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What is the name for the average of the largest and the smallest values in a given data set?

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14

$begingroup$

What do you call a statistical mean that is calculated from upper and lower extremes in any given dataset?

For example, if you have a set:

 -2, 0 , 8, 9, 1, 50, -2, 6

The upper extreme of this set is 50 and lower extreme is -2. So, average of the extremes would be (-2 + 50 / 2) = 48/2 = 24

Is there a term for this kind of statistical mean?

mean terminology average range

share|cite|improve this question

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

$endgroup$

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  $begingroup$
  It's the "midrange".
  $endgroup$
  – jbowman
  Jul 18 at 22:07
  

  
  
  
  

add a comment
 | 

14

$begingroup$

What do you call a statistical mean that is calculated from upper and lower extremes in any given dataset?

For example, if you have a set:

 -2, 0 , 8, 9, 1, 50, -2, 6

The upper extreme of this set is 50 and lower extreme is -2. So, average of the extremes would be (-2 + 50 / 2) = 48/2 = 24

Is there a term for this kind of statistical mean?

mean terminology average range

share|cite|improve this question

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

$endgroup$

• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  $begingroup$
  It's the "midrange".
  $endgroup$
  – jbowman
  Jul 18 at 22:07
  

  
  
  
  

add a comment
 | 

$begingroup$

What do you call a statistical mean that is calculated from upper and lower extremes in any given dataset?

For example, if you have a set:

 -2, 0 , 8, 9, 1, 50, -2, 6

The upper extreme of this set is 50 and lower extreme is -2. So, average of the extremes would be (-2 + 50 / 2) = 48/2 = 24

Is there a term for this kind of statistical mean?

mean terminology average range

share|cite|improve this question

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

$endgroup$

 -2, 0 , 8, 9, 1, 50, -2, 6

share|cite|improve this question

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

share|cite|improve this question

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

edited Jul 19 at 11:06

amoeba says Reinstate Monica

68.1k1919 gold badges231231 silver badges280280 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

asked Jul 18 at 22:01

blackbeardblackbeard

7344 bronze badges

1 Answer
1

active

oldest

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18

$begingroup$

It's called the midrange and while it's not the most widely used statistic in the world it does have some relevance to the uniform distribution.

Let's introduce the order statistic notation: if have $n$ i.i.d. random variables $X_1, ..., X_n$, then the notation $X_(i)$ is used to refer to the $i$-th largest of the set $X_1, ..., X_n$. Thus we have:

$$ X_(1) ≤ X_(2) ≤···≤ X_(n) tag1 $$

Where $X_(1)$ is the minimum and $X_(n)$ is the maximum element. Then range and midrange are defined as:

$$ beginalign
R & = X_(n) - X_(1) tag2 \
A & = fracX_(1) + X_(n)2 tag3 \
endalign
$$

These formulas are taken from CRC Standard Probability and Statistics Tables and Formulae, section 4.6.6.

If $X_i$ is assumed to have a uniform distribution $X_i sim U(alpha, beta)$, where $alpha$ and $beta$ are the lower and upper bounds respectively, then we can give the MLE estimates in terms of these formulas:

$$
beginalign
hatalpha & = X_(1) tag4 \
hatbeta & = X_(n) tag5
endalign
$$

The mean of the resulting distribution is the same as the midrange:

$$
beginalign
mu & = A = fracX_(1) + X_(n)2 tag6 \
endalign
$$

This is probably the only use for this particular statistic.

share|cite|improve this answer

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Historicaly, the mean air temperature of a day was given as the midrange.
  $endgroup$
  – Maxter
  Jul 19 at 17:05
  

  
  
  
  

add a comment
 | 

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18

$begingroup$

It's called the midrange and while it's not the most widely used statistic in the world it does have some relevance to the uniform distribution.

Let's introduce the order statistic notation: if have $n$ i.i.d. random variables $X_1, ..., X_n$, then the notation $X_(i)$ is used to refer to the $i$-th largest of the set $X_1, ..., X_n$. Thus we have:

$$ X_(1) ≤ X_(2) ≤···≤ X_(n) tag1 $$

Where $X_(1)$ is the minimum and $X_(n)$ is the maximum element. Then range and midrange are defined as:

$$ beginalign
R & = X_(n) - X_(1) tag2 \
A & = fracX_(1) + X_(n)2 tag3 \
endalign
$$

These formulas are taken from CRC Standard Probability and Statistics Tables and Formulae, section 4.6.6.

If $X_i$ is assumed to have a uniform distribution $X_i sim U(alpha, beta)$, where $alpha$ and $beta$ are the lower and upper bounds respectively, then we can give the MLE estimates in terms of these formulas:

$$
beginalign
hatalpha & = X_(1) tag4 \
hatbeta & = X_(n) tag5
endalign
$$

The mean of the resulting distribution is the same as the midrange:

$$
beginalign
mu & = A = fracX_(1) + X_(n)2 tag6 \
endalign
$$

This is probably the only use for this particular statistic.

share|cite|improve this answer

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Historicaly, the mean air temperature of a day was given as the midrange.
  $endgroup$
  – Maxter
  Jul 19 at 17:05
  

  
  
  
  

add a comment
 | 

18

$begingroup$

It's called the midrange and while it's not the most widely used statistic in the world it does have some relevance to the uniform distribution.

Let's introduce the order statistic notation: if have $n$ i.i.d. random variables $X_1, ..., X_n$, then the notation $X_(i)$ is used to refer to the $i$-th largest of the set $X_1, ..., X_n$. Thus we have:

$$ X_(1) ≤ X_(2) ≤···≤ X_(n) tag1 $$

Where $X_(1)$ is the minimum and $X_(n)$ is the maximum element. Then range and midrange are defined as:

$$ beginalign
R & = X_(n) - X_(1) tag2 \
A & = fracX_(1) + X_(n)2 tag3 \
endalign
$$

These formulas are taken from CRC Standard Probability and Statistics Tables and Formulae, section 4.6.6.

If $X_i$ is assumed to have a uniform distribution $X_i sim U(alpha, beta)$, where $alpha$ and $beta$ are the lower and upper bounds respectively, then we can give the MLE estimates in terms of these formulas:

$$
beginalign
hatalpha & = X_(1) tag4 \
hatbeta & = X_(n) tag5
endalign
$$

The mean of the resulting distribution is the same as the midrange:

$$
beginalign
mu & = A = fracX_(1) + X_(n)2 tag6 \
endalign
$$

This is probably the only use for this particular statistic.

share|cite|improve this answer

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Historicaly, the mean air temperature of a day was given as the midrange.
  $endgroup$
  – Maxter
  Jul 19 at 17:05
  

  
  
  
  

add a comment
 | 

$begingroup$

It's called the midrange and while it's not the most widely used statistic in the world it does have some relevance to the uniform distribution.

Let's introduce the order statistic notation: if have $n$ i.i.d. random variables $X_1, ..., X_n$, then the notation $X_(i)$ is used to refer to the $i$-th largest of the set $X_1, ..., X_n$. Thus we have:

$$ X_(1) ≤ X_(2) ≤···≤ X_(n) tag1 $$

Where $X_(1)$ is the minimum and $X_(n)$ is the maximum element. Then range and midrange are defined as:

$$ beginalign
R & = X_(n) - X_(1) tag2 \
A & = fracX_(1) + X_(n)2 tag3 \
endalign
$$

These formulas are taken from CRC Standard Probability and Statistics Tables and Formulae, section 4.6.6.

If $X_i$ is assumed to have a uniform distribution $X_i sim U(alpha, beta)$, where $alpha$ and $beta$ are the lower and upper bounds respectively, then we can give the MLE estimates in terms of these formulas:

$$
beginalign
hatalpha & = X_(1) tag4 \
hatbeta & = X_(n) tag5
endalign
$$

The mean of the resulting distribution is the same as the midrange:

$$
beginalign
mu & = A = fracX_(1) + X_(n)2 tag6 \
endalign
$$

This is probably the only use for this particular statistic.

share|cite|improve this answer

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

$endgroup$

share|cite|improve this answer

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges

answered Jul 18 at 23:16

olooneyolooney

2,36999 silver badges2020 bronze badges