statistic

statistic

A

statistic

, or

sample statistic

,

S

is simply a

function

, usually real-valued, of a set of (sample) data or

observations

𝑿

=

(

X

1

,

X

2

,

…

,

X

n

)

:

S

=

S

⁢

(

𝑿

)

. More formally, let

Ω

be the

sample space

of the data

𝑿

, then

S

is a function from

Ω

to some set

T

, usually a subset of

ℝ

k

. The data

𝑿

is usually considered as a vector of iid

random variables

X

i

.

Examples

1.

100 light bulbs out of 1,000,000 are tested for their functionality. Then the number

n

, of defective light bulbs in the 100 samples is a statistic. To see this, define, for each

i

from 1 to 100,

x

i

=

{

1

if the event

⁢

X

i

=

{

the

⁢

i

⁢

th light bulb is defective

}

0

otherwise.

Then

n

=

∑

i

=

1

100

x

i

, a function of the data. Similarly, the number of operating light bulbs is also a statistic if we switch the 1 and 0 in the above definitions for the

x

i

’s. If we make all

x

i

=

1

, then

n

is just the
count of the observations, one of the simplest forms of sample statistics. If we make all

x

i

=

0

, then

n

=

0

is a statistic that is not at all useful.

2.

Let

w

1

,

w

2

,

…

,

w

20

be the weights of 20 students from a particular college. Then the

average

weight defined by

w

¯

=

1

20

⁢

∑

i

=

1

20

w

i

is a statistic. It is commonly called the

sample mean

. It is often used to

estimate

E

⁡

[

X

]

, the

expectation

of a particular random variable, which, in this case, is the weight of a student in the college. Of course, other averages, such as medians, mode,

trimmed mean

, are also examples of (sample) statistics.

3.

Using the same example as in (2), we can define

s

2

=

1

20

-

1

⁢

∑

i

=

1

20

(

w

i

-

w

¯

)

2

.

This is also a statistic, for, after some substitution and rewriting,

s

2

=

1

20

-

1

⁢

[

∑

i

=

1

20

w

i

2

-

1

20

⁢

(

∑

i

=

1

20

w

i

)

2

]

,

which is a function explicitly in terms of the

w

i

’s. This statistic is known as the

sample variance

, which is a common estimate of

Var

⁡

[

X

]

, the

variance

of the random variable

X

. Again, in this example, the

X

is the weight of a student in the college.

4.

Again, borrowing from the same example above, we can simply order the weights of the 20 students in an

ascending order

, so we get a vector of 20

real numbers

(

w

(

1

)

,

w

(

2

)

,

…

,

w

(

20

)

)

. This is also a statistic, called an

order statistic

. It is not real-valued and its range is a subset of

ℝ

20

.

5.

Given a set of numeric observations

X

1

,

X

2

,

…

,

X

n

, without knowing the

distribution

of these observations, one can define what is known as the

empirical distribution function

F

^

, which is a real-valued function, based on the observations. This is an example of a statistic whose range is a

function space

.

Remarks

.

•

Any function of a statistic is again a statistic.

•

Since the underlying data is assumed to be random, a statistic is necessarily a random variable.

•

Although mostly real-valued, a statistic can be vector-valued, or even function-valued as we have seen in earlier examples.

Title

statistic

Canonical name

Statistic

Date of creation

2013-03-22 14:46:18

Last modified on

2013-03-22 14:46:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Defines

sample mean

Defines

sample variance

Generated on Sat Feb 10 12:03:12 2018 by

LaTeXML