AP Statistics Formula Sheet (2027), Explained

Using the sheet on exam day

The exam is fully digital for 2027, and the formula sheet and tables are available in the testing app. Look through them before exam day so you know where each formula and table is and don't lose time hunting for it.
In free-response answers you can use the testing app's toolbar for symbols, or type keyboard versions: x-bar for $\bar{x}$ , p-hat for $\hat{p}$ , mu for $\mu$ , sigma for $\sigma$ , alpha for $\alpha$ , H_0 and H_a for the hypotheses, and an underscore for a subscript, like p-hat_1 or mu_A.
To show a calculation, type it the way you'd enter it in a calculator: ^ for a power, sqrt() for a square root, +/- for $\pm$ , <= and >= for $\le$ and $\ge$ , and != for $\ne$ . For example, a binomial probability can be typed as 10*(0.3)^2*(0.7)^3, and an interval as 0.62+/-1.96*sqrt(0.62*0.38/400). Write a finished interval as (0.572, 0.668) or "0.572 to 0.668".
The sheet asks you to show your work on every free-response question. Write the formula with your numbers plugged in, not just the final answer or a calculator command.
The sheet never tells you which procedure to use, which conditions to check or how many degrees of freedom to use. It also doesn't list common critical values, but the last row of Table B (df = ∞) gives the normal ones: $z^* = 1.645$ for 90%, 1.960 for 95% and 2.576 for 99%.

Descriptive statistics

Summaries of one quantitative variable, and the form of a regression line.

Sample mean

\displaystyle \bar{x} = \frac{1}{n}\sum x_i = \frac{\sum x_i}{n}

What the symbols mean

$\bar{x}$: the sample mean (read "x-bar"); Unit: same units as the data
$x_i$: each data value in the sample; Unit: same units as the data
$n$: the number of values in the sample; Unit: none (a count)
$\sum$: add up every value that follows; Unit: none

Use it when: Use it when you have a list of data values and need the average, the usual measure of center for a roughly symmetric distribution.

Watch out: Reporting the mean as the center of a strongly skewed distribution or one with outliers. The mean gets pulled toward the tail, so use the median there.

Learn it: 1.7 Summary Statistics for One Quantitative Variable

Sample standard deviation

\displaystyle s = \sqrt{\frac{1}{n-1}\sum\left(x_i - \bar{x}\right)^2} = \sqrt{\frac{\sum\left(x_i - \bar{x}\right)^2}{n-1}}

What the symbols mean

$s$: the sample standard deviation: a typical distance between the data values and their mean; Unit: same units as the data
$x_i - \bar{x}$: the deviation of one value from the mean; Unit: same units as the data
$n - 1$: the sample size minus one; Unit: none

Use it when: Use it to measure the spread of sample data around the mean. Square each deviation, add them, divide by $n - 1$ and take the square root.

Watch out: Dividing by $n$ instead of $n - 1$ , or stopping before the square root. Without the root you have the variance, which is in squared units of the data.

Try it: Find the mean and standard deviation of the sample 2, 4, 6, 8.

Answer: $\bar{x} = 20/4 = 5$ . The deviations are −3, −1, 1, 3, whose squares add to 20. Then $s = \sqrt{20/3} \approx 2.58$ .

Learn it: 1.7 Summary Statistics for One Quantitative Variable

Least-squares regression line

\displaystyle \hat{y} = a + bx

What the symbols mean

$\hat{y}$: the predicted value of the response variable (read "y-hat"); Unit: units of y
$a$: the y-intercept: the predicted y when x = 0; Unit: units of y
$b$: the slope: the predicted change in y for each one-unit increase in x; Unit: units of y per unit of x
$x$: the value of the explanatory variable; Unit: units of x

Use it when: Use it to predict a response from an explanatory value once you have a and b, usually from technology or computer output. You also interpret b and a in context.

Watch out: Writing the line with $y$ instead of $\hat{y}$ , or interpreting the slope as a certain change. It's a predicted change, on average, and only within the range of the data.

Try it: A line is $\hat{y} = 12 + 3.5x$ . Predict y when x = 4. If the actual y is 29, what's the residual?

Answer: $\hat{y} = 12 + 3.5(4) = 26$ . The residual is $y - \hat{y} = 29 - 26 = 3$ , so the point is 3 units above the line.

Learn it: 5.3 Linear Regression Models · 5.5 Least-Squares Regression

Probability

Two rules for combining events.

Addition rule for A or B

\displaystyle P\left(A \cup B\right) = P\left(A\right) + P\left(B\right) - P\left(A \cap B\right)

What the symbols mean

$P\left(A \cup B\right)$: the probability that A or B (or both) happens; Unit: none
$P\left(A \cap B\right)$: the probability that A and B both happen; Unit: none

Use it when: Use it when a question asks for the probability of "A or B" and you know (or can find) each probability and the probability of both.

Watch out: Forgetting to subtract $P(A \cap B)$ , which counts the overlap twice. You can skip that term only when A and B are mutually exclusive, since then $P(A \cap B) = 0$ .

Try it: $P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.2$ . Find $P(A \cup B)$ and $P(A \mid B)$ .

Answer: $P(A \cup B) = 0.5 + 0.4 - 0.2 = 0.7$ . $P(A \mid B) = 0.2/0.4 = 0.5$ . That equals $P(A)$ , so A and B are independent.

Learn it: 2.7 Independent Events and Unions of Events · 2.5 Mutually Exclusive Events

Conditional probability

\displaystyle P\left(A \mid B\right) = \frac{P\left(A \cap B\right)}{P\left(B\right)}

What the symbols mean

$P\left(A \mid B\right)$: the probability of A given that B happened; Unit: none
$P\left(B\right)$: the probability of the condition, B; Unit: none

Use it when: Use it for "given that" questions, or to check independence: A and B are independent when $P(A \mid B) = P(A)$ . With a two-way table, you can also just divide within the row or column for B.

Watch out: Dividing by the wrong event. The event after the bar, the one you're told happened, always goes in the denominator, so $P(A \mid B)$ and $P(B \mid A)$ are usually different.

Learn it: 2.6 Conditional Probability · 2.7 Independent Events and Unions of Events

Probability distributions

The mean and standard deviation of a discrete random variable, and the binomial distribution.

Mean of a discrete random variable

\displaystyle \mu_X = E\left(X\right) = \sum x_i \cdot P\left(x_i\right)

What the symbols mean

$\mu_X$: the mean of the random variable X; Unit: same units as X
$E\left(X\right)$: the expected value of X, another name for its mean; Unit: same units as X
$x_i$: each possible value of X; Unit: same units as X
$P\left(x_i\right)$: the probability of that value; Unit: none

Use it when: Use it when you have a probability distribution table and need the long-run average outcome.

Watch out: Averaging the values without weighting them by their probabilities, or saying the mean is the most likely outcome. It's the average over many, many trials.

Learn it: 2.9 Parameters of Random Variables · 2.8 Introduction to Random Variables and Probability Distributions

Standard deviation of a discrete random variable

\displaystyle \sigma_X = \sqrt{\sum\left(x_i - \mu_X\right)^2 \cdot P\left(x_i\right)}

What the symbols mean

$\sigma_X$: the standard deviation of X: how much outcomes typically vary from the mean; Unit: same units as X
$x_i - \mu_X$: how far one possible value is from the mean; Unit: same units as X
$P\left(x_i\right)$: the probability of that value; Unit: none

Use it when: Use it with a probability distribution table to find the spread of the outcomes. Find $\mu_X$ first.

Watch out: Forgetting the square root (that's the variance) or forgetting to multiply each squared deviation by its probability. There's no $n - 1$ here.

Try it: X takes the values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2. Find $\mu_X$ and $\sigma_X$ .

Answer: $\mu_X = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 1.7$ . The weighted squared deviations are 0.289, 0.147, 0.036 and 0.338, which add to 0.81, so $\sigma_X = \sqrt{0.81} = 0.9$ .

Learn it: 2.9 Parameters of Random Variables

Binomial probability

\displaystyle P\left(X = x\right) = \binom{n}{x} p^x \left(1 - p\right)^{n-x}, \quad \text{where } x = 0, 1, 2, 3, \ldots, n

What the symbols mean

$X$: the number of successes in n trials; Unit: none (a count)
$x$: the particular number of successes you want; Unit: none (a count)
$n$: the number of trials; Unit: none (a count)
$p$: the probability of success on each trial; Unit: none
$\binom{n}{x}$: the number of ways to arrange x successes among n trials ("n choose x"); Unit: none

Use it when: Use it for the probability of exactly x successes when there's a fixed number of independent trials, each a success or failure with the same p. For "at most" or "at least", add several of these, or use a cumulative calculator command.

Watch out: Using it when the conditions fail, for example drawing without replacement from a small group, so the trials aren't independent. Also, "at least 1" is easiest as $1 - P(X = 0)$ .

Try it: A free throw goes in with probability 0.3. In 5 independent shots, find $P(X = 2)$ .

Answer: $\binom{5}{2}(0.3)^2(0.7)^3 = 10(0.09)(0.343) \approx 0.309$ .

Learn it: 2.10 The Binomial Distribution

Mean of a binomial random variable

\displaystyle \mu_X = np

What the symbols mean

$\mu_X$: the mean number of successes; Unit: none (a count)
$n$: the number of trials; Unit: none (a count)
$p$: the probability of success on each trial; Unit: none

Use it when: Use it for the expected number of successes in a binomial setting, instead of the long sum.

Watch out: Using it for a variable that isn't binomial. Check that n is fixed, trials are independent and p is the same every time.

Learn it: 2.10 The Binomial Distribution

Standard deviation of a binomial random variable

\displaystyle \sigma_X = \sqrt{np\left(1 - p\right)}

What the symbols mean

$\sigma_X$: the standard deviation of the number of successes; Unit: none (a count)
$1 - p$: the probability of failure on each trial; Unit: none

Use it when: Use it for the spread of the number of successes in a binomial setting.

Watch out: Leaving off the square root and reporting $np(1 - p)$ , which is the variance.

Try it: For 50 shots with $p = 0.3$ , find the mean and standard deviation of the number made.

Answer: $\mu_X = 50(0.3) = 15$ shots and $\sigma_X = \sqrt{50(0.3)(0.7)} = \sqrt{10.5} \approx 3.24$ shots.

Learn it: 2.10 The Binomial Distribution

Sampling distributions and inferential statistics

The two patterns every z, t test and interval follows. The tables below fill in the standard error.

Standardized test statistic

\displaystyle \text{test statistic} = \frac{\text{statistic} - \text{parameter}}{\text{standard error of the statistic}}

What the symbols mean

$\text{statistic}$: the value from your sample, like $\hat{p}$ or $\bar{x}$; Unit: units of the statistic
$\text{parameter}$: the value the null hypothesis claims, like $p_0$ or $\mu_0$; Unit: units of the statistic
$\text{standard error of the statistic}$: how much the statistic typically varies from sample to sample; Unit: units of the statistic

Use it when: Use it in a significance test to find z or t: how many standard errors your statistic is from the null value. Then find the p-value from Table A (z) or Table B (t).

Watch out: Plugging in the sample statistic where the null value goes, or using the wrong spread. For a one-proportion z test, the denominator uses $p_0$ : $\sqrt{p_0(1 - p_0)/n}$ , not $\hat{p}$ .

Try it: Test $H_0: p = 0.5$ against $H_a: p > 0.5$ with $n = 100$ and $\hat{p} = 0.58$ .

Answer: The standard error under $H_0$ is $\sqrt{0.5(0.5)/100} = 0.05$ , so $z = (0.58 - 0.5)/0.05 = 1.60$ . Table A gives 0.9452 to the left, so the p-value is $1 - 0.9452 \approx 0.055$ .

Learn it: 3.7 Carrying Out a Test for a Population Proportion · 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference · 3.13 Carrying Out a Test for the Difference Between Two Population Proportions

Confidence interval

\displaystyle \text{confidence interval} = \text{statistic} \pm \left(\text{critical value}\right) \cdot \left(\text{standard error of the statistic}\right)

What the symbols mean

$\text{statistic}$: the point estimate from your sample, like $\hat{p}$ or $\bar{x}$; Unit: units of the statistic
$\text{critical value}$: $z^*$ or $t^*$ for your confidence level, from Table A or Table B; Unit: none
$\text{standard error of the statistic}$: the estimated standard deviation of the statistic, computed from sample data; Unit: units of the statistic

Use it when: Use it to estimate a parameter. The part after $\pm$ is the margin of error. Pick the standard error from the tables below to match your parameter.

Watch out: Using $z^*$ for a mean when $\sigma$ is unknown (use $t^*$ ), or using a test statistic like 1.60 as the critical value. The critical value comes only from the confidence level (and df).

Try it: In a random sample of 400, $\hat{p} = 0.62$ . Build a 95% confidence interval for p.

Answer: $SE = \sqrt{0.62(0.38)/400} \approx 0.0243$ , and $z^* = 1.96$ , so the margin of error is about 0.048. The interval is $0.62 \pm 0.048$ , or about (0.572, 0.668).

Learn it: 3.3 Constructing a Confidence Interval for a Population Proportion · 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference · 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means

Sampling distributions for proportions

Standard deviation columns use the true parameter p. Standard error columns use the sample's $\hat{p}$ , because in practice you don't know p.

Mean of a sample proportion

\displaystyle \mu_{\hat{p}} = p

What the symbols mean

$\mu_{\hat{p}}$: the mean of the sampling distribution of $\hat{p}$; Unit: none
$p$: the true population proportion; Unit: none

Use it when: Use it to describe the center of the sampling distribution of $\hat{p}$ . It says $\hat{p}$ is an unbiased estimator of p.

Watch out: Saying any one sample's $\hat{p}$ equals p. Only the average of $\hat{p}$ over all possible samples equals p.

Learn it: 3.2 Sampling Distributions for Sample Proportions · 3.1 Estimators

Standard deviation of a sample proportion

\displaystyle \sigma_{\hat{p}} = \sqrt{\frac{p\left(1 - p\right)}{n}}

What the symbols mean

$\sigma_{\hat{p}}$: the standard deviation of $\hat{p}$ from sample to sample; Unit: none
$p$: the true (or claimed) population proportion; Unit: none
$n$: the sample size; Unit: none (a count)

Use it when: Use it when p is known or assumed: in probability questions about $\hat{p}$ , and in a one-proportion z test, where you use the null value $p_0$ .

Watch out: Using it without checking the 10% condition ( $n \le 0.10N$ ) when sampling without replacement. Also, larger samples shrink it by a factor of $\sqrt{n}$ , so 4 times the sample size halves it.

Try it: If $p = 0.30$ and $n = 200$ , what is $\sigma_{\hat{p}}$ ?

Answer: $\sigma_{\hat{p}} = \sqrt{0.30(0.70)/200} \approx 0.032$ .

Learn it: 3.2 Sampling Distributions for Sample Proportions · 3.7 Carrying Out a Test for a Population Proportion

Standard error of a sample proportion

\displaystyle SE_{\hat{p}} = \sqrt{\frac{\hat{p}\left(1 - \hat{p}\right)}{n}}

What the symbols mean

$SE_{\hat{p}}$: the estimated standard deviation of $\hat{p}$; Unit: none
$\hat{p}$: the sample proportion (read "p-hat"); Unit: none

Use it when: Use it in a one-proportion z interval, when you don't know p and have to estimate the spread from your sample.

Watch out: Mixing it up with $\sigma_{\hat{p}}$ . Intervals use $\hat{p}$ in the standard error; a one-proportion test uses $p_0$ .

Learn it: 3.3 Constructing a Confidence Interval for a Population Proportion

Mean of a difference in sample proportions

\displaystyle \mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2

What the symbols mean

$\hat{p}_1 - \hat{p}_2$: the difference between the two sample proportions; Unit: none
$p_1 - p_2$: the difference between the two population proportions; Unit: none

Use it when: Use it to describe the center of the sampling distribution of $\hat{p}_1 - \hat{p}_2$ , with two independent samples or two groups in an experiment.

Watch out: Switching the order partway through. Decide which group is 1 and which is 2, and keep that order in the hypotheses, the statistic and the conclusion.

Learn it: 3.9 Sampling Distributions for the Difference Between Sample Proportions

Standard deviation of a difference in sample proportions

\displaystyle \sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1\left(1 - p_1\right)}{n_1} + \frac{p_2\left(1 - p_2\right)}{n_2}}

What the symbols mean

$p_1, p_2$: the true proportions in populations 1 and 2; Unit: none
$n_1, n_2$: the two sample sizes; Unit: none (a count)

Use it when: Use it when the true proportions are known, to find probabilities about the difference $\hat{p}_1 - \hat{p}_2$ .

Watch out: Subtracting under the root because the statistic is a difference. The variances always add: each sample adds its own chance variation.

Learn it: 3.9 Sampling Distributions for the Difference Between Sample Proportions

Standard error of a difference in sample proportions

\displaystyle SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{\hat{p}_1\left(1 - \hat{p}_1\right)}{n_1} + \frac{\hat{p}_2\left(1 - \hat{p}_2\right)}{n_2}}

What the symbols mean

$\hat{p}_1, \hat{p}_2$: the two sample proportions; Unit: none
$n_1, n_2$: the two sample sizes; Unit: none (a count)

Use it when: Use it in a two-proportion z interval for $p_1 - p_2$ . Each sample keeps its own $\hat{p}$ , with no pooling.

Watch out: Using the pooled version for a confidence interval. An interval doesn't assume $p_1 = p_2$ , so it uses this unpooled standard error.

Try it: 60 of 200 people in group 1 and 50 of 250 in group 2 say yes. Build a 95% interval for $p_1 - p_2$ .

Answer: $\hat{p}_1 = 0.30$ and $\hat{p}_2 = 0.20$ . $SE = \sqrt{0.30(0.70)/200 + 0.20(0.80)/250} \approx 0.0411$ , so the interval is $0.10 \pm 1.96(0.0411)$ , about (0.019, 0.181).

Learn it: 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions

Pooled standard error for two proportions

\displaystyle SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\hat{p}_c\left(1 - \hat{p}_c\right)}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}, \quad \text{where } \hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}

What the symbols mean

$\hat{p}_c$: the combined (pooled) proportion: all successes from both samples over the total sample size; Unit: none
$n_1\hat{p}_1 + n_2\hat{p}_2$: the total number of successes in both samples; Unit: none (a count)
$n_1 + n_2$: the total number of individuals in both samples; Unit: none (a count)

Use it when: The sheet labels this one "when $p_1 = p_2$ is assumed". That's only the two-proportion z test with $H_0: p_1 = p_2$ , never an interval: if the null is true, both samples estimate the same p, so you combine them.

Watch out: Averaging $\hat{p}_1$ and $\hat{p}_2$ to get $\hat{p}_c$ when the sample sizes differ. Add the successes and divide by the total, which is what the formula does.

Try it: With the same data (60 of 200 and 50 of 250), test $H_0: p_1 = p_2$ against $H_a: p_1 > p_2$ .

Answer: $\hat{p}_c = 110/450 \approx 0.244$ , and $SE = \sqrt{0.244(0.756)}\sqrt{1/200 + 1/250} \approx 0.0408$ . Then $z = 0.10/0.0408 \approx 2.45$ , and Table A gives 0.9929 to the left, so the p-value is about 0.007.

Learn it: 3.12 Setting Up a Test for the Difference Between Two Population Proportions · 3.13 Carrying Out a Test for the Difference Between Two Population Proportions

Sampling distributions for means

As with proportions, standard deviations use the population's $\sigma$ and standard errors use the sample's s.

Mean of a sample mean

\displaystyle \mu_{\bar{x}} = \mu

What the symbols mean

$\mu_{\bar{x}}$: the mean of the sampling distribution of $\bar{x}$; Unit: same units as the data
$\mu$: the population mean; Unit: same units as the data

Use it when: Use it to describe the center of the sampling distribution of $\bar{x}$ . It says $\bar{x}$ is an unbiased estimator of $\mu$ .

Watch out: Confusing the distribution of individual values with the distribution of sample means. They have the same center but different spreads and possibly different shapes.

Learn it: 4.1 Sampling Distributions for Sample Means

Standard deviation of a sample mean

\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

What the symbols mean

$\sigma_{\bar{x}}$: the standard deviation of $\bar{x}$ from sample to sample; Unit: same units as the data
$\sigma$: the population standard deviation; Unit: same units as the data
$n$: the sample size; Unit: none (a count)

Use it when: Use it when $\sigma$ is known, to find probabilities about a sample mean. If the population is normal, or n is large (the central limit theorem), $\bar{x}$ is approximately normal.

Watch out: Using $\sigma$ instead of $\sigma/\sqrt{n}$ when the question is about the mean of a sample rather than one individual.

Try it: A population has $\sigma = 15$ . What is the standard deviation of $\bar{x}$ for samples of 25?

Answer: $\sigma_{\bar{x}} = 15/\sqrt{25} = 3$ , in the same units as the data.

Learn it: 4.1 Sampling Distributions for Sample Means · 2.12 Sampling Distributions and the Central Limit Theorem

Standard error of a sample mean

\displaystyle SE_{\bar{x}} = \frac{s}{\sqrt{n}}

What the symbols mean

$SE_{\bar{x}}$: the estimated standard deviation of $\bar{x}$; Unit: same units as the data
$s$: the sample standard deviation; Unit: same units as the data

Use it when: Use it in one-sample t intervals and tests for $\mu$ , and for a mean difference with paired data (where s and n come from the list of differences). Use $df = n - 1$ .

Watch out: Treating paired data as two independent samples. If each pair is measured twice, find the differences first and use this formula on them.

Try it: A random sample of 16 has $\bar{x} = 52$ and $s = 8$ . Build a 95% confidence interval for $\mu$ .

Answer: $SE = 8/\sqrt{16} = 2$ . With $df = 15$ , Table B gives $t^* = 2.131$ , so the interval is $52 \pm 2.131(2) = 52 \pm 4.262$ , about (47.74, 56.26).

Learn it: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference · 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference

Mean of a difference in sample means

\displaystyle \mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2

What the symbols mean

$\bar{x}_1 - \bar{x}_2$: the difference between the two sample means; Unit: same units as the data
$\mu_1 - \mu_2$: the difference between the two population means; Unit: same units as the data

Use it when: Use it to describe the center of the sampling distribution of $\bar{x}_1 - \bar{x}_2$ , with two independent samples or two groups in an experiment.

Watch out: Using it for paired data. Paired data give one list of differences, so use the one-sample formulas instead.

Learn it: 4.6 Sampling Distributions for the Difference Between Two Sample Means

Standard deviation of a difference in sample means

\displaystyle \sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}

What the symbols mean

$\sigma_1, \sigma_2$: the two population standard deviations; Unit: same units as the data
$n_1, n_2$: the two sample sizes; Unit: none (a count)

Use it when: Use it when both population standard deviations are known, to find probabilities about $\bar{x}_1 - \bar{x}_2$ .

Watch out: Adding or subtracting the standard deviations themselves. Square each, divide by its n, add (never subtract), then take the square root.

Learn it: 4.6 Sampling Distributions for the Difference Between Two Sample Means

Standard error of a difference in sample means

\displaystyle SE_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

What the symbols mean

$s_1, s_2$: the two sample standard deviations; Unit: same units as the data
$n_1, n_2$: the two sample sizes; Unit: none (a count)

Use it when: Use it in two-sample t intervals and tests for $\mu_1 - \mu_2$ . Get the degrees of freedom from technology.

Watch out: Forgetting to square s, or adding $s_1/\sqrt{n_1}$ and $s_2/\sqrt{n_2}$ directly. Variances add, standard deviations don't.

Try it: Sample 1 has $s_1 = 10$ and $n_1 = 25$ . Sample 2 has $s_2 = 12$ and $n_2 = 36$ . Find the standard error of $\bar{x}_1 - \bar{x}_2$ .

Answer: $SE = \sqrt{10^2/25 + 12^2/36} = \sqrt{4 + 4} \approx 2.83$ , in the units of the data.

Learn it: 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means · 4.10 Carrying Out a Test for the Difference Between Two Population Means

Chi-square statistic

For two-way tables of counts: tests for homogeneity and for independence.

Chi-square statistic

\displaystyle \chi^2 = \sum\frac{\left(\text{observed} - \text{expected}\right)^2}{\text{expected}}

What the symbols mean

$\chi^2$: the chi-square statistic: how far the observed counts are from what the null hypothesis predicts; Unit: none
$\text{observed}$: the observed count: the actual count in one cell of the table; Unit: none (a count)
$\text{expected}$: the expected count: the count you'd expect in that cell if the null hypothesis were true; Unit: none (a count)

Use it when: Use it in a chi-square test for homogeneity or independence. Add one term for every cell, then find the p-value from Table C with $df = (r - 1)(c - 1)$ .

Watch out: Using proportions or percents instead of counts, or forgetting to square. The expected count, not the observed count, goes in the denominator.

Try it: Group A: 30 yes and 70 no. Group B: 50 yes and 50 no. Find $\chi^2$ and the p-value.

Answer: Expected counts are 40 yes and 60 no in each group. $\chi^2 = 100/40 + 100/60 + 100/40 + 100/60 \approx 8.33$ with $df = 1$ . In Table C, 8.33 is between 7.88 and 9.14, so $0.0025 < p < 0.005$ .

Learn it: 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence · 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

Tables

Three tables of probabilities and critical values. Technology gives the same numbers more precisely.

Table A: standard normal probabilities

\displaystyle \text{area to the left of } z = P\left(Z < z\right)

What the symbols mean

$Z$: a standard normal random variable, with mean 0 and standard deviation 1; Unit: none
$z$: a z-score, found from the row (ones and tenths) and column (hundredths); Unit: none

Use it when: Each entry is the area to the left of z. Use it for normal probabilities, p-values for z tests, and (reading it backward) to find $z^*$ or a percentile.

Watch out: Forgetting the table only gives area to the left. For "greater than", subtract from 1, and for a two-sided p-value, double the tail area.

Try it: Find $P(Z < -1.96)$ and $P(Z > 1.60)$ .

Answer: Row −1.9, column .06 gives 0.0250. Row 1.6, column .00 gives 0.9452, so $P(Z > 1.60) = 1 - 0.9452 = 0.0548$ .

Learn it: 2.11 The Normal Distribution · 3.6 p-Values · 3.7 Carrying Out a Test for a Population Proportion

Table B: t distribution critical values

\displaystyle \text{area to the right of } t^* = p

What the symbols mean

$t^*$: the critical value: probability p lies above it, and probability C lies between $-t^*$ and $t^*$; Unit: none
$p$: the tail probability, listed across the top; Unit: none
$C$: the confidence level, listed across the bottom; Unit: none
$df$: degrees of freedom, one per row, going up to 1000 and ∞; Unit: none

Use it when: Use it to find $t^*$ for a confidence interval (read the confidence level at the bottom) or to bracket a p-value for a t test (find where your t falls in the df row). The ∞ row gives the $z^*$ values.

Watch out: Reading the wrong column: the top row is one tail's probability, not the confidence level. If your df isn't listed, use the next smaller df.

Try it: Find $t^*$ for a 95% interval with $df = 15$ , and the matching $z^*$ .

Answer: Row 15, column .025 (95% at the bottom) gives $t^* = 2.131$ . The ∞ row in the same column gives $z^* = 1.960$ .

Learn it: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference · 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference · 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means

Table C: chi-square critical values

\displaystyle \text{area to the right of } \chi^2 = p

What the symbols mean

$\chi^2$: the critical value with probability p above it; Unit: none
$p$: the tail probability, listed across the top; Unit: none
$df$: degrees of freedom, one per row; Unit: none

Use it when: Use it to bracket the p-value for a chi-square test. Find your df row, then see which two critical values your $\chi^2$ falls between.

Watch out: Doubling the p-value. A chi-square test only uses the right tail, so the column heading is already the p-value bound.

Try it: What $\chi^2$ value is needed to reject at $\alpha = 0.05$ with $df = 2$ ?

Answer: Row 2, column .05 gives 5.99. A $\chi^2$ above 5.99 gives a p-value below 0.05.

Learn it: 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

Not on the sheet: know these

The exam expects you to know these without being given them.

z-score
$\displaystyle z = \frac{x - \mu}{\sigma}$
It tells you how many standard deviations a value is above or below the mean. You need it to compare values from different distributions and to use Table A. With sample data, use $\bar{x}$ and s.
Learn it: 1.9 Comparisons of the Distributions for One Quantitative Variable · 2.11 The Normal Distribution
Interquartile range and the outlier rule
$\displaystyle \text{IQR} = Q_3 - Q_1$
A value is an outlier if it's more than $1.5 \times \text{IQR}$ below $Q_1$ or above $Q_3$ . For example, with $Q_1 = 10$ and $Q_3 = 22$ , the IQR is 12, so values below −8 or above 40 are outliers.
Learn it: 1.7 Summary Statistics for One Quantitative Variable · 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable
Residual
$\displaystyle \text{residual} = y - \hat{y}$
Actual minus predicted. A positive residual means the point is above the line. Residual plots with no pattern support a linear model.
Learn it: 5.4 Residuals
The regression line goes through the point of means
$\displaystyle \left(\bar{x}, \bar{y}\right)$
The least-squares line always passes through $(\bar{x}, \bar{y})$ , so plugging in $\bar{x}$ predicts $\bar{y}$ .
Learn it: 5.5 Least-Squares Regression
Coefficient of determination
$\displaystyle r^2$
It's the proportion of the variation in y that's explained by the linear model with x. Interpret it in context, for example "about 95% of the variation in exam scores is explained by the linear model with hours studied".
Learn it: 5.5 Least-Squares Regression
Multiplication rule for independent events
$\displaystyle P\left(A \cap B\right) = P\left(A\right) \cdot P\left(B\right)$
It works only when A and B are independent. Don't confuse independent with mutually exclusive: mutually exclusive events have $P(A \cap B) = 0$ .
Learn it: 2.7 Independent Events and Unions of Events · 2.5 Mutually Exclusive Events
Expected counts and degrees of freedom for chi-square
$\displaystyle \text{Expected count} = \frac{\left(\text{row total}\right)\left(\text{column total}\right)}{\text{table total}}, \quad df = \left(r - 1\right)\left(c - 1\right)$
The sheet gives the $\chi^2$ formula but not how to get the expected counts or the df, where r and c are the numbers of rows and columns.
Learn it: 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence · 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence
Degrees of freedom for t procedures
$\displaystyle df = n - 1$
Use $df = n - 1$ for a one-sample t procedure, and for paired data, where n is the number of pairs. For two independent samples, take the df from technology.
Learn it: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference · 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference · 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means
Conditions for inference
Random: a random sample or random assignment. 10%: when sampling without replacement, $n \le 0.10N$ . For proportions, large counts: at least 10 successes and 10 failures (in a one-proportion test, check $np_0$ and $n(1 - p_0)$ ; in a two-proportion test, use $\hat{p}_c$ for both groups). For means, the population is normal or $n \ge 30$ ; with a small sample, the data should show no strong skew or outliers. For chi-square, every expected count is greater than 5.
Learn it: 3.2 Sampling Distributions for Sample Proportions · 4.1 Sampling Distributions for Sample Means · 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence
Probability of a Type I error
$\displaystyle P\left(\text{Type I error}\right) = \alpha$
A Type I error rejects a true null hypothesis, and its probability is the significance level. A Type II error fails to reject a false one, and its probability is 1 minus the power.
Learn it: 3.8 Potential Errors When Performing Tests

Using the sheet on exam day

Descriptive statistics

Sample mean

What the symbols mean

Sample standard deviation

What the symbols mean

Least-squares regression line

What the symbols mean

Probability

Addition rule for A or B

What the symbols mean

Conditional probability

What the symbols mean

Probability distributions

Mean of a discrete random variable

What the symbols mean

Standard deviation of a discrete random variable

What the symbols mean

Binomial probability

What the symbols mean

Mean of a binomial random variable

What the symbols mean

Standard deviation of a binomial random variable

What the symbols mean

Sampling distributions and inferential statistics

Standardized test statistic

What the symbols mean

Confidence interval

What the symbols mean

Sampling distributions for proportions

Mean of a sample proportion

What the symbols mean

Standard deviation of a sample proportion

What the symbols mean

Standard error of a sample proportion

What the symbols mean

Mean of a difference in sample proportions

What the symbols mean

Standard deviation of a difference in sample proportions

What the symbols mean

Standard error of a difference in sample proportions

What the symbols mean

Pooled standard error for two proportions

What the symbols mean

Sampling distributions for means

Mean of a sample mean

What the symbols mean

Standard deviation of a sample mean

What the symbols mean

Standard error of a sample mean

What the symbols mean

Mean of a difference in sample means

What the symbols mean

Standard deviation of a difference in sample means

What the symbols mean

Standard error of a difference in sample means

What the symbols mean

Chi-square statistic

Chi-square statistic

What the symbols mean

Tables

Table A: standard normal probabilities

What the symbols mean

Table B: t distribution critical values

What the symbols mean

Table C: chi-square critical values

What the symbols mean

Not on the sheet: know these

z-score

Interquartile range and the outlier rule

Residual

The regression line goes through the point of means

Coefficient of determination

Multiplication rule for independent events

Expected counts and degrees of freedom for chi-square

Degrees of freedom for t procedures

Conditions for inference

Probability of a Type I error