1. Plus One Obvious Album

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I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.

My professor asks me how can mean plus one standard deviation exceed the maximum.

I showed her many examples about this, but she doesn't understand. I need some reference to show her. It could be any chapter or paragraph from a statistics book that talks particularly about this.

4 Answers

Certainly the mean plus one sd can exceed the largest observation.

Consider the sample 1, 5, 5, 5 -

it has mean 4 and standard deviation 2, so the mean + sd is 6, one more than the sample maximum. Here's the calculation in R:

It's a common occurrence. It tends to happen when there's a bunch of high values and a tail off to the left (i.e. when there's strong left skewness and a peak near the maximum).

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The same possibility applies to probability distributions, not just samples - the population mean plus the population sd can easily exceed the maximum possible value.

Here's an example of a $text{beta}(10,frac{1}{2})$ density, which has a maximum possible value of 1:

In this case, we can look at the Wikipedia page for the beta distribution, which states that the mean is:

$operatorname{E}[X] = frac{alpha}{alpha+beta}!$ Facebook handler download mobile9 android for pc.

and the variance is:

$operatorname{var}[X] = frac{alphabeta}{(alpha+beta)^2(alpha+beta+1)}!$

(Though we needn't rely on Wikipedia, since they're pretty easy to derive.)

So for $alpha=10$ and $beta=frac{1}{2}$ we have mean$approx 0.9523$ and sd$approx 0.0628$, so mean+sd$approx 1.0152$, more than the possible maximum of 1.

That is, it's easily possible to have a value of mean+sd that cannot be observed as a data value.

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For any situation where the mode was at the maximum, the Pearson mode skewness need only be $<,-1$ for mean+sd to exceed the maximum. It can take any value, positive or negative, so we can see it's easily possible.

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A closely related issue is often seen with confidence intervals for a binomial proportion, where a commonly used interval, the normal approximation interval can produce limits outside $[0,1]$.

For example, consider a 95.4% normal approximation interval for the population proportion of successes in Bernoulli trials (outcomes are 1 or 0 representing success and failure events respectively), where 3 of 4 observations are '$1$' and one observation is '$0$'.

Then the upper limit for the interval is $hat p + 2 times sqrt{frac{1}{4}hat p left(1 - hat p right)} = hat p + sqrt{hat p (1 - hat p )} = 0.75 + 0.433=1.183$

This is just the sample mean + the usual estimate of the sd for the binomial .. and produces an impossible value.

The usual sample sd for 0,1,1,1 is 0.5 rather than 0.433 (they differ because the binomial ML estimate of the standard deviation $hat p(1-hat p)$ corresponds to dividing the variance by $n$ rather than $n-1$). But it makes no difference - in either case, mean + sd exceeds the largest possible proportion.

This fact - that a normal approximation interval for the binomial can produce 'impossible values' is often noted in books and papers. However, you're not dealing with binomial data. Nevertheless the problem - that mean + some number of standard deviations is not a possible value - is analogous.

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In your case, the unusual '0' value in your sample is making the sd large more than it pulls the mean down, which is why the mean+sd is high.

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(The question would instead be - by what reasoning would it be impossible? -- because without knowing why anyone would think there's a problem at all, what do we address?)

Logically of course, one demonstrates it's possible by giving an example where it happens. You've done that already. In the absence of a stated reason why it should be otherwise, what are you to do?

If an example isn't sufficient, what proof would be acceptable?

There's really no point simply pointing to a statement in a book, since any book may make a statement in error - I see them all the time. One must rely on direct demonstration that it's possible, either a proof in algebra (one could be constructed from the beta example above for example*) or by numerical example (which you have already given), which anyone can examine the truth of for themselves.

* whuber gives the precise conditions for the beta case in comments.

Per Chebyshev's inequality, less than k^-2 points can be more than k standard deviations away. So, for k=1 that means less than 100% of your samples can be more than one standard deviation away.

It's more interesting to look at the low bound. Your professor should be more surprised there are points which are about 2.5 standard deviations below mean. But we now know that only about 1/6th of your samples can be 0.

The essence of the problem may be that your distribution is not a normal distribution which a standard deviation assumes. Your distribution is likely left skewed, so you need to transform your set into a normal distribution first by picking a suitable transform function, this process is called transformation to normality. One such function candidate in your case might be a mirrored log transform. Once your set satisfies a normality test you may then take the standard deviation. Then to use your 1$sigma$ or 2$sigma$ values you must transform them back into your original data space using the inverse of your transform function. I'm thinking this is what your professor was hinting at.

In general for the Bernoulli random variable $X$, that takes the value $1$ with probability $0<p<1$ and the value $0$ with probability $1-p$, we have

$$E(X) = p,;; SE(X) = sqrt {p(1-p)}$$

And we want

$$E(X)+ SE(X) > 1 Rightarrow p +sqrt {p(1-p)} >1$$

$$Rightarrow sqrt {p(1-p)} > (1-p)$$

Square both sides to obtain

$$p(1-p) > (1-p)^2 Rightarrow p > 1-p Rightarrow p > frac 12$$

In words, for any Bernoulli random variable with $p>1/2$ the theoretical expression $E(X)+ SE(X) > max X$ holds.

So for example, for any i.i.d. sample drawn from a Bernoulli with, say, $p=0.7$, in most cases the sample mean plus the sample standard deviation will exceed the value $1$, which will be the maximum value observed (bar the case of an all-zeros sample!).

Plus One Obvious Album

For other distributions we always have the opposite direction in the inequality, e.g. for a Uniform $U(a,b)$, it is always the case that $E(U)+ SE(U) < max U=b$.
Therefore, no general rule exists.

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Plus One was an American Christian popboy band which formed in 1999, disbanded in 2004, and briefly reunited in 2014. The group's line-up consisted of Nate Cole, Gabe Combs, Nathan Walters, Jeremy Mhire, and Jason Perry. They enjoyed their greatest success with the release of their debut album, The Promise (2000) which featured the hits 'Written on My Heart' and 'God Is in This Place'. The group then released their second album Obvious (2002). Jason Perry and Jeremy Mhire left the band in 2002, and the remaining three members signed a deal with Inpop Records. A year later they released their first album as a trio, Exodus (2003). After their Exodus '04 tour, they disbanded in 2004 to pursue other interests. Combs, Walters, and Perry briefly reunited in 2014, releasing a song titled 'My All'.

• 1History
• 2Discography

History[edit]

Formation and early years: 1999–2000[edit]

Plus One began to take shape as early as 1999 and formed by a manager named Mitchell Solarek, when 18-year-old Sacramento, California-boy Nate Cole was joined by 21-year-old Gabe Combs of Pittsburg, California, 22-year-old Nathan Walters of Lakeland, Florida and the 19-year-old Jeremy Mhire, who had been engrossed in vocal studies as a junior on full scholarship to Southwest Missouri State University. Cole and Combs had known each other from mutual music circles while growing up in Northern California and Nathan had initially heard about the new group through an acquaintance of Nate's. The group came to complete fruition with the arrival of Jason Perry, who up to that moment had been focusing on his role as starting fullback and defensive end for his high school football team. The youngest member of the group, The name Plus One refers to the members of the group, plus God. Atlantic Records pursued the formation of a boy band motivated by the success of such secular bands as 'N Sync and Backstreet Boys. Auditions were held, and these five young men, who were all raised in the Assembly of God Church, three of them being pastors' sons, came together and melded as a group, all firmly believing that the Lord had brought them together to be in Christian ministry, to do great things for the Gospel. Plus One got together as a quintet.

The Promise and 'Remarkable Success': 2000-2001[edit]

Their successful debut album, The Promise, was released in May 2000, and enjoyed great success in the Christian Pop scene. The Promise went gold with more than 500,000 album sales. Grammy Award Winning Producer David Foster was one of the producers on Plus One's albums. They had two No. 1 Christian radio hits, Four of the album's tracks is 'Written on My Heart', which was No. 1 on the charts for two months, and 'God Is in This Place' followed by 'Last Flight Out', 'Here in My Heart', and other radio singles are 'Run To You' and 'Soul Tattoo'. They found themselves touring with such Christian Artists as Jaci Velasquez, Rachael Lampa, and Stacie Orrico. In 2000, they appeared in an episode of hit soap opera Days of Our Lives to perform on the show. They performed with Monica on ABC's Christmas Eve special, 'Twas the Night Before Christmas'. In addition, Plus One was featured as one of the new artists of 2000 on TNN's annual television special, 'Class of 2000.' They gave performances at the 2000 Democratic National Convention, and several major league sporting events including games with the Los Angeles Lakers, Chicago Bulls, and Atlanta Hawks. The band was featured on Teen Magazine Online and performed at Atlanta Fest and the Raggedy Ann Festival in Arcola, IL, as part of their tour schedule. The following year in 2001, they appeared as juvenile kids who form a group while in jail on the TV series Touched by an Angel and they were nominated for a Dove Award as New Artist of the Year. The group recorded a soundtrack called 'I Need a Miracle' from the Christian Bible-based film Left Behind: The Movie released in 2000 followed by 'With All Your Heart' was also included on the soundtrack of the Warner Brothers film, Pokémon: The Movie 2000. They also appear on Natalie Grant's album Stronger with the song 'Whenever You Need Somebody' in 2001. They also covered the American patriotic song 'America the Beautiful' and they performed in Luis Palau Festival in 2001.

Obvious and Christmas: 2001-2002[edit]

Returning from the Promise tour, Plus One went back into the studio. Helped once again by some of the top producers in the music business, including Peter Zizzo, Peter Kipley, and Matthew Gerrard, the group recorded songs for a new album, this time including some of their own material, tracks like 'Use Me', a ballad called 'You', 'Under the Influence', 'Kick Me', and 'Going Crazy'. A strong beat and guitar riffs defined these new tracks, distinguishing them from the lighter sound of the group's first recording. Obvious was released in February 2002, and sold twice as many copies in its first week of release as Promise had. It hit the number-one spot on the Christian charts immediately. Three of the album's tracks, 'Camouflage,' became hit single occupying spots on the top-ten Christian radio charts simultaneously followed by 'I Don't Care' and 'Let Me Be The One'. In the fall of 2002 in their Obvious Tour, Plus One released Christmas a collection of Christmas music in time for the holiday season. In addition their own renditions of 'Silent Night,' 'O Little Town of Bethlehem,'What Child Is This?, 'O Holy Night', 'It's The Most Wonderful Time of the Year' and other Christmas standards, the album included some original compositions as well, including the opening track, 'This Is Christmas,' and 'I Won't Forget This Christmas' and a merry and charming song titled 'Our Christmas Prayer' and 'A Prayer For Every Year' which was released in WOW Christmas: Red in 2002.

Departure of Perry and Mhire, and Exodus: 2002-2003[edit]

In 2002, after completing touring for their Obvious tour, two members of Plus One were called to other ministries and obligations, Jason Perry left the group in 2002 followed by Jeremy Mhire leaving the remaining three members to carry on Plus One's music ministry. In 2003, the band was dropped from their label. Seeking to revamp their image, they signed a deal with Inpop Records and released their third album, the first time releasing an album without their two former members. Exodus. As a result, the band became a trio. Co-produced by former Newsboys frontman Peter Furler and keyboardist Jeff Frankenstein, Exodus marked a departure from the boy band image, leaving the all-out 'pop' style for a more mature sound featuring keyboards, guitars and the band performing their own materials. Peter Furler said that 'it's some of the best music I've ever been part of' and that he can feel justifiably proud of turning out this album. 'Be Love' is the hit single on this album, and was followed by 'Circle.' The album contains all original songs, with the exception of a rendition of the Tim Hughes hit 'Here I Am to Worship'.

Break-up and Hiatus: 2004-2016[edit]

In 2004, after completing the Exodus 04 tour, Plus One quietly parted ways with Inpop and with each other. On September 23, 2004, Nate Cole announced that Plus One would most likely be breaking up. Nate Cole and Gabe Combs went on to form Indie rock band Castledoor with their wives Liska Cole and Coury Combs and released 5 EP's and one album. In 2008, Nathan Walters went on to pursue a solo career, releasing one album called Escape in Nashville, TN. After Castledoor, Nate Cole and his wife Liska Cole started a duo band called Doom and Gloom and released home recordings.^{[citation needed]} In 2011 Gabe Combs and Nathan Walters started a music production called Multitone Musik to merge their production and songwriting skills. In December 2016, Nate Cole released his first solo album Bad Beat.^[1]

Reunion: 2014[edit]

On April 14, 2014, Plus One announced that they had reunited with three original members: Nathan Walters, Gabe Combs, and Jason Perry, who previously had quit the group in 2002. They also announced the release of their latest single, 'My All', available on their official website and their Facebook page.^[2]

Members[edit]

• Nate Cole (1999–2004) - vocals
• Jason Perry (1999–2002, 2014) - vocals
• Nathan Walters (1999–2004, 2014) - vocals, keyboards
• Jeremy Mhire (1999–2002) - vocals
• Gabe Combs (1999–2004, 2014) - vocals, guitars

Discography[edit]

Compilation contributions[edit]

Singles[edit]

As a featured artist[edit]

On television[edit]

Videography[edit]

References[edit]

1. ^13th, Kevin Bronson on December; 2016 (December 13, 2016). 'Ears Wide Open: Nate Cole'. Retrieved December 29, 2016.
2. ^'Plus One Returns, Offers First Single'. New Release Tuesday. April 14, 2014. Retrieved April 16, 2014.
3. ^'US Billboard 200 Albums Chart'. billboard.com Retrieved March 10, 2010
4. ^'US Billboard Top Christian Albums Chart'. billboard.com Retrieved March 10, 2010
5. ^'AllMusic charts > Chart history'. allmusicguide.com Retrieved May 13, 2010
6. ^'RIAA (Recording Industry Association of America) > Certifications list'. riaa.com Retrieved May 10, 2010