# AP® Statistics Equation and Formula Sheet

The AP® Statistics Formula Sheet will be one of your most important tools for success on the AP Statistics exam. In order to get an edge in your preparation for the exam, it is a great idea to familiarize yourself with the formula sheet beforehand. Here we will describe everything you need to know about the formula sheet in detail.

## What is AP Statistics Formula Sheet and Tables

The AP Statistics Formula Sheet and Tables includes formulas and probability tables that you will need to solve both the multiple-choice and free-response questions on the exam. The formulas cover all units of the course curriculum except Unit 3 (Collecting Data), which does not require formulas. The tables provide left-tailed or right-tailed areas under the normal distribution curve, the t-distribution curve, and the chi-square distribution curve.

It is technically possible to solve every question on the AP Statistics exam using just the formulas on the formula sheet, algebra, and the probability tables. But the formulas on the sheet won’t be helpful if you do not already have a solid understanding of the course content. Plus, many problems are better suited for calculators than using the formulas. Still, along with the material you learn from your AP Statistics course, learning about each of the formulas and tables will help you be more efficient and better understand the material on the exam.

In this guide, we will cover what is included with the AP Statistics Formula Sheet and Tables generally, as well as go over each individual formula in detail. In addition, we will go over many questions you may have about the formulas and table sheet, including what formulas are NOT included and so should be memorized.

### Descriptive Statistics

The first section of the formula sheet provides the formulas for descriptive statistics from Unit 1 (Exploring One-Variable Data) and Unit 2 (Exploring Two-Variable Data). It is rare to use these formulas directly on questions, especially for multiple-choice questions. Moreover, if you do end up with a chance to calculate the statistics mentioned in this section, it will generally be easier to use a calculator. However, you will absolutely need to understand them and reference them for certain types of questions. Therefore, making yourself familiar with this section is still important.

 x̅ = 1 / n ∑ xi = ∑ xi / n sx = √ 1 / n - 1 ∑ (xi - x̅ )2 = √ ∑ (xi - x̅ )2 / n - 1 ŷ = a + bx y̅ = a + b x̅ r = 1 / n - 1 ∑ ( xi - x̅ / sx ) ( yi - y̅ / sy ) b = r sy / sx

### Probability and Distributions

The second section of the formula sheet contains two important formulas from basic probability as well as several formulas to calculate the mean and standard deviation for any discrete random variable that will show up on the exam. Unlike the first section of the formula sheet, you will almost certainly be required to use these formulas for direct calculations. It is also less likely that you will be able to use calculators to automatically solve the questions involving these formulas. For that reason, it will be helpful to specifically practice using these formulas with AP Statistics practice questions.

P (AB ) = P (A ) + P (B ) - P (AB )
P (A | B ) =
P (AB ) / P (B)
Probability Distribution Mean Standard Deviation
Discrete random variable. X µx = E (X ) = xi . P (xi ) σx = √∑ (xi - µx )2 . P (xi )
If X has a binomial distribution with parameters n and p, then:
P (X = x ) = (
n / x
) px ( 1 - p )n-x
where x = 0, 1, 2, 3, ..., n
µx = np σx = √np ( 1 - p )
If X has a geometric distribution with parameter p, then:
P (X = x ) = (1 - p )n - x p
where x = 1, 2, 3, ...
µx =
1 / p
σx =
√1 - p / p

### Sampling Distributions and Inferential Statistics

The third section of the formula sheet contains meta formulas for the test statistics and confidence intervals on the AP Statistics exam. They are meta formulas because they do not make calculations directly, but are combined with specific information for each procedure. These meta formulas are incredibly important for the AP Statistics exam and you should understand them deeply both on a conceptual level and as be able to use them to generate the test statistics and confidence intervals for the exam.

 Standardized test statistic: statistic - parameter / standard error of the statistic Confidence interval: statistic ± (critical value) (standard error of the statistic)
 Chi-square statistic: χ2 = Σ (observed - expected)2 / expected

The information that can be combined with these formulas is given in the following parts of this section. They also contain general information important for understanding the sampling distributions of the important statistics on the example. For example, for procedures involving sample proportions:

Random Variable Parameters of Sampling Distribution Standard Error of Sample Statistic
For one population:
µ = p       σ = √
p ( 1 - p ) / n
S =√
(1 - ) / n
For two populations:
1 - 2
µ 1 - 2 = p1 - p2    σ 1 - 2 = √
p1 ( 1 - p1 ) / n1
+
p2 ( 1 - p2 ) / n2
S 1 - 2 = √
1 ( 1 - 1 ) / n1
+
2 ( 1 - 2 ) / n2

when p1 - p2 is assumed:
S 1 - 2 = √ c ( 1 - c
) (
1 / n 1
+
1 / n 2
)
where c =
X 1 + X 2 / n 1 + n 2

### Probability Tables

Included with the formula sheet are tables that provide probabilities for the normal distribution, t-distribution, and the chi-square distribution. The probabilities these tables provide are different for each distribution, based on their use on the AP Statistics exam. These tables are used to calculate certain probabilities, mostly probabilities over intervals and p-values. However, the information in these tables can also be found using a calculator.

Should I use the probability tables included with the formula sheet or should I use a calculator?

Many students prefer calculators, but it’s really a personal preference. Calculators can give more information, but those good with tables may find they are faster. You really should be familiar with both. A calculator can always fail, but it will give the most precise results. Conversely, using a table may not be easy, but it is a great way to learn the content of the course.

Study smarter and see A's in your future.   ## AP Statistics Formula Sheet & Tables

Need help with AP® Statistics?
We make really hard stuff easy to understand. Study smarter with UWorld. ### What formulas are not included with the AP Statistics formula sheet?

The AP Statistics formula sheet does not include all formulas you’ll need for the AP Statistics exam. Here is a list of statistics and rules on the AP Statistics exam you will need to memorize.

Units Formulas
Unit 1
1. Relative frequency
2. Median
3. First and Third Quartiles
4. Percentiles
5. The range of a dataset
6. The interquartile range
7. Outliers (according to 1.5 x IQR Rule)
8. 68-95-99.7 rule (empirical rule)
9. Z-scores (standardization formula)
Unit 2
1. Joint relative frequency
2. Marginal relative frequency
3. Conditional relative frequency
4. Residual (Note: Keep in mind the order of the elements in the equation)
5. Coefficient of determination (formula based on correlation)
Unit 3
• None
Unit 4
1. Probability of an event (when all outcomes equally likely)
2. Probability of the complement of an event
3. Multiplication rule (another form of conditional probability formula already on the formula sheet)
4. Conditions for independence of two events
5. Mean and standard deviation (variance) of the linear combination of two random variables
6. Mean and standard deviation (variance) of the linear transformation of a random variable
Unit 5
1. Conditions for when sampling distributions are approximately normal distributed
Unit 6 & 7
1. Inverse-square relationship between sample size and margin of error of a confidence interval
2. The appropriate degrees of freedom for each t-test
Unit 8
1. Expected count formula for each chi-square test
2. The degrees of freedom for each chi-square test
Unit 9
1. The degrees of freedom for the t-test for a slope, both for a simple linear regression with a y-intercept and a simple linear regression without a y-intercept