shout"
US only
(Vee-Jay,
Swan,
Tollie,
Capitol,
Apple)
1963
"I Want to Hold Your Hand" / "I Saw Her Standing There"
1964
"Twist and Shout" / "There's a Place"
"Do You Want to Know a Secret" / "Thank You Girl"
"Love Me Do" / "P.S. I Love You"
"Sie liebt dich" / "I'll Get You"
"I'll Cry Instead" / "I'm Happy Just to Dance with You"
"And I Love Her" / "If I Fell"
"A Hard Day's Night" / "I Should Have Known Better"
"Matchbox" / "Slow Down"
1965
"Eight Days a Week" / "I Don't Want to Spoil the Party"
"Yesterday" / "Act Naturally"
1966
"Nowhere Man" / "What Goes On"
1970
"The Long and Winding Road" / "For You Blue"
1976
"Got to Get You into My Life" / "Helter Skelter"
"Ob-La-Di, Ob-La-Da" / "Julia"
Other
countries
(Odeon,
Parlophone,
Apple)
1963
"All My Loving" / "This Boy" (Canada)
1964
"Komm, gib mir deine Hand / Sie liebt dich" (Germany)
1966
"Norwegian Wood (This Bird Has Flown)" / "Nowhere Man" (Australia)
1968
"Ob-La-Di, Ob-La-Da" / "While My Guitar Gently Weeps" (Europe, Japan, Australia)
"Ob-La-Di, Ob-La-Da" / "I Will" (Philippines)
"Back in the U.S.S.R." / "Don't Pass Me By" (Sweden)
1969
"You're Going to Lose That Girl" / "Tell Me What You See" (Japan)
1970
"Oh! Darling" / "Here Comes the Sun" (Japan)
1972
"All Together Now" / "Hey Bulldog" (Europe)
1978
"Sgt. Pepper's Lonely Hearts Club Band/With a Little Help from My Friends" / "Within You Without You" (Germany)
show
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e
Wet Wet Wet
Graeme Clark
Tommy Cunningham
Neil Mitchell
Marti Pellow
Studio albums
Popped In Souled Out
The Memphis Sessions
Holding Back the River
High on the Happy Side
Cloak & Dagger
Picture This
10
Timeless
Singles
"Wishing I Was Lucky"
"Sweet Little Mystery"
"Angel Eyes (Home and Away)"
"Temptation"
"With a Little Help from My Friends"
"Sweet Surrender"
"Broke Away"
"Hold Back the River"
"Goodnight Girl"
"More than Love"
"Lip Service"
"Shed a Tear"
"Cold Cold Heart"
"Love Is All Around"
"Julia Says"
"Don't Want to Forgive Me Now"
"Somewhere Somehow"
"If I Never See You Again"
"Strange"
"Yesterday/Maybe I'm in Love"
"Weightless"
Live albums
Wet Wet Wet: Live
Live at the Royal Albert Hall
Compilations
End of Part One: Their Greatest Hits
The Greatest Hits
Step by Step: The Greatest Hits
Videos
Live Glasgow Green Free Concert
Wets at the Castle
End of Part One
Picture This - All Around and in the Crowd
Playing Away at Home
The Greatest Hits and More
Related articles
Discography
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Categories:
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Student's t-test
From Wikipedia, the free encyclopedia
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The t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.
A t-test is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistics (under certain conditions) follow a Student's t distribution. The t-test can be used, for example, to determine if two sets of data are significantly different from each other.
Contents
[hide]
1 History
2 Uses
3 Assumptions
4 Unpaired and paired two-sample t-tests
4.1 Independent (unpaired) samples
4.2 Paired samples
5 Calculations
5.1 One-sample t-test
5.2 Slope of a regression line
5.3 Independent two-sample t-test
5.3.1 Equal sample sizes, equal variance
5.3.2 Equal or unequal sample sizes, equal variance
5.3.3 Equal or unequal sample sizes, unequal variances
5.4 Dependent t-test for paired samples
6 Worked examples
6.1 Unequal variances
6.2 Equal variances
7 Alternatives to the t-test for location problems
8 Multivariate testing
9 Software implementations
10 See also
11 Notes
12 References
13 Further reading
14 External links
History[edit]
William Sealy Gosset, who developed the "t-statistic" and published it under the pseudonym of "Student".
The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland. "Student" was his pen name.[1][2][3][4]
Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes.[2] Gosset devised the t-test as an economical way to monitor the quality of stout. The t-test work was submitted to and accepted in the journal Biometrika and published in 1908.[5] Company policy at Guinness forbade its chemists from publishing their findings, so Gosset published his statistical work under the pseudonym "Student" (see Student's t-distribution for a detailed history of this pseudonym, which is not to be confused with the literal term student).
Guinness had a policy of allowing technical staff leave for study (so-called "study leave"), which Gosset used during the first two terms of the 1906–1907 academic year in Professor Karl Pearson's Biometric Laboratory at University College London.[6] Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson.[7]
Uses[edit]
Among the most frequently used t-tests are:
A one-sample location test of whether the mean of a population has a value specified in a null hypothesis.
A two-sample location test of the null hypothesis such that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping.[8]
A test of the null hypothesis that the difference between two responses measured on the same statistical unit has a mean value of zero. For example, suppose we measure the size of a cancer patient's tumor before and after a treatment. If the treatment is effective, we expect the tumor size for many of the patients to be smaller following the treatment. This is often referred to as the "paired" or "repeated measures" t-test:[8][9] see paired difference test.
A test of whether the slope of a regression line differs significantly from 0.
Assumptions[edit]
Most t-test statistics have the form t = Z/s, where Z and s are functions of the data. Typically, Z is designed to be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined.
As an example, in the one-sample t-test
t = Z s = ( X ¯ − μ ) / ( σ n ) {\displaystyle t={\frac {Z}{s}}={\left({\bar {X}}-\mu \right)\left/\left({\frac {\sigma }{\sqrt {n}}}\right)\right.}}
where X is the sample mean from a sample X1, X2, …, Xn, of size n, s is the ratio of sample standard deviation over population standard deviation, σ is the population standard deviation of the data, and μ is the population mean.
The assumptions underlying a t-test are that
X follows a normal distribution with mean μ and variance σ2
ps2 follows a χ2 distribution with p degrees of freedom under the null hypothesis, where p is a positive constant
Z and s are independent.
In a specific type of t-test, these conditions are consequences of the population being studied, and of the way in which the data are sampled. For example, in the t-test comparing the means of two independent samples, the following assumptions should be met:
Each of the two populations being compared should follow a normal distribution. This can be tested using a normality test, such as the Shapiro–Wilk or Kolmogorov–Smirnov test, or it can be assessed graphically using a normal quantile plot.
If using Student's original definition of the t-test, the two populations being compared should have the same variance (testable using F-test, Levene's test, Bartlett's test, or the Brown–Forsythe test; or assessable graphically using a Q–Q plot). If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances.[10] Welch's t-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.
The data used to carry out the test should be sampled independently from the two populations being compared. This is in general not testable from the data, but if the data are known to be dependently sampled (that is, if they were sampled in clusters), then the classical t-tests discussed here may give misleading results.
Most two-sample t-tests are robust to all but large deviations from the assumptions.[11]
Unpaired and paired two-sample t-tests[edit]
Type I error of unpaired and paired two-sample t-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1. The significance level is 5% and the number of cases is 60.
Power of unpaired and paired two-sample t-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1 and a deviation of the expected value of 0.4. The significance level is 5% and the number of cases is 60.
Two-sample t-tests for a difference in mean involve independent samples or unpaired samples. Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared.[12] In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.
Independent (unpaired) samples[edit]
The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test. The randomization is not essential here – if we contacted 100 people by phone and obtained each person's age and gender, and then used a two-sample t-test to see whether the mean ages differ by gender, this would also be an independent samples t-test, even though the data are observational.
Paired samples[edit]
Main article: Paired difference test
Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).
A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random between-patient variation has now been eliminated. Note however that an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's t-test has only n/2 − 1 degrees of freedom (with n being the total number of observations).[citation needed] Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom. Normally, there are n − 1 degrees of freedom (with n being the total number of observations).[13]
A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest.[14] The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.
Paired samples t-tests are often referred to as "dependent samples t-tests".
Calculations[edit]
Explicit expressions that can be used to carry out various t-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test.
Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. If the calculated p-value is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.
One-sample t-test[edit]
In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic
t = x ¯ − μ 0 s n {\displaystyle t={\frac {{\bar {x}}-\mu _{0}}{\frac {s}{\sqrt {n}}}}}
where x is the sample mean, s is the sample standard deviation of the sample and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means, x, is assumed to be normal. By the central limit theorem, if the sampling of the parent population is independent and the second moment of the parent population exists then the sample means will be approximately normal in the large sample limit.[15] (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)
Slope of a regression line[edit]
Suppose one is fitting the model
Y = α + β x + ε {\displaystyle Y=\alpha +\beta x+\varepsilon }
where x is known, α and β are unknown, and ε is a normally distributed random variable with mean 0 and unknown variance σ2, and Y is the outcome of interest. We want to test the null hypothesis that the slope β is equal to some specified value β0 (often taken to be 0, in which case the null hypothesis is that x and y are uncorrelated).
Let
α ^ , β ^ = least-squares estimators , S E α ^ , S E β ^ = the standard errors of least-squares estimators . {\displaystyle {\begin{aligned}{\hat {\alpha }},{\hat {\beta }}&={\text{least-squares estimators}},\\SE_{\hat {\alpha }},SE_{\hat {\beta }}&={\text{the standard errors of least-squares estimators}}.\end{aligned}}}
Then
t score = | β ^ − β 0 | S E β ^ ∼ T n − 2 {\displaystyle t_{\text{score}}={\frac {\left|{\hat {\beta }}-\beta _{0}\right|}{SE_{\hat {\beta }}}}\sim {\mathcal {T}}_{n-2}}
has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true. The standard error of the slope coefficient:
S E β ^ = 1 n − 2 ∑ i = 1 n ( y i − y ^ i ) 2 ∑ i = 1 n ( x i − x ¯ ) 2 {\displaystyle SE_{\hat {\beta }}={\frac {\sqrt {{\dfrac {1}{n-2}}\displaystyle \sum _{i=1}^{n}\left(y_{i}-{\hat {y}}_{i}\right)^{2}}}{\sqrt {\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}}}
can be written in terms of the residuals. Let
ε ^ i = y i − y ^ i = y i − ( α ^ + β ^ x i ) = residuals = estimated errors , SSR = ∑ i = 1 n ε ^ i 2 = sum of squares of residuals . {\displaystyle {\begin{aligned}{\hat {\varepsilon }}_{i}&=y_{i}-{\hat {y}}_{i}=y_{i}-\left({\hat {\alpha }}+{\hat {\beta }}x_{i}\right)={\text{residuals}}={\text{estimated errors}},\\{\text{SSR}}&=\sum _{i=1}^{n}{{\hat {\varepsilon }}_{i}}^{2}={\text{sum of squares of residuals}}.\end{aligned}}}
Then tscore is given by:
t score = ( β ^ − β 0 ) n − 2 SSR / ∑ i = 1 n ( x i − x ¯ ) 2 . {\displaystyle t_{\text{score}}={\frac {\left({\hat {\beta }}-\beta _{0}\right){\sqrt {n-2}}}{\sqrt {{\text{SSR}}\left/\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right.}}}.}
Independent two-sample t-test[edit]
Equal sample sizes, equal variance[edit]
Given two groups (1, 2), this test is only applicable when:
the two sample sizes (that is, the number n of participants of each group) are equal;
it can be assumed that the two distributions have the same variance;
Violations of these assumptions are discussed below.
The t statistic to test whether the means are different can be calculated as follows:
t = X ¯ 1 − X ¯ 2 s p 2 n {\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}{\sqrt {\frac {2}{n}}}}}}
where
s p = s X 1 2 + s X 2 2 2 {\displaystyle s_{p}={\sqrt {\frac {s_{X_{1}}^{2}+s_{X_{2}}^{2}}{2}}}}
Here sp is the pooled standard deviation for n = n1 = n2 and s 2
X1 and s 2
X2 are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means.
For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.
Equal or unequal sample sizes, equal variance[edit]
This test is used only when it can be assumed that the two distributions have the same variance. (When this assumption is violated, see below.) Note that the previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: n = n1 = n2.
The t statistic to test whether the means are different can be calculated as follows:
t = X ¯ 1 − X ¯ 2 s p ⋅ 1 n 1 + 1 n 2 {\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}\cdot {\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}}}}}
where
s p = ( n 1 − 1 ) s X 1 2 + ( n 2 − 1 ) s X 2 2 n 1 + n 2 − 2 . {\displaystyle s_{p}={\sqrt {\frac {\left(n_{1}-1\right)s_{X_{1}}^{2}+\left(n_{2}-1\right)s_{X_{2}}^{2}}{n_{1}+n_{2}-2}}}.}
is an estimator of the pooled standard deviation of the two samples: it is defined in this way so that its square is an unbiased estimator of the common variance whether or not the population means are the same. In these formulae, ni − 1 is the number of degrees of freedom for each group, and the total sample size minus two (that is, n1 + n2 − 2) is the total number of degrees of freedom, which is used in significance testing.
Equal or unequal sample sizes, unequal variances[edit]
Main article: Welch's t-test
This test, also known as Welch's t-test, is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The t statistic to test whether the population means are different is calculated as:
t = X ¯ 1 − X ¯ 2 s Δ ¯ {\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{\bar {\Delta }}}}}
where
s Δ ¯ = s 1 2 n 1 + s 2 2 n 2 . {\displaystyle s_{\bar {\Delta }}={\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}.}
Here si2 is the unbiased estimator of the variance of each of the two samples with ni = number of participants in group i (1 or 2). Note that in this case s2
Δ is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's t-distribution with the degrees of freedom calculated using
d . f . = ( s 1 2 n 1 + s 2 2 n 2 ) 2 ( s 1 2 / n 1 ) 2 n 1 − 1 + ( s 2 2 / n 2 ) 2 n 2 − 1 . {\displaystyle \mathrm {d.f.} ={\frac {\left({\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}\right)^{2}}{{\frac {\left(s_{1}^{2}/n_{1}\right)^{2}}{n_{1}-1}}+{\frac {\left(s_{2}^{2}/n_{2}\right)^{2}}{n_{2}-1}}}}.}
This is known as the Welch–Satterthwaite equation. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see Behrens–Fisher problem).
Dependent t-test for paired samples[edit]
This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test.
t = X ¯ D − μ 0 s D n . {\displaystyle t={\frac {{\bar {X}}_{D}-\mu _{0}}{\frac {s_{D}}{\sqrt {n}}}}.}
For this equation, the differences between all pairs must be calculated. The pairs are either one person's pre-test and post-test scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (XD) and standard deviation (sD) of those differences are used in the equation. The constant μ0 is non-zero if we want to test whether the average of the difference is significantly different from μ0. The degree of freedom used is n − 1, where n represents the number of pairs.
Example of repeated measures
Number
Name
Test 1
Test 2
1 Mike 35% 67%
2 Melanie 50% 46%
3 Melissa 90% 86%
4 Mitchell 78% 91%
Example of matched pairs
Pair
Name
Age
Test
1 John 35 250
1 Jane 36 340
2 Jimmy 22 460
2 Jessy 21 200
Worked examples[edit]
This article may not properly summarize its corresponding main article. Please help improve it by rewriting it in an encyclopedic style. (Learn how and when to remove this template message)
Let A1 denote a set obtained by drawing a random sample of six measurements:
A 1 = { 30.02 , 29.99 , 30.11 , 29.97 , 30.01 , 29.99 } {\displaystyle A_{1}=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}}
and let A2 denote a second set obtained similarly:
A 2 = { 29.89 , 29.93 , 29.72 , 29.98 , 30.02 , 29.98 } {\displaystyle A_{2}=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}}
These could be, for example, the weights of screws that were chosen out of a bucket.
We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.
The difference between the two sample means, each denoted by Xi, which appears in the numerator for all the two-sample testing approaches discussed above, is
X ¯ 1 − X ¯ 2 = 0.095. {\displaystyle {\bar {X}}_{1}-{\bar {X}}_{2}=0.095.}
The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two-sample t-test will perform similarly in this example.
Unequal variances[edit]
If the approach for unequal variances (discussed above) is followed, the results are
s 1 2 n 1 + s 2 2 n 2 ≈ 0.04849 {\displaystyle {\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}\approx 0.04849}
and the degrees of freedom
d.f. ≈ 7.031. {\displaystyle {\text{d.f.}}\approx 7.031.}
The test statistic is approximately 1.959, which gives a two-tailed test p-value of 0.09077.
Equal variances[edit]
If the approach for equal variances (discussed above) is followed, the results are
s p ≈ 0.04849 {\displaystyle s_{p}\approx 0.04849}
and the degrees of freedom
d.f. = 10. {\displaystyle {\text{d.f.}}=10.}
The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857.
Alternatives to the t-test for location problems[edit]
The t-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances. (Welch's t-test is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t-test is relatively robust to moderate violations of the normality assumption.[16]
For exactness, the t-test and Z-test require normality of the sample means, and the t-test additionally requires that the sample variance follows a scaled χ2 distribution, and that the sample mean and sample variance be statistically independent. Normality of the individual data values is not required if these conditions are met. By the central limit theorem, sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For non-normal data, the distribution of the sample variance may deviate substantially from a χ2 distribution. However, if the sample size is large, Slutsky's theorem implies that the distribution of the sample variance has little effect on the distribution of the test statistic. If the data are substantially non-normal and the sample size is small, the t-test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.
When the normality assumption does not hold, a non-parametric alternative to the t-test can often have better statistical power. Similarly, in the presence of an outlier, the t-test is not robust. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the Mann–Whitney U test) can have three to four times higher power than the t-test.[16][17][18] The nonparametric counterpart to the paired samples t-test is the Wilcoxon signed-rank test for paired samples. For a discussion on choosing between the t-test and nonparametric alternatives, see Sawilowsky (2005).[19]
One-way analysis of variance (ANOVA) generalizes the two-sample t-test when the data belong to more than two groups.
Multivariate testing[edit]
Main article: Hotelling's T-squared distribution
A generalization of Student's t statistic, called Hotelling's t-squared statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g. the Minnesota Multiphasic Personality Inventory). Because measures of this type are usually positively correlated, it is not advisable to conduct separate univariate t-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (Type I error). In this case a single multivariate test is preferable for hypothesis testing. Fisher's Method for combining multiple tests with alpha reduced for positive correlation among tests is one. Another is Hotelling's T2 statistic follows a T2 distribution. However, in practice the distribution is rarely used, since tabulated values for T2 are hard to find. Usually, T2 is converted instead to an F statistic.
For a one-sample multivariate test, the hypothesis is that the mean vector (μ) is equal to a given vector (μ0). The test statistic is Hotelling's t2:
t 2 = n ( x ¯ − μ 0 ) ′ S − 1 ( x ¯ − μ 0 ) {\displaystyle t^{2}=n({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})'{\mathbf {S} }^{-1}({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})}
where n is the sample size, x is the vector of column means and S is an m × m sample covariance matrix.
For a two-sample multivariate test, the hypothesis is that the mean vectors (μ1, μ2) of two samples are equal. The test statistic is Hotelling's two-sample t2:
t 2 = n 1 n 2 n 1 + n 2 ( x ¯ 1 − x ¯ 2 ) ′ S pooled − 1 ( x ¯ 1 − x ¯ 2 ) . {\displaystyle t^{2}={\frac {n_{1}n_{2}}{n_{1}+n_{2}}}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right)'{\mathbf {S} _{\text{pooled}}}^{-1}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right).}
Software implementations[edit]
Many spreadsheet programs and statistics packages, such as QtiPlot, LibreOffice Calc, Microsoft Excel, SAS, SPSS, Stata, DAP, gretl, R, Python, PSPP, Matlab and Minitab, include implementations of Student's t-test.
Language/Program
Function
Notes
Microsoft Excel pre 2010 TTEST(array1, array2, tails, type) See [1]
Microsoft Excel 2010 and later T.TEST(array1, array2, tails, type) See [2]
LibreOffice TTEST(Data1; Data2; Mode; Type) See [3]
Google Sheets TTEST(range1, range2, tails, type) See [4]
Python scipy.stats.ttest_rel(a, b, axis=0, equal_var=True) See [5]
Matlab ttest(data1, data2) See [6]
Mathematica TTest[{data1,data2}] See [7]
R t.test(data1, data2, var.equal=TRUE) See [8]
SAS PROC TTEST See [9]
Java tTest(sample1, sample2) See [10]
Julia EqualVarianceTTest(sample1, sample2) See [11]
Stata ttest data1 == data2 See [12]
See also[edit]
Statistics portal
Conditional change model
F-test
Noncentral t-distribution in power analysis
Student's t-statistic
Z-test
Mann–Whitney U test
Šidák correction for t-test
Welch's t-test
Analysis of variance (ANOVA)
Notes[edit]
Jump up ^ Mankiewicz, Richard (2004). The Story of Mathematics (Paperback ed.). Princeton, NJ: Princeton University Press. p. 158. ISBN 9780691120461.
^ Jump up to: a b O'Connor, John J.; Robertson, Edmund F., "William Sealy Gosset", MacTutor History of Mathematics archive, University of St Andrews .
Jump up ^ Fisher Box, Joan (1987). "Guinness, Gosset, Fisher, and Small Samples". Statistical Science. 2 (1): 45–52. doi:10.1214/ss/1177013437. JSTOR 2245613.
Jump up ^ http://www.aliquote.org/cours/2012_biomed/biblio/Student1908.pdf
Jump up ^ "The Probable Error of a Mean" (PDF). Biometrika. 6 (1): 1–25. 1908. doi:10.1093/biomet/6.1.1. Retrieved 24 July 2016.
Jump up ^ Raju, T. N. (2005). "William Sealy Gosset and William A. Silverman: Two "students" of science". Pediatrics. 116 (3): 732–5. doi:10.1542/peds.2005-1134. PMID 16140715.
Jump up ^ Dodge, Yadolah (2008). The Concise Encyclopedia of Statistics. Springer Science & Business Media. pp. 234–235. ISBN 978-0-387-31742-7.
^ Jump up to: a b Fadem, Barbara (2008). High-Yield Behavioral Science. High-Yield Series. Hagerstown, MD: Lippincott Williams & Wilkins. ISBN 0-7817-8258-9.
Jump up ^ Zimmerman, Donald W. (1997). "A Note on Interpretation of the Paired-Samples t Test". Journal of Educational and Behavioral Statistics. 22 (3): 349–360. doi:10.3102/10769986022003349. JSTOR 1165289.
Jump up ^ Markowski, Carol A.; Markowski, Edward P. (1990). "Conditions for the Effectiveness of a Preliminary Test of Variance". The American Statistician. 44 (4): 322–326. doi:10.2307/2684360. JSTOR 2684360.
Jump up ^ Bland, Martin (1995). An Introduction to Medical Statistics. Oxford University Press. p. 168. ISBN 978-0-19-262428-4.
Jump up ^ Rice, John A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Duxbury Advanced. [ISBN missing]
Jump up ^ Weisstein, Eric. "Student's t-Distribution". mathworld.wolfram.com.
Jump up ^ David, H. A.; Gunnink, Jason L. (1997). "The Paired t Test Under Artificial Pairing". The American Statistician. 51 (1): 9–12. doi:10.2307/2684684. JSTOR 2684684.
Jump up ^ Box, George; Hunter, William; Hunter, J. Stuart. Statistics for Experimenters. pp. 66–67. ISBN 978-0471093152.
^ Jump up to: a b Sawilowsky, Shlomo S.; Blair, R. Clifford (1992). "A More Realistic Look at the Robustness and Type II Error Properties of the t Test to Departures From Population Normality". Psychological Bulletin. 111 (2): 352–360. doi:10.1037/0033-2909.111.2.352.
Jump up ^ Blair, R. Clifford; Higgins, James J. (1980). "A Comparison of the Power of Wilcoxon's Rank-Sum Statistic to That of Student's t Statistic Under Various Nonnormal Distributions". Journal of Educational Statistics. 5 (4): 309–335. doi:10.2307/1164905. JSTOR 1164905.
Jump up ^ Fay, Michael P.; Proschan, Michael A. (2010). "Wilcoxon–Mann–Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys. 4: 1–39. doi:10.1214/09-SS051. PMC 2857732  . PMID 20414472.
Jump up ^ Sawilowsky, Shlomo S. (2005). "Misconceptions Leading to Choosing the t Test Over The Wilcoxon Mann–Whitney Test for Shift in Location Parameter". Journal of Modern Applied Statistical Methods. 4 (2): 598–600. Retrieved 2014-06-18.
References[edit]
O'Mahony, Michael (1986). Sensory Evaluation of Food: Statistical Methods and Procedures. CRC Press. p. 487. ISBN 0-82477337-3.
Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1992). https://web.archive.org/web/20151128053615/http://numerical.recipes/ |archive-url= missing title (help). Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press. p. 616. ISBN 0-521-43108-5. Archived from the original (PDF) on 2015-11-28.
Further reading[edit]
Boneau, C. Alan (1960). "The effects of violations of assumptions underlying the t test". Psychological Bulletin. 57 (1): 49–64. doi:10.1037/h0041412.
Edgell, Stephen E.; Noon, Sheila M. (1984). "Effect of violation of normality on the t test of the correlation coefficient". Psychological Bulletin. 95 (3): 576–583. doi:10.1037/0033-2909.95.3.576.
External links[edit]
Wikiversity has learning resources about t-test
Hazewinkel, Michiel, ed. (2001) [1994], "Student test", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
A conceptual article on the Student's t-test
Econometrics lecture (topic: hypothesis testing) on YouTube by Mark Thoma
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mw-redirect
John Snow (physician)

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/wiki/Student%27s_t-test
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values: اختبار-تي للطالب – Arabic

values:
interlanguage-link interwiki-ast
display: none;

values:
interlanguage-link interwiki-ca
display: none;

values:
https://cs.wikipedia.org/wiki/T-test
T-test – Czech
cs
cs
interlanguage-link-target

values:
interlanguage-link interwiki-cs
display: none;

values:
https://de.wikipedia.org/wiki/T-Test
T-Test – German
de
de
interlanguage-link-target

values:
https://et.wikipedia.org/wiki/T-test
T-test – Estonian
et
et
interlanguage-link-target

values:
interlanguage-link interwiki-et
display: none;

values: آزمون تی استیودنت – Persian

values:
interlanguage-link interwiki-fa
display: none;

values:
interlanguage-link interwiki-gl
display: none;

values:
https://hr.wikipedia.org/wiki/T-test
T-test – Croatian
hr
hr
interlanguage-link-target

values:
interlanguage-link interwiki-hr
display: none;

values:
https://it.wikipedia.org/wiki/Test_t
Test t – Italian
it
it
interlanguage-link-target

values:
interlanguage-link interwiki-it
display: none;

values:
interlanguage-link interwiki-he
display: none;

values:
https://lv.wikipedia.org/wiki/T_tests
T tests – Latvian
lv
lv
interlanguage-link-target

values:
interlanguage-link interwiki-lv
display: none;

values:
interlanguage-link interwiki-hu
display: none;

values: Студентов т-тест – Macedonian

values:
interlanguage-link interwiki-mk
display: none;

values:
https://nl.wikipedia.org/wiki/T-toets
T-toets – Dutch
nl
nl
interlanguage-link-target

values:
interlanguage-link interwiki-nl
display: none;

values:
https://no.wikipedia.org/wiki/T-test
T-test – Norwegian
no
no
interlanguage-link-target

values:
interlanguage-link interwiki-no
display: none;

values: T-Критерий Стьюдента – Russian

values:
interlanguage-link interwiki-su
display: none;

values:
https://sv.wikipedia.org/wiki/T-test
T-test – Swedish
sv
sv
interlanguage-link-target

values:
interlanguage-link interwiki-sv
display: none;

values: Tek anakütle ortalaması için parametrik hipotez sınaması – Turkish

values:
interlanguage-link interwiki-tr
display: none;

values: T-критерій Стьюдента – Ukrainian

values:
interlanguage-link interwiki-uk
display: none;

values:
interlanguage-link interwiki-zh-yue
display: none;

values:
suggestions
display: none; font-size: 13px;

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