AP Statistics Midterm Review

Statistics

the science of collecting, analyzing, and drawing conclusions from data

Descriptive

methods of organizing and summarizing statistics

Inferential

making generalizations from a sample to the populations

Population

an entire collection of individuals or objects

Sample

a subset of the population selected for study

Data

observations on single or multi-variables

Categorical

basic characteristics; doesn't make sense to take an average

Numerical

measurements or observations of numerical data

Discrete

listable sets (counts)

Continuous

any value over an interval of values (measurements)

Univariate

one variable

Bivariate

two variables

Multivariate

many variables

Symmetrical

data on which both sides are fairly the same shape and size (mean and median are similar)

Uniform

every class has an equal frequency (number) 'a rectangle'

Skewed

one side (tail) is longer than the other side. The skewness is in the direction of the tail (left or right)

Bimodal

data of two or more classes have frequencies separated by another class between them; two humps

Parameter

a numerical value that describes a characteristic of a population (typically unknown)

Statistic

a numerical value that describes a characteristic of a sample

Median

the middle point of the data (50th percentile) when the data is in numerical order. If two values are present, then average them together

Mean

𝜇 is for a population (parameter) and 𝑥̅ is for a sample (statistic)

Variability

allows a statistician to distinguish between usual and unusual occurrences

Range

single value: maximum-minimum

IQR

interquartile range: Q3-Q1

Standard deviation

𝜎 for population (parameter); s for sample (statistic) - measures the typical or average deviation of observations from the mean; sample standard deviation is divided by df = n - 1

Variance

standard deviation squared

Resistant

not affected by outliers

Non-Resistant

Mean, Range, Standard Deviation, Variance, IQR

Z-Score

a standardized score. This tells you how many standard deviations an observation is from the mean.

Coefficient of Determination (𝑟!)

a measure that assesses how well a model explains and predicts future outcomes.

Comparison of mean and median

Mound shaped - mean and median are nearly the same value; Skewed right - mean is larger than the median; Skewed left - mean is less than the median; The mean is always pulled in the direction of the skew away from the median.

Standard Normal Curve

It creates a standard normal curve consisting of z-scores with 𝑁(𝜇, 𝜎) = 𝑁(0,1)

Normal Curve

Symmetrical density curve that follows the empirical rule.

Assess Normality

Use graphs: dotplots, boxplots, histograms, or normal probability plot.

Empirical Rule (68-95-99.7)

Measures 1, 2, and 3 standard deviations (𝜎) from center (𝜇) of a normal curve.

68% of Observations

Fall within 1 𝜎 of 𝜇.

95% of Observations

Fall within 1 𝜎 of 𝜇.

99.7% of Observations

Fall within 1 𝜎 of 𝜇.

Boxplots

For medium or large numerical data. It does not contain original observations.

Modified Boxplots

Used where the outlier cutoffs are 1.5 IQRs from the end of the box (Q1 and Q3).

Outliers

Points more extreme than the cutoffs are considered outliers.

5-Number Summary

Minimum, Q1 (1st quartile), Median, Q3 (3rd quartile), maximum.

Correlation Coefficient (r)

A quantitative assessment of strength and direction of a linear relationship.

Population Parameter

Uses (𝜌) for population parameter.

Correlation Values

Values [-1,1]: 0 - no correlation, (0 ± .5) - weak, [±.5, ±.8] - moderate, [±.8, ±1] - strong.

Least Squares Regression Line (LSRL)

Minimizes the sum of the squared residuals on a scatterplot.

Residuals

Difference between observed and predicted responses.

Residual Plot

Indicates a good model if (1) no discernable pattern and (2) points spread about evenly above and below the LSRL.

Coefficient of Determination (𝑟!)

Gives proportion of variation in responses that is explained by the relationship of x and y.

Slope (b)

For every additional x, the predicted response will in/decrease by about b.

Extrapolation

LSRL cannot be used to predict responses outside the scope (interval) of explanatory values.

Influential Points

Points that if removed significantly change the LSRL.

Outliers (in context)

Points with large residuals and do not follow the trend of the bivariate data.

Census

A complete count of the population.

Sampling Frame

A list of everyone in the population.

Sampling Design

Refers to the method used to choose a sample.

Simple Random Sample (SRS)

Every individual has the same chance of being chosen and every group of size n has the same chance of being chosen.

Stratified Sampling

Divide the population into homogenous groups called strata, then SRS each strata.

Advantages of Stratified Sampling

More precise than SRS and cost reduced if strata already available

Disadvantages of Stratified Sampling

Difficult to divide into groups, more complex formulas, must know population

Cluster Sampling

Based on location; select a random location and sample ALL at that location.

Advantages of Cluster Sampling

Cost is reduced, is unbiased, and don't need to know population.

Disadvantages of Cluster Sampling

May not be representative of population and has complex formulas.

Random Digit Table

Each entry is equally likely and each digit is independent of the rest.

Random Number Generator

Calculator or computer program; RandInt(lower, upper).

Bias

Systematically favors a certain outcome.

Sources of Bias

Factors that can lead to biased results in sampling.

Voluntary Response Bias

People choose themselves to participate; polarized responses.

Convenience Sampling

Ask people who are easy to find, friendly, or comfortable asking.

Undercoverage

Subset of the population is left out of selection process.

Non-response Bias

Someone cannot or does not want to be contacted to participate.

Response Bias

False answers; can be caused by a variety of things.

Wording of the Questions

Leading questions that can influence responses.

Observational Study

Observe outcomes without giving a treatment.

Experiment

Actively imposes a treatment on the subjects; randomly assigns experimental units.

Experimental Unit

Single individual or subject that receives a treatment.

Factor

The explanatory variable; what is being tested.

Level

A specific value of the factor.

Response Variable

What you are measuring with the experiment.

Treatment

Experimental condition applied to each unit.

Control Group

Used to compare the factor to for effectiveness; does NOT have to be a placebo.

Placebo

A treatment with no active ingredients (provides a control).

Blinding

A method used so subjects are unaware of treatment or control group.

Double Blinding

Neither subjects nor evaluators know which treatment is being given.

Principles of Experimental Design

Control, Replication, Randomization, Comparison.

Control in Experimental Design

Isolates effects of treatment variable by keeping all other variables constant.

Replication in Experimental Design

Reduce impact of chance variation due to random assignment to different treatments.

Randomization in Experimental Design

Uses chance to assign subjects to treatments to create similar treatment groups; reduces bias and establishes cause and effect.

Comparison in Experimental Design

Measures responses of control and treatment groups to determine effectiveness of treatment.

Completely Randomized Design

All units are assigned to all of the treatments randomly.

Randomized Block Design

Units are subjectively blocked by similar characteristics and then randomly designed within each block; reduces variation and controls confounding variable.

Matched Pairs Design

Matched up units by characteristics and then randomly assigned.

Confounding Variables

The effect of the variable on the response is indistinguishable from the effects of the factor being tested; happens in observational studies and when blocking should occur.

Law of Large Numbers

As an experiment is repeated, the experimental probability gets closer and closer to the true (theoretical) probability.

Probability

The proportion of time an outcome occurs over a long run of trials.

Sample Space (S)

Collection of all possible outcomes.

Events

Any subset of the sample space; denoted by capital letter.

Complement

All outcomes NOT in the event.

Union

A or B, all the outcomes in both circles (𝐴∪𝐵).

Intersection

A and B, happening in the middle of A and B (𝐴∩𝐵).