Average \ (\ MU \) ; variance \ (\ Sigma ^ 2 \) ; standard deviation \ (\ Sigma \)
Average (also known as mathematical expectation)
For data:
\ [x_1 \ x_2 \ X_3 \ X_4 \ cdots \ x_n \]
Average:
\ [\ mu = \ frac {
1} {n} \ cdot \ sum_ {i = 1} ^ {n} {x_i} \] language interpreter: the average is divided by the number of all data and add up the number n.
Mathematical meaning: middle position data in numerical values.
Variance and standard deviation
Before variance variance and standard deviation detail, brush up on the Pythagorean theorem (also known in the West Pythagorean theorem ) and flat-screen distance between two points formula .
In the right triangle, to the side length a, b, c have the following relationship:
\ [^ 2 = C 2 + A ^ B ^ 2 \]
ie \ (c = \ sqrt {a ^ 2 + b ^ 2}; a = \ sqrt {c ^ 2 -b ^ 2} \)
Xoy plane coordinate system at any two points on \ (P (x_1, y_1) Q (x_2, y_2) \) the distance D between are:
\ [D = \ sqrt {(x_1-x_2) ^ 2- (y_1- y_2) ^ 2} \]
As it can be seen by the Pythagorean theorem and equations plane distance between two points, such as the type \ (A = \ sqrt {C 2} ^ -b ^ 2 \)
\ (D = \ sqrt {(x_1-x_2) ^ 2- ( y_1-y_2) ^ 2} \ ) meaning represented by the distance between the two. The smaller the value, the closer the proof between the two.
A set of data, the average number is this data center, then you can use other data to measure the distance to the mean distance relationship data and average. That this set of data is gathered some of it, or spread some of it.
Variance \ (\ Sigma ^ 2 \)
\ [\ Sigma ^ 2 = \ n-FRAC {} {}. 1 \ CDOT {\ sum_. 1} ^ {n-I = (x_i- \ MU)} ^ 2 \]
Because the distance D prescription is required, so the meaning of the variance is the square of the distance . The square root of the variance is called the standard deviation \ (\ Sigma \) .
\ [\ sigma = \ sqrt { \ sigma ^ 2} = \ sqrt {\ frac {1} {n} \ cdot {\ sum_ {i = 1} ^ n (x_i- \ mu) ^ 2}} \]