Formulas and Tables for AP Statistics
I. Descriptive Statistics
1
i
i
x
xx
nn
∑
=∑=
( )
( )
2
2
1
11
i
xi
xx
s xx
nn
∑−
= ∑− =
−−
ˆ
y a bx= +
y a bx= +
1
1
ii
xy
xxyy
r
n ss
−−
= ∑
−
y
x
br
s
=
s
II. Probability and Distributions
( ) ( ) ( ) ( )
PA B PA PB PA B∪= + − ∩
( )
( )
( )
|
PA B
PAB
PB
∩
=
Probability Distribution Mean Standard Deviation
Discrete random variable, X
µ
=
X
E
(
X
)
= ∑ xP
i
(
x
)
i
σ
X
=∑−x
iX
Px
i
µ
( )
2
( )
If has a binomial distribution
with parameters n and p, then:
n
PX
(
= x
)
=
p
x
(
1 − p
)
nx−
x
where
x= 0, 1, 2, 3, , n
µ
=
X
np
σ
=
X
np
(
1 − p
)
If has a geometric distribution
with parameter p, then:
PX
(
= x
)
=
(
1 − p
)
x−1
p
where
x = 1, 2, 3,
1
µ
=
X
p
1 − p
σ
=
X
p
III. Sampling Distributions and Inferential Statistics
Standardized test statistic:
statistic parameter
standard error of the statistic
−
Confidence interval:
( )( )
statistic critical value standard error of statistic±
Chi-square statistic:
( )
2
observed expected
expected
χ
−
=
∑
2
AP Statistics 2020 Formulas and Tables Sheet
*S
tandard 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.
III. Sampling Distributions and Inferential Statistics (continued)
Sampling distributions for proportions:
Random
Variable
Parameters of
Sampling Distribution
Standard Error
*
of Sample Statistic
For one
population:
p
ˆ
−
(
1
)
p
σ
p
ˆ
=
n
µ
p
ˆ
= p
p
p
ˆ
(
1 −
)
s
p
ˆ
=
n
p
ˆ
For two
populations:
p−
1
p
ˆˆ
2
µ
= p−
pp
ˆˆ
− 1
12
p
p
(
1−
σ
1
pp
12
) ( )
= +
pp
ˆˆ
−
12
n
1
−p1
2
2
n
2
p
ˆ
s
1
1−pp
ˆˆ
12
= +
pp
ˆˆ
−
12
n
1
(
1−
) (
p
ˆ
2
)
n
2
When
p=
1
is assumed:
p
2
s=pp
(
1
ˆˆ
1 −
pp
ˆˆ
− c c
)
12
nn
1
1
+
2
XX
+
p
ˆ
=
1
c
nn
+
where
2
12
Sampling distributions for means:
Random
Variable
Parameters of Sampling Distribution
Standard Error
*
of Sample Statistic
For one
population:
X
µ
=
X
σ
σ
=
X
n
µ
s
s =
X
n
For two
populations:
XX−
12
µ
=
µ
−
XX−
1
12
µ
2
σσ
22
σ
=
12
+
XX−
12
n
12
n
ss
2
s =
1
+
XX−
12
nn
1
2
2
2
Sampling distributions for simple linear regression:
Random
Variable
Parameters of Sampling Distribution
Standard Error
*
of Sample Statistic
For slope:
b
σ
σ
=
b
,
σ
x
n
( )
2
∑−xx
σ
=
i
x
n
where
s
s =
b
,
sn− 1
x
(
ˆ
)
2
∑−yy
s =
ii
−
n
2
( )
2
∑−xx
s =
i
x
n − 1
and
where
AP Statistics 2020 Formulas and Tables Sheet
b
=
µ
β
Probability
z
-1.3
AP Statistics 2020 Formulas and Tables Sheet
Probability
z
Table A
(Connue
AP Statistics 2020 Formulas and Tables Sheet
Table B t distribution critical values
Tail probability p
df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.6
2 .816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60
3 .765 .978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.92
4 .741 .941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 .727 .920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869
6 .718 .906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
7 .711 .896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408
8 .706 .889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041
9 .703 .883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781
10 .700 .879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587
11 .697 .876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437
12 .695 .873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
13 .694 .870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
14 .692 .868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
15 .691 .866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
16 .690 .865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015
17 .689 .863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
18 .688 .862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.611 3.922
19 .688 .861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883
20 .687 .860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
21 .686 .859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819
22 .686 .858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
23 .685 .858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768
24 .685 .857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745
25 .684 .856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725
26 .684 .856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
27 .684 .855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
28 .683 .855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
29 .683 .854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
30 .683 .854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
40 .681 .851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551
50 .679 .849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496
60 .679 .848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460
80 .678 .846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416
100 .677 .845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
1000 .675 .842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
•
.674 .841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291
50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%
Confidence level C
Probability p
t*
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between
t
* and t*.
AP Statistics 2020 Formulas and Tables Sheet
Table C
c
2
critical values
Tail probability p
df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005
1 1.32 1.64 2.07 2.71 3.84 5.02 5.41 6.63 7.88 9.14 10.83 12.12
2 2.77 3.22 3.79 4.61 5.99 7.38 7.82 9.21 10.60 11.98 13.82 15.20
3 4.11 4.64 5.32 6.25 7.81 9.35 9.84 11.34 12.84 14.32 16.27 17.73
4 5.39 5.99 6.74 7.78 9.49 11.14 11.67 13.28 14.86 16.42 18.47 20.00
5 6.63 7.29 8.12 9.24 11.07 12.83 13.39 15.09 16.75 18.39 20.51 22.11
6 7.84 8.56 9.45 10.64 12.59 14.45 15.03 16.81 18.55 20.25 22.46 24.10
7 9.04 9.80 10.75 12.02 14.07 16.01 16.62 18.48 20.28 22.04 24.32 26.02
8 10.22 11.03 12.03 13.36 15.51 17.53 18.17 20.09 21.95 23.77 26.12 27.87
9 11.39 12.24 13.29 14.68 16.92 19.02 19.68 21.67 23.59 25.46 27.88 29.67
10 12.55 13.44 14.53 15.99 18.31 20.48 21.16 23.21 25.19 27.11 29.59 31.42
11 13.70 14.63 15.77 17.28 19.68 21.92 22.62 24.72 26.76 28.73 31.26 33.14
12 14.85 15.81 16.99 18.55 21.03 23.34 24.05 26.22 28.30 30.32 32.91 34.82
13 15.98 16.98 18.20 19.81 22.36 24.74 25.47 27.69 29.82 31.88 34.53 36.48
14 17.12 18.15 19.41 21.06 23.68 26.12 26.87 29.14 31.32 33.43 36.12 38.11
15 18.25 19.31 20.60 22.31 25.00 27.49 28.26 30.58 32.80 34.95 37.70 39.72
16 19.37 20.47 21.79 23.54 26.30 28.85 29.63 32.00 34.27 36.46 39.25 41.31
17 20.49 21.61 22.98 24.77 27.59 30.19 31.00 33.41 35.72 37.95 40.79 42.88
18 21.60 22.76 24.16 25.99 28.87 31.53 32.35 34.81 37.16 39.42 42.31 44.43
19 22.72 23.90 25.33 27.20 30.14 32.85 33.69 36.19 38.58 40.88 43.82 45.97
20 23.83 25.04 26.50 28.41 31.41 34.17 35.02 37.57 40.00 42.34 45.31 47.50
21 24.93 26.17 27.66 29.62 32.67 35.48 36.34 38.93 41.40 43.78 46.80 49.01
22 26.04 27.30 28.82 30.81 33.92 36.78 37.66 40.29 42.80 45.20 48.27 50.51
23 27.14 28.43 29.98 32.01 35.17 38.08 38.97 41.64 44.18 46.62 49.73 52.00
24 28.24 29.55 31.13 33.20 36.42 39.36 40.27 42.98 45.56 48.03 51.18 53.48
25 29.34 30.68 32.28 34.38 37.65 40.65 41.57 44.31 46.93 49.44 52.62 54.95
26 30.43 31.79 33.43 35.56 38.89 41.92 42.86 45.64 48.29 50.83 54.05 56.41
27 31.53 32.91 34.57 36.74 40.11 43.19 44.14 46.96 49.64 52.22 55.48 57.86
28 32.62 34.03 35.71 37.92 41.34 44.46 45.42 48.28 50.99 53.59 56.89 59.30
29 33.71 35.14 36.85 39.09 42.56 45.72 46.69 49.59 52.34 54.97 58.30 60.73
30 34.80 36.25 37.99 40.26 43.77 46.98 47.96 50.89 53.67 56.33 59.70 62.16
40 45.62 47.27 49.24 51.81 55.76 59.34 60.44 63.69 66.77 69.70 73.40 76.09
50 56.33 58.16 60.35 63.17 67.50 71.42 72.61 76.15 79.49 82.66 86.66 89.56
60 66.98 68.97 71.34 74.40 79.08 83.30 84.58 88.38 91.95 95.34 99.61 102.7
80 88.13 90.41 93.11 96.58 101.9 106.6 108.1 112.3 116.3 120.1 124.8 128.3
100 109.1 111.7 114.7 118.5 124.3 129.6 131.1 135.8 140.2 144.3 149.4 153.2
Probability p
(χ
2
)
Table entry for p is the point
(
2
)
with probability p lying
above it.
AP Statistics 2020 Formulas and Tables Sheet
