Mixing
Definition of sample mean
I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which one is correct.
Also if it is defined as the average of random variables why don't we just define it as the average of sample values?
statistics means sampling
asked Nov 11 '18 at 13:27
falcon lover
11
add a comment |
I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which one is correct.
Also if it is defined as the average of random variables why don't we just define it as the average of sample values?
statistics means sampling
asked Nov 11 '18 at 13:27
falcon lover
11
For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51
add a comment |
I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which one is correct.
Also if it is defined as the average of random variables why don't we just define it as the average of sample values?
statistics means sampling
asked Nov 11 '18 at 13:27
falcon lover
11
I've seen two definitions of sample mean on the internet.
One definition defines it as the average of Random variable other defines it as the average of sample values of a sample.
I'm confused which one is correct.
Also if it is defined as the average of random variables why don't we just define it as the average of sample values?
statistics means sampling
statistics means sampling
asked Nov 11 '18 at 13:27
falcon lover
11
asked Nov 11 '18 at 13:27
falcon lover
11
asked Nov 11 '18 at 13:27
falcon lover
11
asked Nov 11 '18 at 13:27
falcon lover
11
asked Nov 11 '18 at 13:27
falcon lover
11
11
For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51
add a comment |
For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51
For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51
For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51
add a comment |
2 Answers
2
active
oldest
votes
Given samples ${x_i}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $bar{x}$ defined as:
$bar{x} = frac{1}{n} sum_{i=1}^n x_i$
is an estimate of the expected value $mathbb{E}[X]$ of that distribution.
On the other hand, given RVs ${X_i}_{i=1}^n$ (drawn independently from distribution $F$), their average $bar{X}$ defined as:
$bar{X} = frac{1}{n} sum_{i=1}^n X_i$
is also a random variable with mean $mathbb{E}[bar{X}] = mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.
It is not meaningful to compare $bar{x}$ and $bar{X}$ directly.
answered Nov 21 '18 at 6:37
Aditya Dua
86418
add a comment |
In statistical probability, a single sampled value is a random variable.
If you have a sample consisting of $N$ values, then you have $N$ random variables.
The words "sample value" are just another name for one of these random variables.
So both methods of obtaining a "sample mean" do exactly the same thing.
The mean value of a random variable is a completely different thing.
Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural).
It changes the possible interpretation of what you write quite dramatically.
answered Nov 21 '18 at 6:46
David K
52.7k340115
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Given samples ${x_i}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $bar{x}$ defined as:
$bar{x} = frac{1}{n} sum_{i=1}^n x_i$
is an estimate of the expected value $mathbb{E}[X]$ of that distribution.
On the other hand, given RVs ${X_i}_{i=1}^n$ (drawn independently from distribution $F$), their average $bar{X}$ defined as:
$bar{X} = frac{1}{n} sum_{i=1}^n X_i$
is also a random variable with mean $mathbb{E}[bar{X}] = mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.
It is not meaningful to compare $bar{x}$ and $bar{X}$ directly.
answered Nov 21 '18 at 6:37
Aditya Dua
86418
add a comment |
Given samples ${x_i}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $bar{x}$ defined as:
$bar{x} = frac{1}{n} sum_{i=1}^n x_i$
is an estimate of the expected value $mathbb{E}[X]$ of that distribution.
On the other hand, given RVs ${X_i}_{i=1}^n$ (drawn independently from distribution $F$), their average $bar{X}$ defined as:
$bar{X} = frac{1}{n} sum_{i=1}^n X_i$
is also a random variable with mean $mathbb{E}[bar{X}] = mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.
It is not meaningful to compare $bar{x}$ and $bar{X}$ directly.
answered Nov 21 '18 at 6:37
Aditya Dua
86418
add a comment |
Given samples ${x_i}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $bar{x}$ defined as:
$bar{x} = frac{1}{n} sum_{i=1}^n x_i$
is an estimate of the expected value $mathbb{E}[X]$ of that distribution.
On the other hand, given RVs ${X_i}_{i=1}^n$ (drawn independently from distribution $F$), their average $bar{X}$ defined as:
$bar{X} = frac{1}{n} sum_{i=1}^n X_i$
is also a random variable with mean $mathbb{E}[bar{X}] = mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.
It is not meaningful to compare $bar{x}$ and $bar{X}$ directly.
answered Nov 21 '18 at 6:37
Aditya Dua
86418
Given samples ${x_i}_{i=1}^n$ (presumed to be drawn from an underlying distribution $F$, density $f$), the sample mean $bar{x}$ defined as:
$bar{x} = frac{1}{n} sum_{i=1}^n x_i$
is an estimate of the expected value $mathbb{E}[X]$ of that distribution.
On the other hand, given RVs ${X_i}_{i=1}^n$ (drawn independently from distribution $F$), their average $bar{X}$ defined as:
$bar{X} = frac{1}{n} sum_{i=1}^n X_i$
is also a random variable with mean $mathbb{E}[bar{X}] = mathbb{E}[X]$ and distribution given by an n-fold convolution of $f$.
It is not meaningful to compare $bar{x}$ and $bar{X}$ directly.
answered Nov 21 '18 at 6:37
Aditya Dua
86418
answered Nov 21 '18 at 6:37
Aditya Dua
86418
answered Nov 21 '18 at 6:37
Aditya Dua
86418
answered Nov 21 '18 at 6:37
Aditya Dua
86418
86418
add a comment |
add a comment |
In statistical probability, a single sampled value is a random variable.
If you have a sample consisting of $N$ values, then you have $N$ random variables.
The words "sample value" are just another name for one of these random variables.
So both methods of obtaining a "sample mean" do exactly the same thing.
The mean value of a random variable is a completely different thing.
Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural).
It changes the possible interpretation of what you write quite dramatically.
answered Nov 21 '18 at 6:46
David K
52.7k340115
add a comment |
In statistical probability, a single sampled value is a random variable.
If you have a sample consisting of $N$ values, then you have $N$ random variables.
The words "sample value" are just another name for one of these random variables.
So both methods of obtaining a "sample mean" do exactly the same thing.
The mean value of a random variable is a completely different thing.
Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural).
It changes the possible interpretation of what you write quite dramatically.
answered Nov 21 '18 at 6:46
David K
52.7k340115
add a comment |
In statistical probability, a single sampled value is a random variable.
If you have a sample consisting of $N$ values, then you have $N$ random variables.
The words "sample value" are just another name for one of these random variables.
So both methods of obtaining a "sample mean" do exactly the same thing.
The mean value of a random variable is a completely different thing.
Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural).
It changes the possible interpretation of what you write quite dramatically.
answered Nov 21 '18 at 6:46
David K
52.7k340115
In statistical probability, a single sampled value is a random variable.
If you have a sample consisting of $N$ values, then you have $N$ random variables.
The words "sample value" are just another name for one of these random variables.
So both methods of obtaining a "sample mean" do exactly the same thing.
The mean value of a random variable is a completely different thing.
Be careful not to write "variable" (singular) when the text you are getting your definitions from said "variables" (plural).
It changes the possible interpretation of what you write quite dramatically.
answered Nov 21 '18 at 6:46
David K
52.7k340115
answered Nov 21 '18 at 6:46
David K
52.7k340115
answered Nov 21 '18 at 6:46
David K
52.7k340115
answered Nov 21 '18 at 6:46
David K
52.7k340115
52.7k340115
add a comment |
add a comment |
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For a sample $X_1,X_2,ldots,X_n$ ($X_i$'s being random variables) of size $n$, the sample mean is the statistic $overline X=frac{1}{n}sum_{i=1}^n X_i$. If we denote the observed values of the sample by $x_1,ldots,x_n$, then the observed value of the sample mean is $bar x=frac{1}{n}sum_{i=1}^n x_i$. I think you are asking the same question as here.
– StubbornAtom
Nov 11 '18 at 13:51