Formulas for AP Statistics

I. Descriptive Statistics

1 ii

xx x

n n ¦¦ � � � �2

211 1

ix i

x xs x x

n n�

� � �

¦¦

y a bx � y a bx �

11

i i

x y

x x y yrn s s

§ ·§ ·� � ¨ ¸¨ ¸¨ ¸� © ¹© ¹

¦ y

x

sb r

s

II. Probability and Distributions

� � � � � � � �P A B P A P B P A B� � � �( )( | )

( )P A BP A BP B�

Probability Distribution Mean Standard Deviation

Discrete random variable, X ( ) ( )X i iE X x P xµ = = ⋅¦ ( ) ( )2X i X ix P xσ µ= − ⋅¦

If X has a binomial distribution with parameters n and p, then:

� � � �1 n xxnP X x p p

x�§ ·

�¨ ¸© ¹

where x 0, 1, 2, 3, ! , n

X npP � �1X np pV �

If X has a geometric distribution with parameter p , then:

� � � � 11 xP X x p p� �

where x 1, 2 , 3 , !

1X p

P 1X

pp

V�

III. Sampling Distributions and Inferential Statistics

Standardized test statistic: statistic parameterstandard error of the statistic

−

Confidence interval: ( )( )statistic critical value standard error of statistic±

Chi-square statistic:

( )22 observed expected

expectedχ

−=¦

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Appendix V.1 | 254AP Statistics Course and Exam Description

III. Sampling Distributions and Inferential Statistics (continued)Sampling distributions for proportions:

Random Variable Parameters of Sampling DistributionStandard Error* of Sample Statistic

For one population:p p pP

� �ˆ

1p

p pn

V�

� �

ˆ

ˆ ˆ1p

p ps

n�

For two populations:1 2

ˆ ˆp p− 1 2ˆ ˆ 1 2p p p pP � �

� � � �1 2

1 1 2 2ˆ ˆ

1 2

1 1p p

p p p pn n

V �

� � �

� � � �1 2

1 1 2 2ˆ ˆ

1 2

ˆ ˆ ˆ ˆ1 1p p

p p p ps

n n�

� � �

When 1 2p p is assumed:

( )1 2ˆ ˆ

1 2

1 1ˆ ˆ1c cp ps p pn n−

§ ·= − +¨ ¸

© ¹

where 1 2

1 2

ˆc

X Xpn n�

�

Sampling distributions for means:

Random Variable Parameters of Sampling DistributionStandard Error* of Sample Statistic

For one population:X XP P X n

VV Xssn

For two populations:1 2X X�

1 2 1 2X XP P P� �1 2

2 21 2

1 2X X n n

V VV � �

1 2

2 21 2

1 2X X

s ssn n� �

Sampling distributions for simple linear regression:

Random Variable Parameters of Sampling DistributionStandard Error* of Sample Statistic

For slope:b bP E V V

b V x n ,

where � �2i

x

x xn

V�

¦

1bx

sss n

�

,

where � �2

2i iy y

sn�

�

¦ �

and � �2

1i

x

x xs

n�

�

¦

*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.

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Appendix V.1 | 255AP Statistics Course and Exam Description