STATISTICS FORMULAS (30001) PDF

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DESCRIPTIVE STATISTICSMean𝑥̅=!!"!""⋯"!null$Range𝑚𝑎𝑥𝑖𝑚𝑢𝑚−𝑚𝑖𝑛𝑖𝑚𝑢𝑚Interquartile range𝑄!−𝑄"Outliers are at 𝑄!+1(𝑄!−𝑄") 𝑎𝑛𝑑 𝑄"−1(𝑄!−𝑄")Variance𝑠%=( ! !'!̅)""⋯"( ! #'!̅)"$'*Standard deviation𝑠 =%(!!'!̅)""⋯"(!null'!̅)"$'*Coefficient of variationCV = |!̅ |Concentration curve𝐹 -= $𝑄 -=!!"!""⋯"!$! !"! ""⋯"! #...

Description

DESCRIPTIVE STATISTICS Mean 𝑥" =

!! "!" "⋯"!# $

Range 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚

Interquartile range 𝑄! − 𝑄" Outliers are at 𝑄! + 1.5(𝑄! − 𝑄" )////𝑎𝑛𝑑 ////𝑄" − 1.5(𝑄! − 𝑄" ) Variance 𝑠% =

(!! '!()" "⋯"(!# '!()" $'*

Standard deviation 𝑠=%

( !! '!()" "⋯"(!# '!()" $'*

Coefficient of variation CV =

+

|!( |

Concentration curve

𝐹- = )) $ -

𝑄- =

!! "!" "⋯"!$ !! "!" "⋯"!#

Gini’s index

𝑅=

(.! '/!)"⋯"(.#%! '/#%! ) .! "." "⋯".#%!

Pietra’s index 𝑃) =

$

$'*

∗ max)(𝐹- − 𝑄- )

Conditional relative distribution 01-$2345672-853945:;5$74?-$763@-+24-A;2-1$

Covariance 𝐶𝑜𝑣 (𝑋, 𝑌) =

(!! '!()(=! '=B)"⋯" (!# '!()(=# '=B) $'*

Pearson’s correlation index

𝑟=

C18(D,F) +& +'

PROBABILITY Discrete random variable

𝐸(𝑋) = 𝑥* 𝑝* + 𝑥% 𝑝% + ⋯ + 𝑥G 𝑝G 𝑉𝑎𝑟 ( 𝑋) = (𝑥* − µ) % p* + ⋯ + (xH − µ ) % 𝑝G

Bernoulli random variable

Notation: 𝑋~𝐵𝑒𝑟(𝑝) 𝐸 (𝑋 ) = 𝑝 𝑉𝑎𝑟(𝑋) = p(1 − p)

Linear transformation of one random variable Linear transformation of X: 𝑌 = 𝑎 + 𝑏𝑋 𝐸 (𝑌) = 𝑎 + 𝑏𝐸(𝑋) 𝑉𝑎𝑟(𝑌) = b% 𝑉𝑎𝑟(𝑋)

Standardization of one random variable Standardization of X: 𝑍 = 𝐸(𝑍) = 0 𝑉𝑎𝑟(𝑍) = 1

D'I J

Sum of two random variables

𝐸(𝑋 + 𝑌) = ) 𝜇! + 𝜇= 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝜎%! + 𝜎=% + 2𝐶𝑜𝑣 (𝑋, 𝑌) 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝜎%! + 𝜎=%

(if correlated) (if not correlated)

Difference of two random variables

𝐸 (𝑋 − 𝑌 ) = ) 𝜇! − 𝜇= 𝑉𝑎𝑟(𝑋 − 𝑌) = 𝜎%! + 𝜎=% − 2𝐶𝑜𝑣 (𝑋, 𝑌) 𝑉𝑎𝑟(𝑋 − 𝑌) = 𝜎%! + 𝜎=%

(if correlated) (if not correlated)

Sum of random variables and Central Limit Theorem 𝐸(𝑋) = 𝑛𝜇 𝑉𝑎𝑟 ( 𝑋) = 𝑛𝜎%

Average of random variables and Central Limit Theorem 𝐸(𝑋) = 𝜇 𝑉𝑎𝑟 ( 𝑋) = 𝜎% /𝑛

Sum of Bernoulli random variables and Central Limit Theorem 𝐸 (𝑋 ) = 𝑛𝑝 𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 − 𝑝)

Average of Bernoulli random variables and Central Limit Theorem 𝐸(𝑋) = 𝑝 𝑉𝑎𝑟(𝑋) = 𝑝(1 − 𝑝)/𝑛

INFERENTIAL STATISTICS CONFIDENCE INTERVALS Normal population, unknown mean, known variance $ %& # Standard normal distribution: K Confidence interval: (𝑥1− 𝑧N

Margin of error: 𝑧N O

'

O

√)

/ , 𝑥1+ 𝑧N )

'

√M

√

O

'

√)

/)

Normal population with unknown mean, unknown variance Student’s t distribution :

$ %& # P

Confidence interval: (𝑥1− 𝑡)%", N √) / , 𝑥1+ 𝑡)%", N √)/) √M

O

+

O

+

Unknown population, estimated mean, estimated variance #$ %&

Standard normal distribution: Confidence interval: (𝑥1− 𝑧N

+

P √M

O √)

/ , 𝑥1+ 𝑧N O

+

√)

)

Bernoulli distribution and population proportion , -%. Standard normal distribution: Confidence interval: (𝑝6 − 𝑧N7 Standard error: 7

.("%.) )

O

/Q(RSQ) M

, 𝑝6 + 𝑧N 7

0 ("%.0) . )

O

)

.0("%.0) )

INFERENTIAL STATISTICS HYPOTHESIS TESTING One-sided test with normal population and known variance 𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≤ 𝜇3 // 𝐻" :/𝜇 > 𝜇3 ' 𝑥1> 𝜇3 + 𝑧4 Rejects H0 if: 56%&T K

√)

> 𝑧4

p-value (reject null hypothesis. if p-value < 𝛼): √M

#$%& Pr(𝑋D > 𝑥 1| 𝐻3 : 𝜇 = 𝜇3 ) = Pr F K T > √M

𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≥ 𝜇3 // 𝐻" :/𝜇 < 𝜇3 ' 𝑥1< 𝜇3 − 𝑧4 Rejects H0 if: √) 56 %&T K √M

56 %&T K √M

G𝐻3 : 𝜇 = 𝜇3H

< −𝑧4

#$%& 𝐻3 : 𝜇 = 𝜇3) = Pr F K T < p-value: Pr (𝑋D < 𝑥1| √M

𝐻3 : 𝜇 = 𝜇3 /// 𝐻" :/𝜇 ≠ 𝜇3 Rejects H0 if:

𝑥1> 𝜇3 + 𝑧4

|56 %&T | K

p-value: Pr F

$ %&T | |# K √M

>

√M

> 𝑧U

|56%&T | K √M

'

√)

56%&T K √M

or

O

G𝐻3 : 𝜇 = 𝜇3H

𝑥1< 𝜇3 − 𝑧4

G𝐻3 : 𝜇 = 𝜇3H = 2 Pr F

$ %&T | |# K √M

>

'

√)

|56%&T | K √M

G𝐻3 : 𝜇 = 𝜇3 H

One-sided test with normal population and unknown variance 𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≤ 𝜇3 // 𝐻" :/𝜇 > 𝜇3 56 %&T Rejects H0 if: > 𝑡)%",4 P

p-value (reject null hypothesis. if p-value < 𝛼): √M

Pr F

#$%&T P √M

>

56%&T P √M

G𝐻3 : 𝜇 = 𝜇3 H

𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≥ 𝜇3 // 𝐻" :/𝜇 < 𝜇3

Rejects H0 if:

56%&T P √

< −𝑡)%",4

#$%& p-value: Pr (𝑋D < 𝑥1𝐻| 3 : 𝜇 = 𝜇3) = Pr F P T < M

𝐻3 : 𝜇 = 𝜇3 /// 𝐻" :/𝜇 ≠ 𝜇3 Rejects H0 if:

√

|56%&T | K √M

6 5%&T P

√M

M

> 𝑡)%",4

#$%& p-value: Pr (𝑋D < 𝑥1𝐻| 3 : 𝜇 = 𝜇3) = 2 Pr F P T <

G𝐻3 : 𝜇 = 𝜇3H

G𝐻3 : 𝜇 = 𝜇3H

56 %&T P √M

√M

One-sided asymptotic test and estimated mean and variance 𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≤ 𝜇3 // 𝐻" :/𝜇 > 𝜇3 56 %&T Rejects H0 if: > 𝑧4 P p-value (reject null hypothesis. if p-value < 𝛼): √M

Pr F

#$%&T P √M

>

56%&T P √M

G𝐻3 : 𝜇 = 𝜇3 H

𝐻3 : 𝜇 = 𝜇3 /𝑜𝑟/𝜇 ≥ 𝜇3 // 𝐻" :/𝜇 < 𝜇3 56 %&T < −𝑧4 Rejects H0 if: P √M

p-value: Pr (𝑋D < 𝑥1𝐻| 3 : 𝜇 = 𝜇3) = Pr F 𝐻3 : 𝜇 = 𝜇3 /// 𝐻" :/𝜇 ≠ 𝜇3 Rejects H0 if:

|56%&T | K √M

> 𝑧U O

#$%&T P √M

p-value: Pr (𝑋D < 𝑥1𝐻| 3 : 𝜇 = 𝜇3) = 2 Pr F

<

#$%&T P √M

56%&T

<

P √M

G𝐻3 : 𝜇 = 𝜇3H

G𝐻3 : 𝜇 = 𝜇3H

56 %&T P √M

One-sided Bernoulli distribution with parameter p 𝐻3 : 𝜇 = 𝜇3 / 𝐻" :/𝜇 > 𝜇3 Rejects H0 if:

,$%.T

Q (RSQT ) / T √M

> 𝑧4

p-value (reject null hypothesis. if p-value < 𝛼):Pr M

,$ %.T

Q (RSQT ) / T √M

>

.6 %.T

Q (RSQT ) / T √M

N𝐻3 : 𝑝 = 𝑝3O

𝐻3 : 𝜇 = 𝜇3 / 𝐻" :/𝜇 < 𝜇3

Rejects H0 if: p-value: Pr M

𝐻3 : 𝜇 = 𝜇3 /// 𝐻" :/𝜇 ≠ 𝜇3

,$ %.T

/

Q (RSQT ) / T √M

, $ %.T

QT (RSQT ) √ M

<

.6 %.T

Q (RSQT ) / T √M

|,$ %.T |

Rejects H0 if:

Q (RSQT ) / T √M

p-value: 2 Pr M

|, $ %.T |

Q (RSQT ) / T √M

< −𝑧4

≥

N𝐻3 : 𝑝 = 𝑝3O

> 𝑧U

O

|.6 %.T |

N𝐻3 : 𝜇 = 𝜇3 O

Q (RSQT ) / T √M

Both normal populations 𝐻3 : 𝜇# − 𝜇8 = 0/𝑜𝑟/𝜇# − 𝜇8 ≤ 0// 𝐻" :/𝜇# − 𝜇8 > 0 #$%8$ > 𝑡)W :)X%;,4 Rejects H0 if: P P 9 V :: V O

O

𝐻3 : 𝜇# − 𝜇8 = 0/𝑜𝑟/𝜇# − 𝜇8 ≥ 0// 𝐻" :/𝜇# − 𝜇8 < 0 MW

Rejects H0 if:

𝐻3 : 𝜇# − 𝜇8 = 0// 𝐻" :/𝜇# − 𝜇8 ≠ 0

Rejects H0 if:

MX

#$%8$

O PO P 9 V :: V MW MX

|#$ %8$ |

PO PO 9 V :: V MW MX

< −𝑡)W :)X%;,4

> 𝑡)W :)X%;,4

Two populations 𝐻3 : 𝜇# − 𝜇8 = 0/𝑜𝑟/𝜇# − 𝜇8 ≤ 0/ 𝐻" :/𝜇# − 𝜇8 > 0 #$%8$ > 𝑧4 Rejects H0 if: P 9 V :: PV O

MW

O

MX

𝐻3 : 𝜇# − 𝜇8 = 0/𝑜𝑟/𝜇# − 𝜇8 ≥ 0// 𝐻" :/𝜇# − 𝜇8 < 0

Rejects H0 if:

#$%8$

P P 9 V :: V < O

O

MW

𝐻3 : 𝜇# − 𝜇8 = 0// 𝐻" :/𝜇# − 𝜇8 ≠ 0 Rejects H0 if:

MX

|#$ %8$ |

O PO P 9 V :: V MW MX

−𝑧4

> 𝑧U

O

Chi square Null hypothesis: 𝐹𝑟(𝑋 = 𝑖, 𝑌 = 𝑗) = 𝐹𝑟(𝑋 = 𝑖) ∗ 𝑃𝑟(𝑌 = 𝑗) = < Ideal independence condition: 𝑛𝐹𝑟(𝑋 = 𝑖, 𝑌 = 𝑗) = 𝑛𝐹𝑟(𝑋 = 𝑖) ∗ 𝑃𝑟(𝑌 = 𝑗 ) = 𝑛 )Y ∗ )Y

Expected absolute frequency in case of independence: 𝐸>? =

E" ∑D?E" 𝐻3 : 𝑋/𝑎𝑛𝑑 /𝑌/𝑎𝑟𝑒/𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 / 𝐻" :/𝑋/𝑎𝑛𝑑 /𝑌/𝑎𝑟𝑒/𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 Rejects H0 if: 𝑋 = ∑F>E" ∑D?E"

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