AP Stats CRAM Review!

By Addison Wood || AP Stat student 2025

Please note that this exam sheet is not broken down into the typical units that AP review breaks it down into. These are the units used by stats medic and go more in depth to each section. NOT FINISHED

Chapter 1: Exploring One Variable Data

Terms to know:

• Quantitative variable: takes numerical values for a measured or counted quantity

  • Discrete: countable number of values

  • continuous: infinite values

• Categorical variable: takes on values that are category names or group labels

  • these are typically used by bar graphs

• Mean: mean is the average of the data set

• Median: the middle value in the data set; when the data is arranged in ascending order; if there is an even number of observations, the median is the average of the two middle values.

• Association: If knowing the value of one variable helps in predicting the value of another variable, there is said to be an association between them.

1.2: Representing Categorical Data

When given 2 categorical variables, here is the best way to represent them

• Segmented bar graph: stack up bars to make 100%

• Mosaic Plot: Segmented bar graph were the width of the bars is proportional to the group size.

1.3: Describing Quantitative Data

• Dot plots & Stem plots: show every individual value

• Histogram shows general shape

4 shapes:

• Skewed left: looks like your left foot

• Skewed right: looks like your right foot

• Symmetric: looks like a hill

• Bimodal: looks like a two humped camel

To describe:

S: shape (see above)

O: outliers (are there any points that don’t “fit in” with the distribution)

C:center (median / mean)

V: variability (standard deviation)

When describing always use context. Context makes the problem real. Try to use “ly” words so your words aren’t held against you.

1.4: Measuring variability:

When asked to interpret Standard Deviation: “The context typically varies by Standard Deviation from the mean of x.”

If asked for variance, square the standard deviation

Outliers: these greatly affect the mean and the standard deviation, meaning they are nonresistant. The median however is not affected, so it is resistant to any present outliers.

If the distribution is symmetric use mean and standard deviation

If the distribution is skewed or has outliers use the median and IQR

1.5: Comparing Quantitative Data

To find outliers:

• 1.5 x IQR

  • low outlier <Q1 - 1.5IQR

  • high outlier > Q3 + 1.5IQR

Boxplots show the 5 number summary

To compare use SOCV + context (see 1.3)

Chapter 2: Modeling the Data

2.1: Percentiles

• the n^th percentile is the value that has n% of the data less than or equal to it

• key term is “at” a certain percentile; for example, if a student scores at the 80th percentile, they performed better than 80% of the peers who took the same assessment.

• Cumulative Relative Frequency Graph:

  Q1 is the 25th percentile

• Q3 is the 75th percentile

• Median is the 50th percentile

2.2: Location in a Distribution

• Z scores: (value - mean) / standard deviation

• “Context is z-score standard deviations above / below the mean

• Z-scores show position relative to other values in the distribution

2.3: Linear Transformations of Quantitative Data

• if adding or subtracting by a constant

  • shape is the same

  • center + or - constant

  • variability is the same

• if multiplying or dividing by a constant

  • shape is the same

  • center is x or / by constant

  • variability is x or / by constant

2.4: Normal Distributions and the Empirical Rule

Density Curves:

• total area = 1

• area = a proportion of values

Skewed right: mean > median

Skewed left: mean < median

Symmetric: mean = median

Empirical Rule:

• 68% of data is found within 1 SD of the mean

• 95% within 2 SDs

• 99.7% within 3 SDs

2.5 Normal Distribution Calculations:

• Step 1: find z score

• Step 2: use table A by finding the proportion of the z score

If given a z score, back calculate using your algebra skills

Chapter 3: Two Variable Data

Terms to know:

explanatory variable: explains response variable

response variable: measures outcome

3.1: Scatterplots

Scatterplots: explanatory variable on the x axis, response variable on the y axis.

To describe a scatterplot relationship: DUFS + context

Direction (positive / negative / none)

Unusual features (outliers / clusters)

• unusual points can strengthen r if in pattern

• unusual points can weaken r if out of pattern

Form (linear or non-linear)

Strength (how close to the form)

& context always!

3.2 Correlation

Correlation R:

• direction (+ or -)

• form: linear

• strength: between -1 and 1

• “The linear relationship between x and y is strength and direction”

Coefficient of determination r^²

• “The percent of the variation in y is explained by the linear relationship with x.”

NOTE: CORRELATION DOES NOT EQUAL CAUSATION

3.3 Making Predictions

Predictions:

• y= a + bx ; where y = predicted y; a = y int; and b = slope

• ** be cautious with extrapolation

Residuals:

• residual = actual - predicted → R = A-P

• “The actual context was residual above / below the predicted values for x = #”

Interpretation:

• “When x = 0 context, the predicted y-context is y-int.”

• “For each additional x-context the predicted y context increases/decreases by slope.”

3.4 Residual Plots

LSRL

• the least squares regression line minimizes the sum of the squared residuals

Residual Plots:

• randomness is good 🙂

• patterns are bad ☹

  • tree shapes

  • frowny faces

  • smiley faces

3.5 Outliers, High Leverage, and Influential Points

Outliers: out of pattern (large residuals)

High Leverage: very large or very small x-values

Influential: if removed, big changes to slope, y-intercept, r

Outliers & LSRL:

• Horizontal Outlier → tilt the line

• Vertical Outliers → shift line up or down

3.6 Transforming Non-linear Data

transformations:

• original data to transform to linear

• linear → graph x vs y

• exponential → graph x vs log y

• power → graph log x vs log y

3.7 Choosing the Best Model

1. Check the scatterplot for a linear pattern

2. check the residual plot for no leftover pattern

3. check for the r² that is closest to 1

Chapter 4: Collecting Data

4.1 Simple Random Samples

• convenience samples and voluntary response samples can lead to bias

• SRS limit bias

• All of these are taken from the population

To take a simple random sample:

1. Label individuals (ex: assign numbers or slips of paper)

2. Randomize (ex: random number generator or names in a hat)

3. Select

4.2 Stratified Random Samples

Stratified Random samples happen by splitting the population into groups (strata) then choosing an SRS from each strata.

Note: each strata has individuals with shared attributes or characteristics (homogenous grouping)

4.3 Cluster and Systematic Random Samples

Cluster samples are when the groups are split into strata, but instead of sampling a few from all groups, a few groups are sampled as a whole. Groups are heterogenous in this method, meaning that the groups do not need to be similar.

Systematic Random Sample is when a random starting point is chosen and a sample is taken at every nth individual until the sample size is met.

4.4 Potential Problems With Sampling

Under coverage: some people are less likely to be chosen

• calling landlines, surveying homeowners

Non response: people cannot be reached or refuse to answer

• don’t answer or hang up on phone call

Response bias: problems in the data gathering instrument or process

• people lie (self reported responses) or the wording of the question

4.5 Observational Studies and Experiments

• If a confounding variable is present, that means there is something present that is related to the explanatory variable that influences the response variable

Observational study: no treatment imposed

Experiment: treatments imposed, which allows us to show causation

Experimental Units: what/who treatment is imposed on

Treatments: what is done (or not done) to experimental units; levels or a combination of levels of the explanatory variable

• Control group: a baseline group that does not receive the treatment, used for comparison

• Random assignment: the process of assigning experimental units to treatments at random, minimizes bias.

4.6 Designing Experiments

A well designed experiment has 4 key features:

1. Comparison → there are two or more treatments imposed

2. Random assignment → this allows us to draw causation

   1. label

   2. randomize

   3. assign

3. Replication → there is more than one individual in each treatment group

4. control → keeps the other variables constant and allows for a basis of comparison

The placebo effect is when a fake treatment works

Blinding is when subjects (single blind) and / or experimenters (double blind) don’t know about treatments

4.7 Selecting and Experimental Design

Block Design:

• Blocks are a group of experimental units that are similar

• Randomized Block Design is when subjects are separated into blocks and then randomly assigned treatments within each block.

Matched Pairs Design:

• subjects are paired (blocks of size 2) and then randomly assigned to a treatment

• each subject receives two treatments

• the order of the treatments must be randomized

4.8 Interference and Experiments

Statistically significant

• when results of an experiment are unlikely (less than 5%) to happen purely by chance

• if statistically significant results are obtained, there is convincing evidence the treatment caused the difference.

4.9 Scope of Interference

Random Sample allows us to generalize our conclusions to the population from which we sampled.

Random Assignment allows us to conclude a treatment causes changes in the response variable

Chapter 5: Probability

5.1 Introducing Probability

A probability is a long run frequency.

• always between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• short run is unpredictable

• long run is predictable

Law of Large Numbers:

• simulated probabilities tend to get closer to the true probability as the number of trials increases

5.2 Simulation

Simulation is a way to model random events, such that simulated outcomes match real world outcomes.

• ex: dice roll, coin toss, applet, random number generator

Evidence for a claim:

assuming a claim is true, find the probability of getting the observed result or more extreme

• < 5%

  • statistically significant

  • convincing evidence against the claim

5.3 Rules for Probability

Sample Space: List of all possible outcomes

• P(x) = (# of outcomes in X) / (total # of outcomes in sample space)

• ALL OUTCOMES MUST BE EQUALLY LIKELY

Rules

• Complement Rule: P(not A) = 1- P(A)

Notation:

• P(A and B) = P (A n B) → both occur

• P(A or B) = P (A u B) → one or the other or both

5.4 The Addition Rule

• Two way tables and venn diagrams are used a lot, so make sure you are able to interpret and apply them

Addition Rule

• P(A or B) = P(A) + P(B) - P(A and B)

• if events A and B are mutually exclusive, they cannot occur together meaning

  P(A and B) = 0 therefore P(A or B) = P(A) + P(B)

5.5 Conditional Probability

Conditional Probability is the probability of one event given another event has occurred

• P(A | B) = P(A and B) / P(B)(the given condition)

Independent events:

• knowing whether or not one event occurs does not change the probability of the other event

• If P(A)= P(A|B)= P(A| not B) → A and B are independent

5.6 Tree Diagrams

General Multiplication Rule:

• P(A and B) = P(A) x P(B|A)

• IF A AND B ARE INDEPENDENT

  • P(B|A) = P(B) so…

  • P(A and B) = P(A) x P(B)

Tree diagrams:

• the probability must add up to one.

  • first branches is probability of of event happening / not happening

  • next branches is the probability of that event happening / not happening given an event has already happened

Chapter 6: Random Variables

6.1 Discrete Random Variables

Probability distribution always adds to one

Discrete Random Variable: takes a countable number of values with gaps between

6.2 Continuous Random Variable

Random Variables:

• discrete: has a countable number of values with gaps

• continuous: has infinite values with no gaps

Probabilities:

• find the area under the curve

• uniform:

  • 1/k → y axis

  • k → x axis

  • b x h

• Normal curves:

  • table A

6.3 Transforming Random Variables

Add / Subtract a constant c:

• shape: stays the same

• center (mean or median): add / subtract c

• variability (range, IQR, SD, variance): stays the same

• shape: stays the same

• center: multiply / divide by c

• variability:

  • range, IQR, SD: multiply / divide by c

  • variance: multiply / divide by c²

6.4 Combining Random Variables

Combining Random variables

• DO NOT ADD STANDARD DEVIATIONS

• Addition

  • M_x+Y = M_x+ M_Y

  • SD_x+Y = sqrt(SD_x²+ SD_Y²)

• Subtraction:

  • M_x-Y = M_x- M_Y

  • SD_x-Y = sqrt(SD_x²+ SD_Y²)

Normal Distribution calculations

• use formulas to find M + SD

• proceed as usual

6.5 Introduction to the Binomial Distribution

Binomial random variable:

• x → number of successes

1. Binary? Is there success or failure

2. Independent trials?

3. Number of trials is fixed? (n)

4. Same probability of success (p)

Binomial formula:

• P(x=k) = nCkp^k(1-p)^(n-k)

  • nCk→ the # of ways to get k successes

  • p^k→ probability of successes where k= # of successes

  • (1-p) → probability of failure

  • n-k → # of failures

6.6 Parameters for Binomial Distribution

Using technology:

P(x=k) → binompdf (n,p,k)

P(x <= k) → binomcdf (n,p,k)

Mean + Standard Deviation for binomial

• M=np “

  • After many, many groups of n trials, the average number of successes is M”

• SD = sqrt of (np(1-p))

  • “The number of successes typically varies by a SD from the mean of M”

6.7 Conditions for Inference

10 % condition:

• when taking a random sample (without replacement) of size n from a population of size N we can use a binomial distribution if n <= .10N.

Large Counts Conditions:

• use a normal distribution to model a binomial distribution if np >= 10 and n(1-p) >= 10

• np = # of successes

• n(1-p) = # of failures

6.8 The Geometric Distribution

BITS:

• Binary: success or failure

• Independent trials

• Trials until success

• Same probability of success

Geometric formula: P(X = k) = (1 - p)^{k-1} p, where p is the probability of success and k is the number of trials.

• shape: skewed right

• center: M = 1/p

• Variability: SD = (sqrt of (1-p)) / p

Chapter 7: Sampling Distribution

7.1 Sampling Distributions

A statistic is used to estimate a parameter

• Statistic → parameter

• p hat → p

• x bar → M

• s → SD

Populations = parameters

Samples = statistics

Sampling Distribution:

• The distribution of values for a statistic for all possible samples of a given size from a given population

7.2 Bias and Variability

Biased VS Unbiased Estimator:

• Biased:

  • consistently overestimates or

  • consistently underestimates the true population parameter

• Unbiased:

  • mean of the sampling distribution is equal to the population parameter

Sample Size:

• as n increases, variability of the sampling distribution decreases

A good statistic has:

• low bias (accurate)

• low variability (precise)

7.3 Sample Proportions

Sampling Distribution of p hat:

• shape: approximately normal is np >= 10 & n(1-p) >= 10

• center: M_p-hat= p

• variability: SD_p-hat= sqrt(p(1-p)/n)

  • check 10% condition if sampling without replacement

Normal Probability:

• z = (p-hat - p) / sqrt (p(1-p)/ n)

7.4 Differences in Sample Proportions

Sampling Distributions of p-hat1 - p-hat2

• shape: approximately normal if

  • n₁p₁>= 10 & n₂p₂>= 10

  • n₁(1-p₁)>= 10 & n₂(1-p₂)>= 10

• center:

  • M_{(p-hat 1 - p-hat 2)}= p1 - p2

• variability