Stat 101 Formulas Sample Statistics - University of Illinois ...

Stat 101 Formulas

Sample Statistics

Sample mean

5-Number summary

?0 = ???????

?1 = 1?? ????????

?2 = ??????

?3 = 3?? ????????

?4 = ???????

Range

????? = ??????? ? ???????

= ?4 ? ?0

?

?? =

1

�� ??

?

?=1

Sample variance

?

�� ??2 ? ??? 2

1

2

2

? =

��(?? ? ?? ) =

??1

??1

?=1

Sample standard deviation

?

?=

��? 2

1

=��

��(?? ? ?? )2

??1

Inter-Quartile Range

??? = ?3 ? ?1

Fences for Outliers

?1 ? 1.5 ? ???, ?3 + 1.5 ? ???

?=1

Simple Linear Regression

Sample Covariance

Regression Model

?? = ? + ??

Slope

??

?=?

??

Intercept

? = ?? ? ???

Residual

?????? = ?? ? ??? = ?? ? (? + ??? )

�� ?? ?? ? ??? ??

???(?, ?) =

(? ? 1)

Sample Correlation

???(?, ?) �� ?? ?? ? ??? ??

?=

=

?? ??

(? ? 1)?? ??

Normal Distribution

Standardize

?=

???

?

Un-Standardize

? = ? + ??

68/95/99.7 Rule

?(?1 < ? < 1) �� .68

?(?2 < ? < 2) �� .95

?(?3 < ? < 3) �� .997

kth Percentile

? ???? ???? ?(? < ?) = ?%

Brian Powers, Summer 2014

Stat 101 Formulas

Probability

Complement Rule

?(?? ) = 1 ? ?(?)

General Addition Rule

?(? �� ?) = ?(?) + ?(?) ? ?(? �� ?)

Multiplication Rule for Independent Events

?(? �� ?) = ?(?) ? ?(?)

General Multiplication Rule

?(? �� ?) = ?(?) ? ?(?|?) = ?(?) ? (?(?|?)

Conditional Probability

?(? �� ?)

?(?|?) =

?(?)

A and B are Independent if:

1) ?(? �� ?) = ?(?) ? ?(?)

2) ?(?) = ?(?|?)

3) ?(?) = ?(?|?)

Random Variables

Expected Value

?

?

? = ?(?) = �� ?? ? ?(? = ?? ) = �� ?? ? ??

?=1

?=1

Variance

?

2

? = ???(?) = ?((? ? ?)

2)

= ?(?

2)

2

? ? = ��(? ? ?)2 ? ??

?=1

Linearity of Expected Value

?(??) = ??(?)

?(? + ?) = ?(?) + ?

?(? + ?) = ?(?) + ?(?)

Variance of a Linear Combination

???(??) = ?2 ???(?)

???(? + ?) = ???(?)

???(?? + ??) = ?2 ???(?) + ? 2 ???(?) + 2?????(?, ?)

Variance of Linear Combination of Independent X,Y

???(?? + ??) = ?2 ???(?) + ? 2 ???(?)

??(?1 + ?2 + ? + ?? ) = ��???(?)

Brian Powers, Summer 2014

Stat 101 Formulas

Special Distributions

Bernoulli(p)

?(? = 1) = ?

?(? = 0) = ? = 1 ? ?

?(?) = ?

???(?) = ??

Binomial(n,p)

Sum of n independent Bernoullis

?

?(? = ?) = ( ) ?? ? ??? = ????????(?, ?, ?)

?

?

?

?(? �� ?) = �� ( ) ?? ? ??? = ????????(?, ?, ?)

?

?=0

?(?) = ??

???(?) = ???

Central Limit Theorem

If ?1 , �� , ?? independent, come from a distribution with mean ? and standard deviation ?

?

?? approximately follows a Normal distribution with mean ? and standard deviation

.

��?

Sampling Distributions (assuming CLT applies)

If x1,��,xn ~Bernoulli(p)

�� ?? ~?????(?, ?) �� ?(??, ��???)

�� ??

??

~? (?, �� )

?

?

If x1,��,xn ~ have mean �� and standard deviation ��

?? =

�� ?? ~?(??, ��??)

?? =

�� ??

?

~? (?, )

?

��?

Confidence Intervals

(1-��)100% Confidence Interval

Estimate �� Margin of Error

Margin of Error = (# of Standard errors)*(Size of Standard Error)

Population proportion p (n large)

?? ??

?? �� ??/2 ? ��

?

Population difference p1-p2 (n1, n2 large)

??1 ??1 ?? 2 ??2

??1 ? ?? 2 �� ??/2 ? ��

+

?

?

?

Population mean �� (n��30, �� known)

?? �� ??/2 ?

��?

Brian Powers, Summer 2014

Stat 101 Formulas

Population mean �� (n ................
................

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