AP Statistics Unit 1-8 Comprehensive 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 (cannot be quantified numerically).

Two-Way Tables for Categorical Data

• Representation of two variables and their intersections.

• Example:

  • Variables: Math students, English students (rows) vs. Internal, External (columns).

  • The table shows the number of students in each category (e.g., math students who are internal).

• Vocab:

  • Marginal Relative Frequency:

    • Percentage of data in a single row or column compared to the total.

    • Calculated as $b/d$ (row total / total) or $c/d$ (column total / total).

  • Joint Relative Frequency:

    • Percentage of data in a single group compared to the total.

    • Calculated as $a/d$ (single group / total).

    • Example: Math internal (3/23).

    • Example: external English (9/23).

  • Conditional Relative Frequencies:

    • Percentage of data in a single category when given a specific group.

    • Calculated as $a/b$ or $a/c$.

    • Example: Given the student is in math (total 10), the percentage that is internal is 3/10.

Describing Quantitative Data (CSOCS Acronym)

• C: Context

• S: Shape

  • Symmetrical or skewed.

  • Number of peaks (unimodal, bimodal).

• O: Outliers

  • Data points that are far from the rest of the data.

• C: Center

  • Mean or median.

• S: Spread

  • Range, standard deviation, IQR (interquartile range).

• Tips:

  • Use descriptive language (strongly, roughly).

  • Use comparative language.

Basic Terms for Quantitative Data

• Mean:

  • Sum of all values divided by the number of values (average).

• Standard Deviation:

  • Measure of variation.

  • Interpretation: "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.

  • The value in the middle when data is organized from least to greatest.

• Range:

  • Value of max minus 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):

  • $Q3 - Q1$

• Outliers:

  • Low-end outlier: Any value less than $Q1 - 1.5 Imes IQR$

  • High-end outlier: Any value greater than $Q3 + 1.5 Imes IQR$

Percentiles and Cumulative Relative Frequency

• Percentile: Percentage of values that are less than or equal to a specific value.

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

• Relative Frequency: Chance of something occurring. Calculated as occurrence/frequency over total.

• Visual Representation:

  • Data points are graphed. When no data exists, the graph plateaus.

Z-Scores

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

• Equation: $z=(value-mean)/standardD_{}eviation$

Transforming Data

• Adding/Subtracting a Constant:

  • Shape and variability stay the same.

  • Center (mean, median) moves up or down by that amount.

• Multiplying/Dividing by a Constant:

  • Shape stays the same.

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

Density Curves and Normal Distributions

• Density Curve:

  • On or above the horizontal axis.

  • Area of one.

  • Shows probability distribution.

  • Normal distribution is a type of density curve.

• Uniform Density Curve: Rare, total area of one.

• Normal Distribution:

  • 68-95-99.7 Rule:

    • 68% of values are within one standard deviation of the mean.

    • 95% of values are within two standard deviations of the mean.

    • 99.7% of values are within three standard deviations of the mean.

• Calculator Commands:

  • normpdf: Probability at a specific value.

  • normcdf: Probability between a set interval.

  • Inverse normal: Finds a value that corresponds to a given percentile (area).

• Normal Probability Plot:

  • Plots actual values versus theoretical z-values.

  • Shows how well the data fits a normal distribution.

  • Roughly linear = roughly normal distribution.

  • Not linear = roughly not a normal distribution.

Describing Scatter Plots (SEDAW Acronym)

• S: Context

• E: Direction

  • Positive or negative.

• D: State Outliers

• A: Form

  • Linear or nonlinear.

• W: Strength

  • Strongly linear, weakly linear.

Correlation Coefficient (r-Value)

• Ranges from $-1$ to $1$.

  • Closer to $-1$ or $1$ = stronger linear correlation.

• Examples:

  • $r = -0.97$: Strong linear correlation.

  • $r = 0$: No correlation.

  • $r = 0.21$: Weak linear correlation.

• Changing units or switching x and y axis: does not impact the correlation coefficient.

Effect of Outliers on r-Value

• Outliers within the pattern of data: strengthen $r$.

• Outliers outside the pattern of data: weaken $r$.

• Correlation does not equal causation.

Regression Lines

• Best fit line used to estimate values.

• Equation: $\hat{y} = a + bx$

  • $\hat{y}$: Predicted value.

  • a: Constant.

  • b: Slope.

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

  • Calculated as actual value minus predicted value.

  • Negative residual: Overestimation.

  • Positive residual: Underestimation.

Interpreting Regression Lines

• Focus on using proper language when interpreting slope, y-intercept, and residuals.

• Explanatory Variable: Independent variable.

• Response Variable: Dependent variable.

Least Squares Regression Line

• Line that minimizes the sum of the squared residuals.

• s-value: Average distance the predicted values are away from the LSR.

• r-squared value: Coefficient of determination.

  • Percent of response variables that can be explained with the explanatory variable.

• Optimal LSRL: Low s-value, high r-squared value.

Computer Printouts

• Memorize the location of y-intercept, slope, s-value, r-squared value.

Effect of Outliers on the Least Squares Regression Line

• Outliers decrease correlation.

• Outliers far away from the mean of y (horizontal line):

  • Decrease slope.

  • Increase y-intercept.

• Outliers above the mean of x (vertical line):

  • Slope stays the same.

  • Y-intercept decreases.

• Outliers below the mean of x (vertical line):

  • Slope stays the same.

  • Y-intercept increases.

Residual Plots

• Plots residual values versus the explanatory variable.

• Clear pattern: Linear function is unlikely to be best fit.

• Unclear pattern: Linear function is likely to be best fit.

Sampling Methods

• Simple Random Sample (SRS):

  • Randomly selected subset of the population.

  • All members have an equal chance of being selected.

  • Process:

    • Label individuals.

    • Randomize.

    • Select members.

• Stratified Random Sample:

  • Split population into groups/strata based on shared traits (homogeneous groups).

  • Randomly select samples from each group.

  • Example: Divide school by grade level and pick students from each grade.

• Cluster Sample:

  • Split population into groups/clusters.

  • Randomly select ENTIRE clusters to sample.

  • Example: Split a city into neighborhoods and survey everyone in a few randomly chosen neighborhoods.

• Systematic Random Sample:

  • Select individuals at regular/set intervals starting at a random point.

  • Example: Survey every fifth person that walks into a building.

Bad Sampling Methods

• Convenience Sample:

  • Choose people who are easy to reach.

  • Example: Surveying people at a nearby mall.

• Voluntary Response Sampling:

  • Allow people to choose to participate.

  • Example: Online poll.

Shortcomings of Sampling

• Undercoverage: Some groups are left out or underrepresented.

  • Example: Survey sent only to people with internet access..Those lacking internet access will be excluded.

• Non-response: Selected individuals don't/can't respond.

• Response Bias: People give false or misleading answers.

• Wording in the question: Poorly phrased/biased question influences answers.

Observational Studies vs. Experiments

• Observational Study:

  • Observe and collect data without influencing the subjects.

  • Example: Recording seatbelt usage.

• Experiment:

  • Manipulate variables/apply treatments to observe and measure effects.

  • Principles: comparison, random assignment, controls, replication.

Key Vocab for Experiments

• Factor: Explanatory/independent variable(s).

• Level: Specific value/category of the factor.

  • Example: low vs. high levels of sunlight

• Confounding: Another variable affects the results.

• Placebo: Fake treatment where participants may react favorably.

• Single Blind: Subjects don't know which group they're in, researchers do.

• Double Blind: Neither subjects nor researchers know who gets what treatment.

Experimental Designs

• Randomized Block Design:

  • Subjects divided into blocks/groups based on specific characteristics.

  • Each block is randomly assigned a treatment.

  • Block: Group with experimental units with same characteristic.

    • Example: PrepWorks subscribers vs. not subscribers.

• Matched Pairs Design:

  • Subjects are paired based on specific characteristics.

  • Each pair is randomly assigned a treatment, or one subject in each pair is a control.

    • Example: Pairing male vs. female

Definition and Interpretation of Probability

• Chance something happens. Scales $0$ to $1$.

  • $0$: Impossible.

  • $1$: 100% chance of happening.

• Unpredictable in the short term due to randomness.

• Predictable in the long term after many repeated trials. Results approach true probability.

Simulation

• Used to model and mimic real events to estimate probabilities in a variety of situations. For example, predicting the weather forecast.

• Four-step process:

  • Define the problem questions.

  • Use random numbers to model the situation.

  • Run/ perform simulation.

  • Base on results to estimate probabilities.

Mutually Exclusive Events

• Events have no overlap and cannot occur at the same and the same probability.

  • Probability of A: is the number of outcomes in A divided by the total number of outcomes.

  • Probability of B: is the number of outcome in B divided by the total number of outcomes.

  • Independent: The outcome of one event doesn't affect the outcome of the other.

Independence

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

• The visualization methods:

  • Venn diagram: represents a set of objects; shows probabilities in and between events.

  • Two-way table: visual tool that is useful for analyzing categorical data; presents a summary of the relationship between the two.

  • Probability tree: display all possible outcome combinations and their associated probabilities, but must start with probability of the first event in tree.

Transformation of Probability Distributions

• Add or subtract the same constant c to each dataset, shape unchanged. Center increase or decrease by c variability remains unchanged, if you multiply or divide by the same constant c to each value in the dataset, your shape stay the same, center multiply is divided by c; variability multiply or divide by c

• Equations:$\mu x + y = \mu x + \mu y$,$\mu x-y=\mu x - \mu y$

Types of Random Variables (DISCRETE)

• Discrete random variable.

  • Take up a specific countable value. For example, number of heads or car by the day.

  • $\mu x=x\cdotp(p)$

• Continuous random variable.

  • time or height.

  • Any value within a range or interval. So it's not just 1 or 2, you can also count what is in between one and two.

• Binomial Random

  • Has a binomial of the fixed number of trails.

  • Independent event with fixed number of trials.

  • BINS: binary, independence, number of trials and success rate for each trial.

  • Calculator commands: binompdf, binomcdf.

• Geometric Random.

  • Have a certain chance or probability to achieve for the first time (first success in a series of independent trials) . For example, flipping a coin trying to see the number of flips it may require to get the head.

  • Calculator commands: geomatric pdf and geomatric cdf with success/ trail has to be constant(p= success to trial and be independent.

Differentiating Statistic vs. Parameter

• Statistic:

  • A number that describes something from sample data.

• Parameter:

  • A number that describes something from the entire population of data.

Core concept to remember

• Population mean should equal sample mean of sample size if the sampling distribution is equal to that of the parameter

Sampling Distribution of Proportions

• Repeated samples of the same size from the population, finding the proportion of each sample, then plotting info on a distribution.

• Formulas:

  • $\text{Mean}: \mu_{\hat{p}} = p$

  • $\text{Standard Deviation}: \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

  • Increasing the sample size will decrease your variability

Random Sampling

• Need 1random sampling and assignment

  • 10% sample size from size: n<((=10%)N.

  • Large counts. $Np>=10, N(1-p)>=10

Probability calculation with z value

• Sampling distribution is approximately normal.

• Z value formulas:

  • Calculator command normal cdf

• Mean formulas $M\mu x+\muy+m; \mu x-y= \mu x - \mu y$</p></li><li><p>$\sigma2=((\mu x+y = \mu \sqrtx^2+ \mu\sqrt^y2); \sigma2=((\mu x-y = \mu \sqrtx^2+ \mu\sqrt^y2)$</p></li></ul><h3 id="5d2c433a-e5e0-4b35-9e61-e64bb659e48e" data-toc-id="5d2c433a-e5e0-4b35-9e61-e64bb659e48e" collapsed="false" seolevelmigrated="true">Key components of the Sampling distribution</h3><ul><li><p>Repeated the sample taking from population to calculate a mean to reapply of each sample into a distribution(probability distribution)</p></li><li><p>With the increase of sample size = variability get decrease.</p></li></ul><h3 id="36418356-af3c-4d42-80cf-f09c15c3259c" data-toc-id="36418356-af3c-4d42-80cf-f09c15c3259c" collapsed="false" seolevelmigrated="true">10$n<((=10%)N$.</h3><ul><li><p>Large counts.$$Np>=10, N(1-p)>=10$$
  **CENTRAL LIMIT THEOREM:

• Sample size must be at least 30

What parameter of interest

*Point estimate
*confidence interval
*Confidence levels- increasing will increase margin of errors. bias is affected by marginal errors
*Increasing sample size = reduces marginal errors, make narrower

Panic

*Check parameter
*Assumption of conclusions must be random-check 10% sample , then large counts (equal to sample 10+)

One sample z/test z intervals for P. (2 samples means P1-P2) and random for both-same condition to consider for both samples

*Name the test, stated distribution and assumption of conditions
What’s alpha value?

Check hypothesis with HO, and alternative in contact= make decisions wether to reject or to fail to rejected HO.

*Type 1 error: false negative - probability is related to alpha values.
*Type 2 error: false +.
*Power = 1+p(false negative)- probability is always less and HO is false.

• increase alpha value

• increase samples size and

• increase the distance between null vs alternative hypothesis.

Types of Chi squared

***Test for goodness- differ from expected distribution?. Hypotheses needs not to have a variable. (random large count +5)
Goodness of fit- degree of freedom = category (N-1)
***Chi squares is higher and higher = higher discrepancy - means m and m is not telling the truth
***Test of homogeneity - shared same group distribute and what is being used:

• Random samples = 1 + variables.
  *Hypotheses =no difference vs alternative is change in groups (chi squared for homeg)
  *Formula for sample = raw + column divide by total numbers and +equal to 5
  *Then reject all + conclusion with 4 chi squared or association- what variables is the chi quared and hypothesis : + variable but has different samples.
  Conclusion reject.

Assumptions of conditions

All point must has Random, assignment ,10% less N. equal variables +5

Inference Of slopes formulas

***Assumptions of conditions is 5 variables. With the data has x and y should is slope.

***The intervals and data does random and not random =no increasing patterns.
***If not the should that to use any confident data or sample.
***There are experiments= the number has assignments for experiments