Prep2CollegeAP Statistics Practice

Formulas and tables

The formula sheet and the three tables given with the AP Statistics exam (the same information as the printed sheet; table values computed for this site). A graphing calculator is allowed on the whole exam.

I. Descriptive statistics

xˉ=1n∑xi=∑xin\bar{x} = \frac{1}{n}\sum x_i = \frac{\sum x_i}{n}
sx=1n−1∑(xi−xˉ)2=∑(xi−xˉ)2n−1s_x = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
y^=a+bx\hat{y} = a + bx
yˉ=a+bxˉ\bar{y} = a + b\bar{x}
r=1n−1∑(xi−xˉsx)(yi−yˉsy)r = \frac{1}{n-1}\sum \left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)
b=r sysxb = r\,\frac{s_y}{s_x}

II. Probability and distributions

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Probability distributionMeanStandard deviation
Discrete random variable X\displaystyle X
x1,x2,x3,…\displaystyle x_1, x_2, x_3, \ldots with probabilities P(x1),P(x2),P(x3),…\displaystyle P(x_1), P(x_2), P(x_3), \ldots		μX=E(X)=∑xi⋅P(xi)\displaystyle \mu_X = E(X) = \sum x_i \cdot P(x_i)σX=∑(xi−μX)2⋅P(xi)\displaystyle \sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}
Binomial: P(X=x)=(nx)px(1−p)n−x\displaystyle P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}
x=0,1,2,…,n\displaystyle x = 0, 1, 2, \ldots, n		μX=np\displaystyle \mu_X = npσX=np(1−p)\displaystyle \sigma_X = \sqrt{np(1-p)}
Geometric: P(X=x)=(1−p)x−1p\displaystyle P(X = x) = (1-p)^{x-1} p
x=1,2,3,…\displaystyle x = 1, 2, 3, \ldots		μX=1p\displaystyle \mu_X = \frac{1}{p}σX=1−pp\displaystyle \sigma_X = \frac{\sqrt{1-p}}{p}

III. Sampling distributions and inferential statistics

Standardized test statistic: statistic−parameterstandard error of the statistic\text{Standardized test statistic: } \frac{\text{statistic} - \text{parameter}}{\text{standard error of the statistic}}
Confidence interval: statistic±(critical value)(standard error of the statistic)\text{Confidence interval: } \text{statistic} \pm (\text{critical value})(\text{standard error of the statistic})
Chi-square statistic: χ2=∑(observed−expected)2expected\text{Chi-square statistic: } \chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}

Sampling distributions for proportions

Random variableParameters of the sampling distributionStandard error* of the sample statistic
For one population: p^\displaystyle \hat{p}μp^=pσp^=p(1−p)n\displaystyle \mu_{\hat p} = p \qquad \sigma_{\hat p} = \sqrt{\frac{p(1-p)}{n}}sp^=p^(1−p^)n\displaystyle s_{\hat p} = \sqrt{\frac{\hat p(1-\hat p)}{n}}
For two populations: p^1−p^2\displaystyle \hat{p}_1 - \hat{p}_2μp^1−p^2=p1−p2\displaystyle \mu_{\hat p_1 - \hat p_2} = p_1 - p_2
σp^1−p^2=p1(1−p1)n1+p2(1−p2)n2\displaystyle \sigma_{\hat p_1 - \hat p_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}		sp^1−p^2=p^1(1−p^1)n1+p^2(1−p^2)n2\displaystyle s_{\hat p_1 - \hat p_2} = \sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1} + \frac{\hat p_2(1-\hat p_2)}{n_2}}
When p1=p2\displaystyle p_1 = p_2 is assumed:
sp^1−p^2=p^c(1−p^c)(1n1+1n2)\displaystyle s_{\hat p_1 - \hat p_2} = \sqrt{\hat p_c(1-\hat p_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}
where p^c=X1+X2n1+n2\displaystyle \hat p_c = \frac{X_1 + X_2}{n_1 + n_2}

Sampling distributions for means

Random variableParameters of the sampling distributionStandard error* of the sample statistic
For one population: Xˉ\displaystyle \bar{X}μxˉ=μσxˉ=σn\displaystyle \mu_{\bar x} = \mu \qquad \sigma_{\bar x} = \frac{\sigma}{\sqrt n}sxˉ=sn\displaystyle s_{\bar x} = \frac{s}{\sqrt n}
For two populations: Xˉ1−Xˉ2\displaystyle \bar{X}_1 - \bar{X}_2μxˉ1−xˉ2=μ1−μ2\displaystyle \mu_{\bar x_1 - \bar x_2} = \mu_1 - \mu_2
σxˉ1−xˉ2=σ12n1+σ22n2\displaystyle \sigma_{\bar x_1 - \bar x_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}		sxˉ1−xˉ2=s12n1+s22n2\displaystyle s_{\bar x_1 - \bar x_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

Sampling distributions for simple linear regression

Random variableParameters of the sampling distributionStandard error* of the sample statistic
For slope: b\displaystyle bμb=βσb=σσxn\displaystyle \mu_b = \beta \qquad \sigma_b = \frac{\sigma}{\sigma_x \sqrt n}
where σx=∑(xi−xˉ)2n\displaystyle \sigma_x = \sqrt{\frac{\sum (x_i - \bar x)^2}{n}}		sb=ssxn−1\displaystyle s_b = \frac{s}{s_x \sqrt{n-1}}
where s=∑(yi−y^i)2n−2\displaystyle s = \sqrt{\frac{\sum (y_i - \hat y_i)^2}{n-2}}
and sx=∑(xi−xˉ)2n−1\displaystyle s_x = \sqrt{\frac{\sum (x_i - \bar x)^2}{n-1}}

*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.

Table A · Standard normal probabilities

Table entry for z\displaystyle z is the probability lying below z\displaystyle z: P(Z<z)\displaystyle P(Z < z).

z\displaystyle z.00.01.02.03.04.05.06.07.08.09
−3.4.0003.0003.0003.0003.0003.0003.0003.0003.0003.0002
−3.3.0005.0005.0005.0004.0004.0004.0004.0004.0004.0003
−3.2.0007.0007.0006.0006.0006.0006.0006.0005.0005.0005
−3.1.0010.0009.0009.0009.0008.0008.0008.0008.0007.0007
−3.0.0013.0013.0013.0012.0012.0011.0011.0011.0010.0010
−2.9.0019.0018.0018.0017.0016.0016.0015.0015.0014.0014
−2.8.0026.0025.0024.0023.0023.0022.0021.0021.0020.0019
−2.7.0035.0034.0033.0032.0031.0030.0029.0028.0027.0026
−2.6.0047.0045.0044.0043.0041.0040.0039.0038.0037.0036
−2.5.0062.0060.0059.0057.0055.0054.0052.0051.0049.0048
−2.4.0082.0080.0078.0075.0073.0071.0069.0068.0066.0064
−2.3.0107.0104.0102.0099.0096.0094.0091.0089.0087.0084
−2.2.0139.0136.0132.0129.0125.0122.0119.0116.0113.0110
−2.1.0179.0174.0170.0166.0162.0158.0154.0150.0146.0143
−2.0.0228.0222.0217.0212.0207.0202.0197.0192.0188.0183
−1.9.0287.0281.0274.0268.0262.0256.0250.0244.0239.0233
−1.8.0359.0351.0344.0336.0329.0322.0314.0307.0301.0294
−1.7.0446.0436.0427.0418.0409.0401.0392.0384.0375.0367
−1.6.0548.0537.0526.0516.0505.0495.0485.0475.0465.0455
−1.5.0668.0655.0643.0630.0618.0606.0594.0582.0571.0559
−1.4.0808.0793.0778.0764.0749.0735.0721.0708.0694.0681
−1.3.0968.0951.0934.0918.0901.0885.0869.0853.0838.0823
−1.2.1151.1131.1112.1093.1075.1056.1038.1020.1003.0985
−1.1.1357.1335.1314.1292.1271.1251.1230.1210.1190.1170
−1.0.1587.1562.1539.1515.1492.1469.1446.1423.1401.1379
−0.9.1841.1814.1788.1762.1736.1711.1685.1660.1635.1611
−0.8.2119.2090.2061.2033.2005.1977.1949.1922.1894.1867
−0.7.2420.2389.2358.2327.2296.2266.2236.2206.2177.2148
−0.6.2743.2709.2676.2643.2611.2578.2546.2514.2483.2451
−0.5.3085.3050.3015.2981.2946.2912.2877.2843.2810.2776
−0.4.3446.3409.3372.3336.3300.3264.3228.3192.3156.3121
−0.3.3821.3783.3745.3707.3669.3632.3594.3557.3520.3483
−0.2.4207.4168.4129.4090.4052.4013.3974.3936.3897.3859
−0.1.4602.4562.4522.4483.4443.4404.4364.4325.4286.4247
−0.0.5000.4960.4920.4880.4840.4801.4761.4721.4681.4641
z\displaystyle z.00.01.02.03.04.05.06.07.08.09
0.0.5000.5040.5080.5120.5160.5199.5239.5279.5319.5359
0.1.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753
0.2.5793.5832.5871.5910.5948.5987.6026.6064.6103.6141
0.3.6179.6217.6255.6293.6331.6368.6406.6443.6480.6517
0.4.6554.6591.6628.6664.6700.6736.6772.6808.6844.6879
0.5.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224
0.6.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549
0.7.7580.7611.7642.7673.7704.7734.7764.7794.7823.7852
0.8.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133
0.9.8159.8186.8212.8238.8264.8289.8315.8340.8365.8389
1.0.8413.8438.8461.8485.8508.8531.8554.8577.8599.8621
1.1.8643.8665.8686.8708.8729.8749.8770.8790.8810.8830
1.2.8849.8869.8888.8907.8925.8944.8962.8980.8997.9015
1.3.9032.9049.9066.9082.9099.9115.9131.9147.9162.9177
1.4.9192.9207.9222.9236.9251.9265.9279.9292.9306.9319
1.5.9332.9345.9357.9370.9382.9394.9406.9418.9429.9441
1.6.9452.9463.9474.9484.9495.9505.9515.9525.9535.9545
1.7.9554.9564.9573.9582.9591.9599.9608.9616.9625.9633
1.8.9641.9649.9656.9664.9671.9678.9686.9693.9699.9706
1.9.9713.9719.9726.9732.9738.9744.9750.9756.9761.9767
2.0.9772.9778.9783.9788.9793.9798.9803.9808.9812.9817
2.1.9821.9826.9830.9834.9838.9842.9846.9850.9854.9857
2.2.9861.9864.9868.9871.9875.9878.9881.9884.9887.9890
2.3.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916
2.4.9918.9920.9922.9925.9927.9929.9931.9932.9934.9936
2.5.9938.9940.9941.9943.9945.9946.9948.9949.9951.9952
2.6.9953.9955.9956.9957.9959.9960.9961.9962.9963.9964
2.7.9965.9966.9967.9968.9969.9970.9971.9972.9973.9974
2.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.9981
2.9.9981.9982.9982.9983.9984.9984.9985.9985.9986.9986
3.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990
3.1.9990.9991.9991.9991.9992.9992.9992.9992.9993.9993
3.2.9993.9993.9994.9994.9994.9994.9994.9995.9995.9995
3.3.9995.9995.9995.9996.9996.9996.9996.9996.9996.9997
3.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998

Table B · t distribution critical values

Table entry for p\displaystyle p and C\displaystyle C is the point t∗\displaystyle t^* with probability p\displaystyle p lying above it and probability C\displaystyle C lying between −t∗\displaystyle -t^* and t∗\displaystyle t^*. Column heads: upper-tail probability p\displaystyle p.

df.25.2.15.1.05.025.02.01.005.0025.001.0005
11.0001.3761.9633.0786.31412.70615.89531.82163.657127.321318.309636.619
20.8161.0611.3861.8862.9204.3034.8496.9659.92514.08922.32731.599
30.7650.9781.2501.6382.3533.1823.4824.5415.8417.45310.21512.924
40.7410.9411.1901.5332.1322.7762.9993.7474.6045.5987.1738.610
50.7270.9201.1561.4762.0152.5712.7573.3654.0324.7735.8936.869
60.7180.9061.1341.4401.9432.4472.6123.1433.7074.3175.2085.959
70.7110.8961.1191.4151.8952.3652.5172.9983.4994.0294.7855.408
80.7060.8891.1081.3971.8602.3062.4492.8963.3553.8334.5015.041
90.7030.8831.1001.3831.8332.2622.3982.8213.2503.6904.2974.781
100.7000.8791.0931.3721.8122.2282.3592.7643.1693.5814.1444.587
110.6970.8761.0881.3631.7962.2012.3282.7183.1063.4974.0254.437
120.6950.8731.0831.3561.7822.1792.3032.6813.0553.4283.9304.318
130.6940.8701.0791.3501.7712.1602.2822.6503.0123.3723.8524.221
140.6920.8681.0761.3451.7612.1452.2642.6242.9773.3263.7874.140
150.6910.8661.0741.3411.7532.1312.2492.6022.9473.2863.7334.073
160.6900.8651.0711.3371.7462.1202.2352.5832.9213.2523.6864.015
170.6890.8631.0691.3331.7402.1102.2242.5672.8983.2223.6463.965
180.6880.8621.0671.3301.7342.1012.2142.5522.8783.1973.6103.922
190.6880.8611.0661.3281.7292.0932.2052.5392.8613.1743.5793.883
200.6870.8601.0641.3251.7252.0862.1972.5282.8453.1533.5523.850
210.6860.8591.0631.3231.7212.0802.1892.5182.8313.1353.5273.819
220.6860.8581.0611.3211.7172.0742.1832.5082.8193.1193.5053.792
230.6850.8581.0601.3191.7142.0692.1772.5002.8073.1043.4853.768
240.6850.8571.0591.3181.7112.0642.1722.4922.7973.0913.4673.745
250.6840.8561.0581.3161.7082.0602.1672.4852.7873.0783.4503.725
260.6840.8561.0581.3151.7062.0562.1622.4792.7793.0673.4353.707
270.6840.8551.0571.3141.7032.0522.1582.4732.7713.0573.4213.690
280.6830.8551.0561.3131.7012.0482.1542.4672.7633.0473.4083.674
290.6830.8541.0551.3111.6992.0452.1502.4622.7563.0383.3963.659
300.6830.8541.0551.3101.6972.0422.1472.4572.7503.0303.3853.646
400.6810.8511.0501.3031.6842.0212.1232.4232.7042.9713.3073.551
500.6790.8491.0471.2991.6762.0092.1092.4032.6782.9373.2613.496
600.6790.8481.0451.2961.6712.0002.0992.3902.6602.9153.2323.460
800.6780.8461.0431.2921.6641.9902.0882.3742.6392.8873.1953.416
1000.6770.8451.0421.2901.6601.9842.0812.3642.6262.8713.1743.390
10000.6750.8421.0371.2821.6461.9622.0562.3302.5812.8133.0983.300
∞0.6740.8421.0361.2821.6451.9602.0542.3262.5762.8073.0903.291
50%60%70%80%90%95%96%98%99%99.5%99.8%99.9%

Bottom row: confidence level C.

Table C · χ² critical values

Table entry for p\displaystyle p is the point (χ2)\displaystyle (\chi^2) with probability p\displaystyle p lying above it. Column heads: upper-tail probability p\displaystyle p.

df.25.2.15.1.05.025.02.01.005.0025.001.0005
11.321.642.072.713.845.025.416.637.889.1410.8312.12
22.773.223.794.615.997.387.829.2110.6011.9813.8215.20
34.114.645.326.257.819.359.8411.3412.8414.3216.2717.73
45.395.996.747.789.4911.1411.6713.2814.8616.4218.4720.00
56.637.298.129.2411.0712.8313.3915.0916.7518.3920.5222.11
67.848.569.4510.6412.5914.4515.0316.8118.5520.2522.4624.10
79.049.8010.7512.0214.0716.0116.6218.4820.2822.0424.3226.02
810.2211.0312.0313.3615.5117.5318.1720.0921.9523.7726.1227.87
911.3912.2413.2914.6816.9219.0219.6821.6723.5925.4627.8829.67
1012.5513.4414.5315.9918.3120.4821.1623.2125.1927.1129.5931.42
1113.7014.6315.7717.2819.6821.9222.6224.7226.7628.7331.2633.14
1214.8515.8116.9918.5521.0323.3424.0526.2228.3030.3232.9134.82
1315.9816.9818.2019.8122.3624.7425.4727.6929.8231.8834.5336.48
1417.1218.1519.4121.0623.6826.1226.8729.1431.3233.4336.1238.11
1518.2519.3120.6022.3125.0027.4928.2630.5832.8034.9537.7039.72
1619.3720.4721.7923.5426.3028.8529.6332.0034.2736.4639.2541.31
1720.4921.6122.9824.7727.5930.1931.0033.4135.7237.9540.7942.88
1821.6022.7624.1625.9928.8731.5332.3534.8137.1639.4242.3144.43
1922.7223.9025.3327.2030.1432.8533.6936.1938.5840.8843.8245.97
2023.8325.0426.5028.4131.4134.1735.0237.5740.0042.3445.3147.50
2124.9326.1727.6629.6232.6735.4836.3438.9341.4043.7846.8049.01
2226.0427.3028.8230.8133.9236.7837.6640.2942.8045.2048.2750.51
2327.1428.4329.9832.0135.1738.0838.9741.6444.1846.6249.7352.00
2428.2429.5531.1333.2036.4239.3640.2742.9845.5648.0351.1853.48
2529.3430.6832.2834.3837.6540.6541.5744.3146.9349.4452.6254.95
2630.4331.7933.4335.5638.8941.9242.8645.6448.2950.8354.0556.41
2731.5332.9134.5736.7440.1143.1944.1446.9649.6452.2255.4857.86
2832.6234.0335.7137.9241.3444.4645.4248.2850.9953.5956.8959.30
2933.7135.1436.8539.0942.5645.7246.6949.5952.3454.9758.3060.73
3034.8036.2537.9940.2643.7746.9847.9650.8953.6756.3359.7062.16
4045.6247.2749.2451.8155.7659.3460.4463.6966.7769.7073.4076.09
5056.3358.1660.3563.1767.5071.4272.6176.1579.4982.6686.6689.56
6066.9868.9771.3474.4079.0883.3084.5888.3891.9595.3499.61102.69
8088.1390.4193.1196.58101.88106.63108.07112.33116.32120.10124.84128.26
100109.14111.67114.66118.50124.34129.56131.14135.81140.17144.29149.45153.17