AP Statistics Unit 1-8 Review Notes

Categorical vs. Quantitative Data

• Quantitative data: Deals with numbers (quantity).

  • Examples: Heights, class size, population size.

• Categorical data: Deals with names and labels.

  • Examples: Eye color, hair color.

Categorical Data: Two-Way Tables

• Two variables on either side show intersections between variables.

• Example: Math students (10 total), split into internal (3) and external (7) variables. English students similarly split.

• Marginal Relative Frequency: Percentage of data in a single row or column compared to the total.

  • Calculation: $\frac{b}{d}$ (row total over total) or $\frac{c}{d}$ (column total over total).

• Joint Relative Frequency: Percentage of data in a single group compared to the total.

  • Group refers to a singular variable or intersection.

  • Calculation: $\frac{A}{D}$, e.g., Math Internal (3/23).

• Conditional Relative Frequencies: Percentage of data in a single category given a specific group.

  • Calculation: $\frac{a}{b}$ or $\frac{a}{c}$, e.g., given the student is in Math (10 total), the percentage that is internal (3/10).

Quantitative Data: Describing Data (CSOX)

• Context: Provide context for the data.

• Shape: Describe the distribution's shape.

  • Symmetrical or skewed?

  • Number of peaks (unimodal, bimodal)?

• Outliers: Identify any data points far from the rest.

• Center: Measure the center using mean or median.

• Spread: Measure the spread using range, standard deviation, or IQR.

• Use descriptive language (e.g., "strongly," "roughly") and comparative language.

Basic Terms

• Mean: Sum of all values divided by the number of values (average). Average value.

• Standard Deviation: Measure of variation. The value-context typically varies by standard deviation value from the mean of mean.

  • Example: "The average Prepworks subscriber's IQ typically varies by 5 IQ points from the mean of 169 IQ points."

• Median: 50th percentile. Value in the middle when data is organized from least to greatest.

• Range: Value of max - min in the dataset.

Box Plots and Five-Number Summary

• Five-number summary:

  • Minimum: Smallest value.

  • Q1 (25th percentile): Median of the minimum and median.

  • Median (50th percentile).

  • Q3 (75th percentile).

  • Maximum value.

• IQR (Interquartile Range): Q3 - Q1.

• Low-End Outliers: Any value less than $Q1-1.5\left(IQR\right)$

• High-end Outliers: Any value that is more than $Q3+1.5\left(IQR\right)$

Additional Terms

• Officially Defined Percentile: Percentage of values less than or equal to a specific value.

• Cumulative Relative Frequency: Cumulative percentages from each interval up through all the data.

• Relative Frequency: Frequency of an occurrence over the total.

Z-Scores

• Number of standard deviations a value is away from the mean.

• Formula: $z = \frac{value - mean}{standard \ deviation}$

Transforming Data

• Adding/Subtracting a Constant:

  • Shape and variability stay the same.

  • Center moves up or down by that amount.

• Multiplying/Dividing:

  • Shape stays the same.

  • Center and variability are multiplied/divided by that amount.

Density Curves and Normal Distributions

• Density curve is on or above the horizontal axis with an area of 1, showing probability distribution. A normal distribution is a type of density curve.

• Uniform Density Curve: Total area of 1.

• Normal Distribution: Follows the 68-95-99.7 rule.

  • 68% within one standard deviation of the mean.

  • 95% within two standard deviations of the mean.

  • 99.7% within three standard deviations of the mean.

• Calculator Commands:

  • normpdf: Probability at a specific value.

  • normcdf: Probability between a set interval.

  • invNorm: Value corresponding to a given percentile (area).

Normal Probability Plot

• Plots actual values versus theoretical z-values.

• Shows how well data fits a normal distribution.

• Roughly linear plot implies roughly normal distribution; nonlinear implies non-normal distribution.

Describing Scatter Plots (SEDAW)

• Context: Define context for data set.

• Direction: Is there a positive or negative correlation?

• Outliers: State clear outliers.

• Form: Is the data linear or nonlinear?

• Strength: How strongly linear is the data?

• R Value (Correlation Coefficient): Ranges from -1 to 1.

  • Closer to -1 or 1 indicates a Stronger correlation.

  • = -1 (Perfect negative linear correlation).

  • = 1 (Perfect positive linear correlation).

  • Ex: -0.97 (Strong corr). 0 (No corr). 0.21 (Weak corr).

  • Changing units, switching x/y axis does NOT impact this value.

  • Outliers within pattern stregthen r, outside the data pattern weaken r.

  • Remember does not equal causation.

Regression Lines

• A best fit line to estimate values for points not given (y hat = a + bx).

• Residual: Degree of error of the regression line prediction.

• Value: $ActualValue - PredictedValue$.

  • If the resultant value is negative, meaning that it was overestimated.

  • If the resultant value is positive, meaning that it was underestimated.

  • Language is super important here.

• Y= a + bx, so remember dependent vs independant variables to estimate values.

Least Squares Regression Line

• Minimizes the sum of squared residuals.

• Can find the values using a calculator function.

• S Value: Average distance the predicted values are away from the LSR.

• R squared Value: Coefficient of determination. (The percent of your response variables that can be explained with explanatory variables).

  • For optimal lines, want lower value s and higher value r-squared.

Computer Printouts

• Know values, so that it all makes sense.

• The lines that you see are the y-intercept, slope, s value, r squared value.

• Outlier in LSR:

  • Decreases Correlation.

    • Outliers added away from horizontal line decrease slope, increase y intercept.

    • Outliers added above vertical line the, slope the is same but y intercept decreases.

    • For outliers added below vertical line, slope is same, but the y intercept increase.

• Residual Plot: if unclear, the linear function is the best fit for the data. But if the is a clear pattern, a linear is not the best. Example of unclear are pretty much random dots.

Sampling methods

• SRS is simple random sample. where there's equal selected chances within a pop.
  * Define the population and randomize, and then actually get your selected samples.

• Stratified Random Sample where you split and split and split with homoge traits but you need the randomly get samples from EACH group.

• Systematic Random where you choose subjects at time-based and set intervals and start at randomly generated points. (Example, picking every 5th person.)

• Convenience samples are where its the easiest samples to grab.

• Voluntary response the samples are all based on the self-selection ones, which gives bias to the selection. (The more the strongly subjects will join).

• Bias short comings include under coverage: there needs to have access for the subjects, the individuals dont or cant response (non-response), the data given are faulty (Response Bias). Worded Questions can also lead to skewed results.

Observation vs Experiment

• Observation, you are strictly only observing the variable of interest, direct opposite from experiments. And Experiments is where you manipulate all variables to observe and measure the effects on the subjects. The guiding experiments are follows, compare to two or more groups, random assign (treatment), Control. Constant All Variables, Replicate. have enough subjects in experiments.

  • Factor (Explanatory or independant variable), and level (the categories within or variable). Confounding is the affect on the experiments when you dont have a proper control group. Placebo is the fake treatments for some participants., Single blind ( where subject is blind of which group, but the researchers can see), double blind ( both subjects and researchers are blind what variable is in affect). Also random block is dividing blocks into characteristic (Stratified Random samples) where then they are randomly assigned a treatment where is then you divide it back into homoge characteristics. In contrast Matched Pair: is where pairs of characteristics, a member of the pair could the control.

Probability

What is probability?

• Chance of something happening.

• Written as $0 - 1$.

• Unpredictable in the short term.

• Predictable in the long term because you have to observe many trials of the same proportion of the random chance process before it actually starts to approach its actual probability value.

• Simulation: A model used to mimic real-world events to estimate probabilities.

Probability Rules Definitions

• Mutually Exclusive: Events have no overlap and cannot occur at the same time.

• Independence: Two events are independent if the outcome of one does not affect the outcome of the other.

To talk about probability rules, we first need to get a couple definitions out of the way. The first one is mutually exclusive. Events have no overlap and cannot occur at the same time is the definition for a mutually exclusive occurrence.So you can see here, this is a nice diagram. We have event a and event b. Right? They're mutually exclusive because there's no overlap. Right? They cannot occur at the same time. But if they are not mutually exclusive, they can occur at the same time. That's why there's the overlap in the Venn diagram. The next one is independence. So two events are independent if the outcome of one does not affect the outcome of the other. So for example, if I flip a coin and I roll a die, the result of one does not change the result of other. If I flip a coin, I get, head, tails, whatever, me rolling a die, because I flipped a head or because I flipped a tails, that doesn't change what I'm gonna get or the probability of what I'm gonna get on the die.

Probability Rules

• Both events are not mutually exclusive:

  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Both events are mutually exclusive:

  • $P(A \cup B) = P(A) + P(B)$

• Both events are not independant:

  • $P(A \cap B) = P(A) x P(B|A)$

• Both events are independant:

  • $P(A \cap B) = P(A) x P(B)$

• Complement Rule:

  • $P(A^c) = 1 - P(A)$

Visualizing probability:

• Venn diagram shows the chance of something happening, event A and event B, and then their overlap.

• Two way table displays the data.

• Probability Tree.

There are four types:

• Discrete: specific countable integers that count what comes in the mean

• Continuous: within any given range in the dataset.

  • Finding a mean and variables: (use 1 variable equation on the calculator).
    Common calculations for: calculating the mean or expected value or the weighed average.
    Value: the average value of the rep of the process for the given random variable. the value x basically describe value of that discreet random variable and the p = probability within the variable. So what you would do is quite literally just multiply across one times 0.1, 2 times 0.3, all the way to the end, and then whatever you get, you add everything and that is your mean weighted average expected value.

Transformation:

• add or subtract same constant c to each dataset, the shape unchanged center increases or decreases by c, variability Remains Unchanged. You times it by the same Constant C it does the following your center multiply or divide by c, and variability multiply or divide by c.

• Equations for adding or subtracting a set of data with 4 random variables with independent events, make sure you have these equations down

**BINS - Binomial random variables

Is it binary, is it success and failure, are the trials independent and fixed/set probability? Address each binomial point and show that it applies to Binomial random variable. These are on the reference sheets you have the mean and standard deviation there. Calculator abuse will help in dealing with binomial random variables.

Binomial PDF: is the N number of trials
Number for successes and it calculates the probability of exactly k from n trials, binary NP lower and upper bound is pretty similar except the cumulative probability