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The Five Heartbeats
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The Five Heartbeats
Directed by
Robert Townsend[1]
Produced by
Christina Schmidlin
Robert Townsend
Kokayi Ampah
Loretha C. Jones[2]
Written by
Robert Townsend
Keenen Ivory Wayans[3]
Starring
Robert Townsend
Michael Wright
Leon
Harry J. Lennix
Tico Wells
Harold Nicholas
Diahann Carroll
[1]
Music by
Stanley Clarke
Cinematography
Bill Dill
Edited by
John Carter
Distributed by
20th Century Fox[4]
Release date
March 29, 1991 (1991-03-29) (U.S.)[5]
Running time
122 minutes
Country
United States
Language
English[4]
Box office
$8,750,400
The Five Heartbeats is a 1991 musical drama film directed by Robert Townsend, who co-wrote the script with Keenen Ivory Wayans. Distributed by 20th Century Fox, the film's main cast includes Townsend, Michael Wright, Leon Robinson, Harry J. Lennix, Tico Wells, Harold Nicholas of the Nicholas Brothers, and Diahann Carroll.[1] The plot of the film (which is loosely based on the lives of several artists: The Dells, The Temptations, Four Tops, Wilson Pickett, James Brown, Frankie Lymon, Sam Cooke and others)[7] follows the three decade career of the rhythm and blues vocal group The Five Heartbeats. The film depicts the rise and fall of a Motown inspired soul act through the eyes of one of the Heartbeats, Donald "Duck" Matthews.
The film was released to most North American audiences on March 29, 1991[8] however it was not made available to audiences in other continents until 2002 when a DVD was released prior to another DVD release in 2006 for the film's 15th anniversary.[9] The movie received mixed reviews from critics.[10][11]
Contents
1 Plot
2 Cast and characters
2.1 The Five Heartbeats
2.2 Other characters
2.3 Other bands
3 Production
3.1 Promotion
4 Soundtrack
5 Reception
6 DVD releases
7 Footnotes
8 External links
Plot[edit]
Donald "Duck" Matthews, Anthony "Choirboy" Stone, J.T. Matthews, Terrence "Dresser" Williams, and Eddie King Jr. perform at a Battle of the Bands contest as The Heartbeats. The group loses to Flash and the Ebony Sparks but pleases the crowd and is noticed by Jimmy Potter. Jimmy offers to manage the group, promising them $100 if they do not win the next month's Battle of the Bands contest. After the group loses, Jimmy pays them. When the owner of the club asks to hire them, they agree to let Jimmy manage them.
Bird and The Midnight Falcons witness the Heartbeats rehearsing for a competition and are concerned they could lose; frontman Victor "Bird" Thomas asks his girlfriend to invite her friends and boo The Heartbeats while cheering The Falcons. The announcer, a cousin of one of the Falcons, forces The Heartbeats to use the house piano player. Duck grows frustrated with the house piano player and shoves him off the piano stool. Eddie leads the group in a number that results in Bird's girlfriend fainting in Eddie's arms. The Heartbeats win the contest and the interest of Big Red Davis, who owns Big Red Records. Big Red offers them a deal, but Jimmy and his wife Eleanor, aware of Big Red's corruption, decline. The group instead releases their first single on Jimmy's own independent label and searches for a record company they can trust, but no one else is interested aside from a label which wants to buy Duck's songs for a group they've already signed, The Five Horsemen, prompting The Heartbeats to sign with Big Red.
The group goes on the road, but the travel is marred by racism and poor living conditions. Dresser's girlfriend visits and reveals she is pregnant, and the group is faced with their first album cover featuring white people, despite the label having earlier approved a photo of the Heartbeats as the cover.
Throughout the mid to late 1960s The Five Heartbeats receive numerous awards, chart several hits, and are featured on magazine covers. Eddie abuses alcohol and cocaine, affecting his performance and prompting his girlfriend Baby Doll to break up with him. Convinced that Jimmy intends to replace him due to his deteriorating condition, he cuts a deal with Big Red to have Jimmy cut out of his contract. Jimmy threatens to go to authorities with information about bootlegged LPs, cooked books and payola that could have Big Red arrested, leading Red to have Jimmy killed. The group learns that Eddie's deceit caused the fallout between Jimmy and Big Red. Big Red is eventually convicted of Jimmy's murder, forcing the group to sign to a new record label, and a guilt-ridden Eddie to leave in disgrace. Eddie later pops up at several Heartbeats concerts, to try to convince his friends to let him back into the group, but his drug and alcohol abuse has taken its toll on him, and all the other group members can do is supply him with money out of sympathy; Duck later hears on the radio that Eddie was involved in an armed robbery and, after a shootout with the police, was in critical condition.
The Heartbeats add Flash as their lead singer. Duck comes to suspect his fiancé, Tanya Sawyer, is having an affair with Choirboy. He follows her to a hotel, only to discover that Tanya is meeting with J.T., not Choirboy. Tanya's relationship with J.T. predates her relationship with Duck, but she says she is now in love with Duck. J.T. urges Tanya to disclose the affair, but she refuses. At an awards ceremony celebrating their success, Flash announces he is going solo. Duck reveals that he knows about Tanya and J.T. and also leaves.
The film then skips ahead to the early 1990s. Choirboy, true to his nickname, has returned to singing in his father's church. After converting to Christianity, Eddie became clean and sober. He is now married to Baby Doll, sings in Choirboy's choir, and manages a group. He asks Duck to write songs for them, to which he agrees. J.T. has a wife and two children, including a son named "Duck". The brothers reconcile. The only member to have maintained a singing career is Flash, who transitioned from doo-wop to pop music, and is part of the group Flash and The Five Horsemen.
At a family gathering, Eleanor Potter, coming to terms with her husband's death, forgives Eddie. The Five Heartbeats reunite in front of their families and friends, trying to remember their old moves.
Cast and characters[edit]
The Five Heartbeats[edit]
Robert Townsend as Donald "Duck" Matthews: Duck hails from a poor family. He is The Five Heartbeats' co-founder and brother of fellow Heartbeat's member J.T. Matthews, and originally was only the composer and musician for the group. He is a permanent vocalist after Bobby disappears. He serves as the movie's narrator, with the film beginning as he reminisces about the group's career.
Leon Robinson as J.T. Matthews: J.T. is the older brother of Duck. A womanizer; he and his brother Duck share a close and sometimes turbulent relationship.
Michael Wright as Eddie King, Jr.: Eddie comes from an area that features predominantly poor individuals, his own father believing his attempts to start a career in the music industry will be unsuccessful. Similar to David Ruffin, Eddie is the lead vocalist of the band and falls into a life of drugs that eventually leads to his expulsion from the group as well as emotional trauma. Eddie is one of the founding members of The Heartbeats and serves as the crowd pleaser whose voice leads to success in many of their performances.
Tico Wells as Anthony "Choirboy" Stone: Stone is given the nickname "Choirboy" (much to his chagrin) for his past as a choir boy in his father's church. Similar to Eddie, Stone's father does not support his decision to become a music artist fearing rock and jazz are "the devil's music." He becomes more of a womanizer as the group becomes more successful, but later goes back to being a part of his father's ministry after the group disbands.
Harry J. Lennix as Terrence "Dresser" Williams: The original choreographer of The Heartbeats, Dresser serves as the groups bass singer and one of the founding members. He is replaced by Ernest "Sarge" Johnson as the choreographer after Sarge out-dances Dresser.
Other characters[edit]
Hawthorne James as Big Red Davis: Corrupt owner of the first record label The Five Heartbeats are signed to.
Chuck Patterson as Jimmy Potter: The Heartbeats' manager, Jimmy is responsible for giving the group the opportunity to perform at more publicized events and receive training from Ernest "Sarge" Johnson.
Diahann Carroll as Eleanor Potter: Jimmy Potter's wife, Eleanor along with her husband are the original supporters of The Five Heartbeats.
John Canada Terrell as Michael "Flash" Turner: Lead singer of Flash and the Ebony Sparks, the Heartbeats' only competition. When Eddie goes into a downward spiral, Flash is brought in as the new lead singer.
Roy Fegan as Victor "Bird" Thomas: Bird is the lead singer of Bird and The Midnight Falcons. Early in the film the character attempts to defeat The Five Heartbeats in a vocal contest. After Big Red orchestrates the murder of Jimmy Potter, Bird, who was previously attacked and intimidated by Big Red during a party, joins members of the Heartbeats in testifying against Red and helps to convict him of the murder.
Harold Nicholas of The Nicholas Brothers as Ernest "Sarge" Johnson: Johnson is The Five Heartbeat's choreographer. Sarge is last seen in the hospital on his birthday.
Troy Beyer as Baby Doll: Eddie's girlfriend, who leaves him after he abuses drugs and alcohol. She later marries Eddie and they are shown singing together in a choir.
Theresa Randle as Brenda: Dresser's girlfriend, who later becomes his wife. She and Dresser have a daughter named Monica.
Lamont Johnson as Bobby Cassanova: Bobby and Eddie are the original co-lead singers of the group. Bobby is only seen in the movie's opening scenes when he and Eddie are caught cheating at a high-stakes poker game. Bobby gets shot in the leg and is replaced by Eddie as the lead singer.
Carla Brothers as Tanya Sawyer: Duck's love interest. His earlier attempts to win her heart are unsuccessful but they reconnect at a party for the group and later date. Tanya and Duck were set to be wed, but Duck's suspicion about Tanya and Choir Boy had him follow her to a hotel only to find out that his brother J.T. was the one seeing her. This led Duck to estrange himself from the Heartbeats and his brother for years.
Tressa Thomas as "Duck's Little Sister": Clara Matthews, Duck's little sister, was also his muse, often helping him compose and arrange songs for the group as they did household chores. She was instrumental in arranging the vocals for "We Haven't Finished Yet".
Other bands[edit]
Bird and The Midnight Falcons portrayed by actors Roy Fegan, Gregory Alexander, Roger Rose, Jimmy Woodard: The group The Five Heartbeats compete against in their second onscreen performance. The Midnight Falcons attempt to cheat by ordering their girlfriends to deliberately cheer against The Heartbeats. They would later sign with Big Red Records along with the Five Heartbeats.
Flash and the Ebony Sparks portrayed by John Canada Terrell, Ron Jackson, Recoe Walker and Wayne "Crescendo" Ward: The Ebony Sparks defeat The Five Heartbeats in singing competitions—many of which are rigged—prior to their mainstream popularity. Flash later becomes the lead singer for the Five Heartbeats.

Production[edit]
After writing (along with Keenen Ivory Wayans), producing, directing, and starring in his first film Hollywood Shuffle, Robert Townsend had attained near-cult status among independent filmmakers due to his dedication to that film—a project which caused him to max out all his credit cards and spend nearly $100,000 of his own money raised through savings and various acting jobs in order to produce the film. When writing Townsend's first studio film The Five Heartbeats, Townsend and Wayans kept comedy an important aspect of the film, but also explored complex characters in a more dramatic way.[12]
The Five Heartbeats was originally set up as a development deal at Warner Bros. in 1988, with Keenan Ivory Wayans, his brother Damon, and others tapped to star.[13] Warner passed on the project, and the Wayans moved on to develop and star in the FOX sketch comedy show In Living Color instead.[13]
Townsend resurrected Heartbeats at 20th Century Fox in 1990.[13] The original screenplay had been inspired by the lives and careers of Motown group The Temptations, and Townsend had met with former lead singers David Ruffin and Eddie Kendrick with the intention of hiring them as technical advisors.[13] Fox vetoed bringing Ruffi and Kendrick onto the production, for fear that Motown founder Berry Gordy, Jr. might sue the studio.[13] Rhythm and blues-singing group The Dells, who were renowned for their four-decade career, became the technical advisors instead.
Townsend used his film to depict a similar story to the careers of the Temptations and the Dells, following the lives of three friends who aspire to musical stardom.[14] Given the setting of the film, he was able to tie in other elements, such as race relations, as well.[14]
Due to the production's budgetary constraints, Townsend used little-known actors of the time, with the exceptions of Leon Robinson, Diahann Carroll and Harold Nicholas of The Nicholas Brothers.[1] Townsend had considered Denzel Washington as Eddie King, Jr. and Whitney Houston as Baby Doll.[13] The former was not cast due to budget concerns and the latter passed on the role as it was felt to be too small.[13]
Promotion[edit]
To promote the film prior to its release, Townsend, along with the other actors who portrayed the fictional musical quartet The Five Heartbeats (Leon Robinson, Michael Wright, Harry J. Lennix, and Tico Wells) performed in a concert with real-life Soul/R&B vocal group The Dells, one of many groups that inspired the film. The Dells sang and recorded the vocals as the actors lip synced.[15]
Soundtrack[edit]
The Five Heartbeats
(Music from the Motion Picture)
Soundtrack album by Various Artists
Released
April 2, 1991
Genre
R&B, soul
Label
Capitol[9][16]
Professional ratings
Review scores
Source
Rating
AllMusic [17]
A soundtrack for the film was released by Virgin Records, featuring original music by various artists. Both "Nights like This" and "A Heart Is a House for Love" became top 20 hits on the U.S. Billboard Hot R&B Singles chart.[17] Many of the tracks are credited to fictional characters in the film as opposed to the actual vocalists.
A Heart Is a House for Love - Billy Valentine/ The Dells
We Haven't Finished Yet - Patti LaBelle, Tressa Thomas, Billy Valentine
Nights like This - After 7
Bring Back the Days - U.S. Male
Baby Stop Running Around - Bird & The Midnight Falcons
In the Middle - Dee Harvey
Nothing but Love - The Dells with Billy Valentine
Are You Ready for Me - Dee Harvey
Stay in My Corner - The Dells
I Feel Like Going On - Andraé Crouch[16][17] (Eddie, Baby Doll and the L.A. Mass Choir)
Reception[edit]
The film grossed approximately $8,500,000[14][18] after being released in 862 theaters throughout North America. However, despite the film's moderate success, it was not well received by a majority of critics.[19] On Rotten Tomatoes The Five Heartbeats accumulated an average of 35%, although only 17 reviews were counted (6 of which were positive, the remaining 11 negative).[19]
Roger Ebert of the Chicago Sun-Times commented that:
“ ...at feature length, Townsend shows a real talent, and, not surprisingly, an ability to avoid most cliches, to go for the human truth in his characters...by the end we really care about these guys...There is one obligatory scene showing racial prejudice against the group, and it seems a little tacked on, as if the only purpose of the Southern trip was to justify the scene.[11] ”
Due to the nature of the film, music montages were often used to progress the plot; critics considered this a major flaw.[20]
The numerous musical performances in the film were highly acclaimed.[20] All Music complimented the Dells' lead singer Marvin Junior (who provided the singing voice for fictional character Eddie Kane Jr.) stating that he was "one of the most underrated voices in pop music."[16] Tressa Thomas' performance of "We Haven't Finished Yet," in particular, was given favorable attention by critics.[21][22] The film received an ASCAP award for Most Performed Songs in a Motion Picture for the song "Nights Like This."[23]
DVD releases[edit]
A DVD was released for the film in 2002, a special edition was also released in 2006 for the film's 15th Anniversary which includes additional content.[9]
Footnotes[edit]
^ Jump up to: a b c d http://www.vh1.com/movies/movie/11969/castcrew.jhtml
Jump up ^ http://www.vh1.com/movies/movie/11969/details.jhtml
Jump up ^ http://www.vh1.com/movies/movie/11969/plot.jhtml
^ Jump up to: a b https://www.imdb.com/title/tt0101891/
Jump up ^ http://search.ew.com/EWSearch/ew/search/search.html?search=the+five+heartbeats&x=0&y=0/[permanent dead link]
Jump up ^ https://www.rollingstone.com/reviews/dvd/5948151/review/5948152/
Jump up ^ http://movies.msn.com/movies/movie/the-five-heartbeats/?pkw=PI&vendor=Paid%20Inclusion&OCID=iSEMPI
Jump up ^ http://search.ew.com/EWSearch/ew/search/search.html?search=the+five+heartbeats&x=0&y=0[permanent dead link]
^ Jump up to: a b c http://www.answers.com/topic/the-five-heartbeats
Jump up ^ http://www.rottentomatoes.com/m/five_heartbeats/?name_order=asc#mo
^ Jump up to: a b http://rogerebert.suntimes.com/apps/pbcs.dll/article?AID=/19910329/REVIEWS/103290301/1023
Jump up ^ https://books.google.com/books?id=5YWgxVkJmT4C&pg=PA220&dq=the+five+heartbeats&lr=&sig=ACfU3U0Nl-quxN9UYyRjd5Q6i_fBRfUcNw#PPA220,M1
^ Jump up to: a b c d e f g Townsend, Robert (Director) (August 27, 2018). Making The Five Heartbeats (Motion picture). Hollywood: Green Lighthouse.
^ Jump up to: a b c http://www.rottentomatoes.com/m/five_heartbeats/#
Jump up ^ http://www.ew.com/ew/article/0,,313030,00.html
^ Jump up to: a b c https://www.allmusic.com/album/r84315
^ Jump up to: a b c http://www.artistdirect.com/nad/store/artist/album/0,,86336,00.html
Jump up ^ http://boxofficemojo.com/movies/?id=fiveheartbeats.htm
^ Jump up to: a b http://www.rottentomatoes.com/m/five_heartbeats/?name_order=asc#
^ Jump up to: a b http://www.dvdtalk.com/reviews/20018/five-heartbeats-15th-anniversary-special-edition-the/
Jump up ^ https://www.washingtonpost.com/wp-srv/style/longterm/movies/videos/thefiveheartbeatsrhinson_a0a9e7.htm
Jump up ^ http://www.ew.com/ew/article/0,,313751,00.html
Jump up ^ https://www.imdb.com/title/tt0101891/awards
External links[edit]
Wikiquote has quotations related to: The Five Heartbeats
The Five Heartbeats on IMDb
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The Five Heartbeats at Box Office Mojo
The Five Heartbeats at AllMovie
hide
v
t
e
Films directed by Robert Townsend
Hollywood Shuffle (1987)
Eddie Murphy Raw (1987)
The Five Heartbeats (1991)
The Meteor Man (1993)
B*A*P*S (1997)
Jackie's Back (1999)
Up, Up and Away (2000)
Little Richard (2000)
Holiday Heart (2000)
Carmen: A Hip Hopera (2001)
10,000 Black Men Named George (2002)
Black Listed (2003)
Phantom Punch (2009)
In the Hive (2012)
Bill Cosby 77 (2014)
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Mode (statistics)
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Mode.
The mode of a set of data values is the value that appears most often.[1] It is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value, so any peak is a mode.[2]
Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points x1, x2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.
When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal (as opposed to unimodal).
In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode.
Contents
[hide]
1 Mode of a sample
2 Comparison of mean, median and mode
2.1 Use
2.2 Uniqueness and definedness
2.3 Properties
2.4 Example for a skewed distribution
2.5 Van Zwet condition
3 Unimodal distributions
4 History
5 See also
6 References
7 External links
Mode of a sample[edit]
The mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] the mode is not unique – the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal.
For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of the intervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.
The following MATLAB (or Octave) code example computes the mode of a sample:
X = sort(x); indices = find(diff([X; realmax]) > 0); % indices where repeated values change [modeL,i] = max (diff([0; indices])); % longest persistence length of repeated values mode = X(indices(i));
The algorithm requires as a first step to sort the sample in ascending order. It then computes the discrete derivative of the sorted list, and finds the indices where this derivative is positive. Next it computes the discrete derivative of this set of indices, locating the maximum of this derivative of indices, and finally evaluates the sorted sample at the point where that maximum occurs, which corresponds to the last member of the stretch of repeated values.
Comparison of mean, median and mode[edit]
See also: Mean and Median
Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
Type
Description
Example
Result
Arithmetic mean Sum of values of a data set divided by number of values: x ¯ = 1 n ∑ i = 1 n x i {\displaystyle \scriptstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}} (1+2+2+3+4+7+9) / 7 4
Median Middle value separating the greater and lesser halves of a data set 1, 2, 2, 3, 4, 7, 9 3
Mode Most frequent value in a data set 1, 2, 2, 3, 4, 7, 9 2
Use[edit]
Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.
Unlike median, the concept of mode makes sense for any random variable assuming values from a vector space, including the real numbers (a one-dimensional vector space) and the integers (which can be considered embedded in the reals). For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median and the centerpoint.
Uniqueness and definedness[edit]
For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain pathological distributions (for example, the Cantor distribution) have no defined mode at all.[citation needed] For a finite data sample, the mode is one (or more) of the values in the sample.
Properties[edit]
Assuming definedness, and for simplicity uniqueness, the following are some of the most interesting properties.
All three measures have the following property: If the random variable (or each value from the sample) is subjected to the linear or affine transformation which replaces X by aX+b, so are the mean, median and mode.
Except for extremely small samples, the mode is insensitive to "outliers" (such as occasional, rare, false experimental readings). The median is also very robust in the presence of outliers, while the mean is rather sensitive.
In continuous unimodal distributions the median often lies between the mean and the mode, about one third of the way going from mean to mode. In a formula, median ≈ (2 × mean + mode)/3. This rule, due to Karl Pearson, often applies to slightly non-symmetric distributions that resemble a normal distribution, but it is not always true and in general the three statistics can appear in any order.[3][4]
For unimodal distributions, the mode is within 3 {\displaystyle {\sqrt {3}}} standard deviations of the mean, and the root mean square deviation about the mode is between the standard deviation and twice the standard deviation.[5]
Example for a skewed distribution[edit]
An example of a skewed distribution is personal wealth: Few people are very rich, but among those some are extremely rich. However, many are rather poor.
Comparison of mean, median and mode of two log-normal distributions with different skewness.
A well-known class of distributions that can be arbitrarily skewed is given by the log-normal distribution. It is obtained by transforming a random variable X having a normal distribution into random variable Y = eX. Then the logarithm of random variable Y is normally distributed, hence the name.
Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X. This is so because X has a symmetric distribution, so its median is also 0. The transformation from X to Y is monotonic, and so we find the median e0 = 1 for Y.
When X has standard deviation σ = 0.25, the distribution of Y is weakly skewed. Using formulas for the log-normal distribution, we find:
mean = e μ + σ 2 / 2 = e 0 + 0.25 2 / 2 ≈ 1.032 mode = e μ − σ 2 = e 0 − 0.25 2 ≈ 0.939 median = e μ = e 0 = 1 {\displaystyle {\begin{array}{rlll}{\text{mean}}&=e^{\mu +\sigma ^{2}/2}&=e^{0+0.25^{2}/2}&\approx 1.032\\{\text{mode}}&=e^{\mu -\sigma ^{2}}&=e^{0-0.25^{2}}&\approx 0.939\\{\text{median}}&=e^{\mu }&=e^{0}&=1\end{array}}}
Indeed, the median is about one third on the way from mean to mode.
When X has a larger standard deviation, σ = 1, the distribution of Y is strongly skewed. Now
mean = e μ + σ 2 / 2 = e 0 + 1 2 / 2 ≈ 1.649 mode = e μ − σ 2 = e 0 − 1 2 ≈ 0.368 median = e μ = e 0 = 1 {\displaystyle {\begin{array}{rlll}{\text{mean}}&=e^{\mu +\sigma ^{2}/2}&=e^{0+1^{2}/2}&\approx 1.649\\{\text{mode}}&=e^{\mu -\sigma ^{2}}&=e^{0-1^{2}}&\approx 0.368\\{\text{median}}&=e^{\mu }&=e^{0}&=1\end{array}}}
Here, Pearson's rule of thumb fails.
Van Zwet condition[edit]
Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold.[6] The inequality
Mode ≤ Median ≤ Mean
holds if
F( Median - x ) + F( Median + x ) ≥ 1
for all x where F() is the cumulative distribution function of the distribution.
Unimodal distributions[edit]
It can be shown for a unimodal distribution that the median X ~ {\displaystyle {\tilde {X}}} and the mean X ¯ {\displaystyle {\bar {X}}} lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other.[7] In symbols,
| X ~ − X ¯ | σ ≤ ( 3 / 5 ) 1 / 2 {\displaystyle {\frac {\left|{\tilde {X}}-{\bar {X}}\right|}{\sigma }}\leq (3/5)^{1/2}}
where | ⋅ | {\displaystyle |\cdot |} is the absolute value.
A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of each other:
| X ~ − m o d e | σ ≤ 3 1 / 2 . {\displaystyle {\frac {\left|{\tilde {X}}-\mathrm {mode} \right|}{\sigma }}\leq 3^{1/2}.}
History[edit]
The term mode originates with Karl Pearson in 1895.[8]
See also[edit]
Statistics portal
Arg max
Bimodal distribution
Central tendency
Mean
Median
Descriptive statistics
Moment (mathematics)
Summary statistics
Unimodal function
References[edit]
Jump up ^ [[Damodar N. Gujarati] f Econometrics. McGraw-Hill Irwin. 3rd edition, 2006: p. 110.probability distribution]]
Jump up ^ Zhang, C; Mapes, BE; Soden, BJ (2003). "Bimodality in tropical water vapour". Q J R. Meteorol Soc. 129: 2847–2866. doi:10.1256/qj.02.16.
Jump up ^ "Relationship between the mean, median, mode, and standard deviation in a unimodal distribution".
Jump up ^ Hippel, Paul T. von (2005). "Mean, Median, and Skew: Correcting a Textbook Rule". J. of Statistics Education. 13 (2).
Jump up ^ Bottomley, H. (2004). "Maximum distance between the mode and the mean of a unimodal distribution" (PDF). Unpublished preprint.
Jump up ^ van Zwet WR (1979) "Mean, median, mode II", Statistica Neerlandica, 33 (1) 1–5
Jump up ^ Basu, Sanjib, and Anirban DasGupta. "The mean, median, and mode of unimodal distributions: a characterization." Theory of Probability & Its Applications 41.2 (1997): 210-223.
Jump up ^ Pearson, Karl (1895). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material", Philosophical Transactions of the Royal Society of London, Ser. A, 186, 343-414
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Mode", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
A Guide to Understanding & Calculating the Mode
Weisstein, Eric W. "Mode". MathWorld.
Mean, Median and Mode short beginner video from Khan Academy
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