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.
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.
1 / n∑ xi =
∑ xi / n
sx = √
1 / n - 1∑ (xi - x̅ )2 = √
∑ (xi - x̅ )2 / n - 1
|ŷ = a + bx||y̅ = a + b x̅|
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 (A | 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, ...
1 / p
√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:
µp̂ = p σp̂ = √
p ( 1 - p ) / n
p̂ (1 - p̂ ) / n
For two populations:
p̂ 1 - p̂ 2
|µ p̂ 1 - p̂ 2 = p1 - p2
σ p̂ 1 - p̂ 2 = √
p1 ( 1 - p1 ) / n1+
p2 ( 1 - p2 ) / n2
S p̂ 1 - p̂ 2 = √
p̂1 ( 1 - p̂1 ) / n1+
p̂2 ( 1 - p̂2 ) / n2
when p1 - p2 is assumed:
S p̂ 1 - p̂ 2 = √ p̂c ( 1 - p̂c ) (
1 / n 1+
1 / n 2)
where p̂c =
X 1 + X 2 / n 1 + n 2
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.
AP Statistics Formula Sheet & Tables
In Words: The left side of the equation is an x with a “bar” (line above the x) that represents the sample mean, while the right side of the equation is equal to the sum of the data divided by the number of data points (n ).
Description: This sample mean is the most important statistic on the AP exam. It represents the “center” or “average” of a set of data.
Application on AP Exam: It is unlikely that you will need this formula for calculation. This formula is conceptually important – think the central limit theorem, sampling distributions, t-tests, and other important topics. The formula is also useful for understanding how certain manipulations affect the sample mean. For example, how does multiplying the values in a dataset by 2 affect the sample mean?
Sample Standard Deviation
In Words: The left side of the equation is the sample standard deviation for the variable “x” (the subscript). The right side of the equation involves a lot of symbols, but it is straightforward if you take it in steps. (1) Subtract the sample mean from every value in the dataset, (2) Square the resulting values, (3) Add up the resulting values, (4) Divide the result by n – 1, and finally (5) Take the square root of the final result.
Description: This sample mean is the second most important statistic on the AP exam. It represents the “spread” or “variability” of a set of data.
Application on AP Exam: It is unlikely that you will need this formula purely for calculation. If it does come up, you should probably use a calculator instead of this formula. Most questions involving standard deviation will either give you the value of the standard deviation or otherwise require you to interpret features of the standard deviation.
Predicted value of the response variable (linear regression)
In Words: The left side of the equation represents the predicted value of the response variable in a simple linear regression model. It is denoted with a “y” with a so-called “hat” on top (the circumflex symbol). The right side of the equation provides the linear equation in which a given value of the explanatory variable (x ) is an input. The value of x is multiplied by the slope b and then y-intercept a is added.
Description: Simple linear regression is a procedure to predict a value of one variable (the response variable: y ) from the value of another variable (the explanatory variable: x ). This formula shows how prediction is done through simple linear regression – through plugging x into the equation for a line.
Application on AP Exam: One common question type on the AP exam is interpolation.. If the slope (a ) and y-intercept (b ).are given (and it’s likely they will be), then you should be able to turn the value of an explanatory variable x into the predicted value of the response variable y. Another type of question involves residuals. You will need to find the predicted value of y using a regression line, then compare to the observed value.
Point on the linear regression line
In Words: The left side of the equation (y with a bar) is the sample mean for a response variable y. The right side of the equation is the simple linear equation in which the mean of the explanatory variable is an input. The mean of x is multiplied by the slope b and then y-intercept a is added.
Description: This formula makes a conceptual point as well as being useful for calculation. The conceptual point is that the predicted value of y when x is equal to its mean is equal to the sample mean of y; that is, the regression line always goes through the point (x̄,ȳ ). This formula is also used for calculating the y-intercept of a linear regression line. If you have the sample mean of x, the sample mean of y, and the slope of the regression line, then this formula can be rearranged to isolate the value of a.
Application on AP Exam: The usage of this equation for the exam is as a reminder about the one point mentioned above – the point (x̄,ȳ ) is always on the regression line. Also, while such a question is highly unlikely, it is possible that you will need to calculate a y-intercept using this formula because it is a notable part of the curriculum.
In Words: The left side of the equation (the “r”) is the correlation coefficient. The right side of the equation involves a lot of symbols, but like the standard deviation it is straightforward if you take it in steps. First, note that the expressions in the values are actually a type of z-score. (1) Calculate the z-scores for the data values for both variable x and variable y, (2) Multiply the paired z-scores, (3) Add up the resulting values, and finally (4) Divide the result by n – 1.
Description: The correlation coefficient, r, gives the direction and strength of the linear association between two quantitative variables.
Application on AP Exam: This is another formula you are unlikely to use directly on the AP exam. Similarly to the standard deviation, if it does come up, you should probably use a calculator instead of this formula. However, understanding this formula (especially the z-score interpretation) can help with several types of problems. For example, how does multiplying one variable by -1 affect the correlation coefficient?
Slope of the linear regression line
In Words: The left side of the equation (the “b”) is the slope of the linear regression line. The right side of the equation includes the correlation coefficient, multiplied by the ratio of the standard deviation of the response variable (y ) to the standard deviation of the explanatory variable (x ). Note: Keep in mind that the standard deviation in the numerator is the standard deviation for the values of y, NOT the standard deviation for the residuals of the linear regression.
Description: The slope of the linear regression line is the amount that the predicted value of the response variable y changes for each unit increase in x.
Application on AP Exam: There are several important applications of this formula on the AP exam. One that has come up several times is to understand the relationship between the correlation coefficient and slope. For example, if the correlation coefficient is positive, the slope must also be positive (and vice-versa). You should also be able to calculate the slope if given the correlation and the standard deviation of both variables involved in a regression.
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.
|Unit 6 & 7||
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