Statistical Formulas

[QMM] Statistical formulas

1. Mean The mean, or average, of a collection of numbers x1 , x2 , . . . , xN  is x ¯=

x1 + x2 + N 

··· + x

N 

=

1 N 



xi .

2. Standard Standard deviation deviation The standard deviation is defined as S  =

 (

x1

2

− x¯)

+

2

· · · + (x − x¯) N  − 1 N 

  1  = ( N 

−1

xi

2

− x¯)

.

One may find in some textbooks an alternative version, with N  in the denomi denominat nator. or. When When the author author wishes wishes to distinguish between between both b oth versions, versions, the ‘ N ’ version is presented as the population standard deviation, while the ‘N  1’ is the sample standard deviation.

−

3. The normal distribution The normal density curve is given by a function of the form f (x) =

1 exp 2πσ

√ 



2

(x

− 2−σ µ) 2



.

In this formula, µ y σ are two parameters which are different for each application of the model. A normal normal densit density y curve curve has a bell shape (Figure (Figure 1). The paramet parameter er µ, called the population mean, has an straightforward interpretation: the density curve peaks at x = µ. The parameter σ , called the population standard deviation measures the spread of the distribution: the higher σ , the flatter the bell. The case µ = 0, σ = is called the standard normal. Probabilities for the normal distribution are calculated as (numerical) integrals of the density. For most people, the only probability needed is



p µ

− 1.96σ < X < µ + 1.96σ = 0.95.

This This formu formula la provid provides es us with with an inter interv val which which contain containss 95% of the populatio population. n. The “tails” “tails” contain the remaining 5%.

4. Confidence Confidence limits for for the mean The formula for the 95% confidence limits for the mean is ¯ x [QMM] Statistical formulas 1

± 1.96 √ S N  . 2012–03–01

        5   .         0

        4   .         0

        3   .         0

        2   .         0

        1   .         0

        0   .         0

−4

−2

0

2

4

Figure 1. Three normal density curves

Here, N  is the number of data points, x¯ the sample mean and S  the sample standard deviation. Textbooks recommend replacing the factor 1.96, derived from the normal distribution, by a factor taken from the Student t distribution, but the correction becomes irrelevant when N  is high.

5. Correlation For two dimensional data ( x1 , y1 ), (x2 , y2 ), . . . , (xN , yN ), the (linear) correlation is

R

  − ¯)( − ¯ =     − ¯  − ¯ xi

xi

Always

x yi

x

2

y

yi

y

2

.

−1 ≤ R ≤ 1. 6. Coefficients of the regression line

Given N  data points (x1 , y1 ), ( x2 , y2 ), . . . , ( xN , yN ), the regression line has an equation y = b0 +b1 x, in which b0 and b1 are the regression coefficients: b1 is the slope, and b0 the intercept. The formulas are b1 = R

S Y   , S X

b0 = y¯

− b x¯. 1

R is the linear correlation. y¯ and x ¯ are the means of  Y  and Y , respectively. S Y   and S X are the

standard deviations.

7. R square statistic In a linear regression equation, the R2 statistic is the proportion of the total variability of the [QMM] Statistical formulas 2

2012–03–01

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Figure 2. Regression lines with R = 0.8 and R =

−1

0

1

2

3

−0.2

dependent variable explained by the equation R2 =

Explained variability . Total variability

More explicitly, if  y1 , y2 , . . . , yN  are the observed valued of the dependent variable Y  , with mean y¯ and yˆ1 , yˆ2 , . . . , yˆN  are the values predicted by the equation, R

2

 ˆ − ¯ =  − ¯ yi

y

yi

y

2 2

.

Always 0 1. In simple regression (a single independent variable), R2 coincides with the R2 square of the correlation.

≤

≤

8. Adjusted R square An adjusted R2 statistic, defined as 2

Adjusted R2 = 1

− 1) , − (1 −N R−)( p N  −1

is used sometimes to compare regression equations. N  is the number of data points and p the number of independent variables in the equation. The adjustment becomes irrelevant when N  is high.

[QMM] Statistical formulas 3

2012–03–01