Ridgefield Public Schools - Statistics

Statistics - High School

Standards: Connecticut has adopted the Common Core State Standards (CCSS) as the Connecticut State Standards (CSS).

Long Term Goals - Math Practice Standards

Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

Types of Data and Data Collection Methods (early September)

VISION OF THE GRADUATE

Collaboration, Innovation, Communication, Knowledge

BIG IDEAS AND ESSENTIAL QUESTIONS

Statistics uses numerical data from a representative sample for the purpose of making inferences and predictions about a population.
The goal of the data analysis determines the appropriate method of data collection (sampling technique or experimental design) as well as the types of variables appropriate for data collection.

LEARNING OUTCOMES Students will be able to . . . Students will be skilled at . . .

identifying types of variables and their associated level of measurement.
identifying variables that may be of interest for a given scenario.
determining which data collection method (survey, observational study, experiment, or simulation) is appropriate for a given scenario.
identifying summary values as parameters or statistics and clearly identifying the sample and population used in a statistical study.
comparing the differences between the different sampling techniques and identifying when each is appropriate.
create appropriate experimental designs for given scenarios that incorporate the key concepts of experimental design as well as the appropriate experimental technique (random, block, matched pairs, blinding, etc.).
determining if the method of data collection has created bias in the data that was collecting and hypothesizing as to how the collected data may differ from the entire population.

CONTENT STANDARDS

Understand and evaluate random processes underlying statistical experiments

HSS-IC.A.1 - Understand that statistics is a process for making inferences about population parameters based on a random sample from that population.

Make inferences and justify conclusions from sample surveys, experiments and observational studies

HSS-IC.B.3 - Recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each.

CONTENT The students will know . . .

the difference between a sample and population; statistics and parameter.
variables can be categorical or quantitative.
statistics is divided into two branches: descriptive and inferential.
variables can be classified by level of measurement: nominal, ordinal, interval or ratio.
data can be collected using an observational study, experiment, simulation, or survey.
sampling techniques used in statistics include random sample, simple random sample, stratified sample, cluster sample, systematic sample, and convenience sample.
experiments must include control, replication and randomization to be considered valid.
convenience samples often lead to biased results.

Summarizing and Displaying Data (early October)

VISION OF THE GRADUATE

Collaboration, Innovation, Communication, Knowledge, Resilience

BIG IDEAS AND ESSENTIAL QUESTIONS

The type of variable determines the appropriate graphical display for a data set in order to effectively present the data and prevent misleading representations.
Statistics of data sets, including measures of center and spread, are calculated in order to summarize data and make comparisons.
Certain summary statistics are resistant to outliers while others are greatly influenced and could lead to invalid conclusions.
Key features and summary statistics of a graph can be used to summarize, compare and contrast data sets.
Using measure of position (percentiles and z-scores), individual data values can be compared to an entire data set and/or different data sets.

LEARNING OUTCOMES Students will be able to . . .

utilize appropriate numerical summaries for categorical variables.
create and interpret graphical displays for categorical variables (bar graphs, pie charts, segmented bar graphs).
compare and contrast key features and advantages/disadvantages of each.
identify, apply, interpret, and create appropriate graphical displays for quantitative variables (dotplots, histograms, boxplots, stemplots).
compare and contrast key features and advantages/disadvantages of each.
analyze quantitative data using appropriate numerical summaries for quantitative variables.
calculate, compare and interpret measures of center.
calculate, compare and interpret measures of spread.
determine if a data set has outliers.
determine the relationship between the measures of center/spread and the shape of a distribution.
effectively describe the key features of a distribution (shape, outliers, center, and spread).

CONTENT STANDARDS

Summarize, represent, and interpret data on a single count or measurement variable

HSS-ID.A.1 - Represent data with plots on the real number line (dot plots, histograms, and box plots).

HSS-ID.A.2 - Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HSS-ID.A.3 - Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

CONTENT The students will know how . . .

to determine what graphs are appropriate for categorical & quantitative variables.
to summarize and describe graphs, in context, using their key features.
to calculate measures of center and spread.
the shape of a distribution and the presence of outliers impacts the measures of center and spread.
to determine if a data set has outliers present.

Regression (mid November)

VISION OF THE GRADUATE

Collaboration, Innovation, Mindfulness, Communication, Knowledge

BIG IDEAS AND ESSENTIAL QUESTIONS

Evaluating the relationship that exists between two quantitative variables (association, correlation, causation) is essential to determine if regression analysis is appropriate and what inferences can be made.
Scatterplots can be used to graphically display the relationships between two quantitative variables.
Correlation coefficients and regression lines are calculated and interpreted to describe the linear relationship between two quantitative variables.
Interpolation uses the regression line to make predictions within the range of data values for the independent variable, whereas extrapolation happens outside the range of available data, leading to less reliable predictions.
Residuals and residual plots can be used to quantify the strength of our predictions and the appropriateness of the linear regression model by comparing predicted values generated through the regression model with actual data values from the data set.
The coefficient of determination, is the fraction of variation accounted for by the regression model and allows us to analyze the appropriateness of a linear model.

LEARNING OUTCOMES Students will be able to . . .

identify the difference between an association, a correlation and a causation between two quantitative variables.
create a scatter plot using technology.
calculate and interpret the correlation coefficient.
calculate and interpret the regression line (in context).
make predictions using the regression line.
calculate and interpret residuals in context.
create a residual plot using technology.
verify the fit of the model using a residual plot.
understand the difference between interpolation vs extrapolation and the limitation of each.
calculate and interpret the coefficient of determination.
perform regression analysis on real-world data.

CONTENT STANDARDS

Summarize, represent, and interpret data on two categorical and quantitative variables

HSS-ID.B.6 - Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

HSS-ID.B.6.a - Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

HSS-ID.B.6.b - Informally assess the fit of a model function by plotting and analyzing residuals.

HSS-ID.B.6.c - Fit a linear function for scatter plots that suggest a linear association.

Interpret linear models

HSS-ID.C.7 - Interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.

HSS-ID.C.8 - Compute (using technology) and interpret the correlation coefficient of a linear fit.

HSS-ID.C.9 - Distinguish between correlation and causation.

CONTENT The students will know . . .

the relationship between correlation and causation.
when regression analysis is appropriate to use.
the appropriate way to complete a scatter plot.
a scatter plot of data points can be used to summarize some features.
the difference between independent and dependent variables.
how to identify the correlation coefficient.
how to identify the least squares regression line, slope and y-intercept.
what a residual and residual plot are.
how to identify the coefficient of determination.
the difference between interpolations and extrapolations.

Probability (early December)

VISION OF THE GRADUATE

Collaboration, Communication, Knowledge, Resilience

BIG IDEAS

Probability quantifies the likelihood that something will happen and enables us to make predictions and informed decisions.
Discussing and determining the likelihood of an event relies on recognizing which type of probability we are working with and when to utilize the fundamental counting principle.
Models may be used to simulate real world situations, enabling us to make predictions.

ESSENTIAL QUESTIONS

What is probability?
How do we express probabilities?
What is a sample space?
What is the fundamental counting principle and how do we use it?
How can one differentiate between the three types of probability?
What is conditional probability?
What is a compliment and when do we use it?
How can one determine if two events will occur in sequence?
How can one determine if two events are mutually exclusive and how do we calculate probabilities of events that are not mutually exclusive?
How can one use counting principles to find probabilities?
How can we use Venn diagrams to create visual displays of disjoint and non-disjoint events?
How can we determine if events are independent or dependent?
How can we use tree diagrams to display and calculate compound and conditional probabilities?

LEARNING OUTCOMES Students will be able to . . . The student will be skilled at . . .

create a probability model for a given probability experiment.
create and interpret tree diagrams to identify sample spaces and find probabilities.
find the probability of the complement of an event.
distinguish between independent and dependent events.
use the multiplication rule to find probabilities of compound events.
create and interpret 2 way tables for finding probabilities.
distinguish between mutually exclusive and non-exclusive events.
use the addition rule to find probabilities for multiple events.
distinguish when it is appropriate to use the addition rule vs. the multiplication rule.

CONTENT STANDARDS

Understand independence and conditional probability and use them to interpret data

HSS-CP.A.1 - Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

HSS-CP.A.2 - Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

HSS-CP.A.3 - Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

HSS-CP.A.4 - Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

HSS-CP.A.5 - Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Use the rules of probability to compute probabilities of compound events in a uniform probability model

HSS-CP.B.6 - Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the model.

HSS-CP.B.7 - Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

HSS-CP.B.8 - Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

CONTENT The students will know . . .

the sample space includes all possible outcomes of a probability experiment.
there are three types of probability.
the complement of an event is the probability it will not occur.
tree diagrams can be created to find outcomes and probabilities.
probabilities must be between 0 and 1.
two events can be mutually exclusive or non-mutually exclusive.
events can be independent or dependent.
rules for multiplication rule, additional rule, and conditional probability.
two-way tables can be used to represent probabilities.

Probability Distributions (early February)

VISION OF THE GRADUATE

Collaboration, Mindfulness, Communication, Knowledge, Resilience

BIG IDEAS

Real world situations can be modeled with discrete and continuous random variables.
Each random variable has a probability distribution which gives us information about the likelihood that a specific event happens and what is expected to happen if the change behavior is repeated many times.
When an experiment meets certain criteria, the binomial and geometric distributions can be used to calculate the probability of specific outcomes as well as expected values.
The Normal distribution plays a large role in statistics, can be used to model many different types of chance behavior, and is a key component of statistical inference.

ESSENTIAL QUESTIONS

How can we differentiate between discrete and continuous random variables? How are they similar? How do they differ?
How does one represent and interpret discrete probability distributions?
How does one represent and interpret binomial probability distributions?
How does one represent and interpret continuous probability distributions?
What does a discrete probability distribution list?
Why can a discrete probability distribution be represented with a table, but a continuous distribution cannot?
What determines if an experiment is binomial?
How are the mean and standard deviation affected by transformations on random variables?
How are a binomial and a geometric random variable similar and different?
How can one identify if a distribution is normal?
What does the area under a curve of a normal distribution describe?
How can the normal distribution be used to calculate and interpret probabilities?
How do we interpret and apply expected value?

LEARNING OUTCOMES Students will be able to . . .

compare and contrast discrete and continuous distributions.
identify normal, binomial and geometric probability settings and verify the conditions of each.
calculate and interpret the mean and standard deviation of a probability distribution in real-world applications.
use probability distribution calculations to make informed decisions, test hypothesis and investigate claims.

CONTENT STANDARDS

Summarize, represent, and interpret data on a single count or measurement variable

HSS-ID.A.4 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.

Calculate expected values and use them to solve problems

HSS-MD.A.1 - Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

HSS-MD.A.2 - Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

HSS-MD.A.3 - Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

HSS-MD.A.4 - Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Use probability to evaluate outcomes of decisions

HSS-MD.B.5 - Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CONTENT The students will know . . .

the properties of a normal distribution.
the difference between a discrete and continuous random variables.
the conditions for a binomial distribution.
how to calculate probabilities for a binomial distribution.
the conditions for a geometric distribution.
how to calculate probabilities of a geometric distribution.
how to calculate and interpret the expected value and standard deviation for a random variable, in context.

Sampling Distributions (early March)

VISION OF THE GRADUATE

Collaboration, Mindfulness, Communication, Knowledge, Resilience

BIG IDEAS

Statistical inference incorporates the theory, methods, and practice of forming judgments about the parameters of a population, usually on the basis of random sampling.
A sampling distribution can be constructed provided specific conditions are satisfied in order to make inferences about a population parameter.
The Central Limit Theorem allows us to use Normal probability calculations even when the population distribution is not Normal and forms the foundation for the inferential branch of statistics.

ESSENTIAL QUESTIONS

How is sample size related to the sampling distribution of sample means and proportions?
What condition is required to calculate the mean of a sampling distribution?
What condition is requires to calculate the standard deviation of a sampling distribution?
What condition(s) is/are required to use Normal probability calculations?
How can the Central Limit Theorem describe the relationship between the sampling distribution of sample means and the population from which the samples are taken?
How can we use sampling distributions to investigate claims about a population parameter?

LEARNING OUTCOMES

Students will be able to . . .

identify the mean and standard deviation of the sample mean of a sampling distribution.
identify the mean and standard deviation of the sample proportion of a sampling distribution.
find normal probabilities of sampling distributions when appropriate.

Students will be skilled at . . .

finding sampling distributions and verifying their probabilities.
using the central limit theorem to determine if a sampling distribution of sample means is normally distributed.
use the large counts rule and 10% rule to determine if a sampling distribution of sample proportions is normally distributed.
explaining how the sample size affects a sampling distribution.
interpreting the validity of a claim about a population using calculated probabilities.

CONTENT STANDARDS

Understand and evaluate random processes underlying statistical experiments

HSS-IC.A.1 - Understand that statistics is a process for making inferences about population parameters based on a random sample from that population.

HSS-IC.A.2 - Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation.

Make inferences and justify conclusions from sample surveys, experiments and observational studies

HSS-IC.B.6 - Evaluate reports based on data.

Use probability to evaluate outcomes of decisions

HSS-MD.B.7 - Analyze decisions and strategies using probability concepts

CONTENT The students will know . . .

the difference between a statistic and a parameter (for both means and proportions).
the basic properties of a sampling distribution.
what it means to be an unbaised estimator.
the impact of sample size on the variability of a sampling distribution.
the minimum sample size required in order for the sampling distribution to be modeled with a normal distribution (Central Limit Theorem and Large Counts Condition).
the conditions under which a sampling distribution is appropriate.
how to calculate the probability of different samples.
how to interpret the calculated probability to determine if a claim about a population is valid.

Confidence Intervals (early April)

VISION OF THE GRADUATE

Collaboration, Communication, Knowledge

BIG IDEAS AND

Inferential statistics is the process of combining descriptive statistics with probability and sampling distributions to make inferences for or about a population.
Using the properties of sampling distributions, we can use sample data to construct a confidence interval of plausible values for a population parameter.
The margin of error for a confidence interval changes based on the sample size and the confidence level.
Confidence intervals can be used to investigate claims about a population and draw conclusions.

ESSENTIAL QUESTIONS

Why should we not use a statistic alone to make an inference about a population?
How can we use the the properties of sampling distributions to make predictions about a population?
What does it mean to be C% confident?
How do you calculate a margin of error?
What is a critical value? What impacts our critical value? How do we calculate a critical value?
What conditions must be satisfied in order for a confidence interval to be valid?
How do we interpret a confidence interval?
How can we use a confidence interval to investigate a claim about a population?

LEARNING OUTCOMES Students will be able to . . .

determine the point estimate and margin of error from a confidence interval.
interpret a confidence interval in context
interpret a confidence interval in context
describe how the sample size and confidence level affect the length of a confidence interval.
verify the conditions necessary for the construction of a confidence interval.
construct and interpret confidence intervals for a population mean and proportion
determine the necessary minimum sample size required to obtain a C% confidence interval for a specified margin of error.

CONTENT STANDARDS

Understand and evaluate random processes underlying statistical experiments

HSS-IC.A.1 - Understand that statistics is a process for making inferences about population parameters based on a random sample from that population.

Make inferences and justify conclusions from sample surveys, experiments and observational studies

HSS-IC.B.4 - Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Use probability to evaluate outcomes of decisions

HSS-MD.B.7 - Analyze decisions and strategies using probability concepts

CONTENT The students will know how to . . .

determine if the conditions of a sampling distribution are satisfied so that inference calculations can be performed.
how sampling distributions relate to statistical inference.
construct and interpret confidence intervals for populations means and proportions.
evaluate reports and claims using confidence intervals.
determine the impact of sample size and confidence levels on critical values and margins of error and the resulting confidence intervals.

Hypothesis Testing (late April) -> in Progress

Chi-Squared Test (early May) -> in Progress

Culminating Research Project (late May)

VISION OF THE GRADUATE

Collaboration, innovation, Mindfulness, Communication, Knowledge

BIG IDEAS

Students will choose a topic that interests them and conduct research on that topic. From their research they will design and conduct a study to gather data. Once the data is collected they will summarize and analyze the data. Students will then construct sampling distributions based on their research and determine if the data they collected is reasonable. Finally, students will perform significance tests and inference calculations to draw conclusions about their sample and the population is represents based on their research and data collection.

ESSENTIAL QUESTIONS

How can we apply apply the various types of statistical inference to our own data collection?
How can we design a study/data collection around our research topic that will yield meaningful data?
How can we apply the foundations of data collection to our topic?
How can we use statistical inference to draw conclusions about our population based on our sample?
How can we extend/modify our process for future analysis?

LEARNING OUTCOMES Students will be able to . . .

research a topic of interest to gather information as the basis of their study.
design a study/experiment using the concepts learned in class to gather their data.
effectively conduct a study to gather data.
analyze and summarize their data.
construct sampling distributions and apply the properties of statistical inference to draw conclusions about a population.
formulate and interpret conclusions and inference based on statistical analysis.
evaluate the results of a study.
constructively critique the ideas, processes, analysis and conclusions of others.

CONTENT STANDARDS

Summarize, represent, and interpret data on a single count or measurement variable

HSS-ID.A.1 - Represent data with plots on the real number line (dot plots, histograms, and box plots).

HSS-ID.A.2 - Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

HSS-ID.A.4 - Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables

HSS-ID.B.5 - Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.

Understand and evaluate random processes underlying statistical experiments

HSS-IC.A.1 - Understand that statistics is a process for making inferences about population parameters based on a random sample from that population.

HSS-IC.A.2 - Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation.

Make inferences and justify conclusions from sample surveys, experiments and observational studies

HSS-IC.B.3 - Recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each.

HSS-IC.B.4 - Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

HSS-IC.B.5 - Use data from a randomized experiment to compare two treatments; justify significant differences between parameters through the use of simulation models for random assignment.

HSS-IC.B.6 - Evaluate reports based on data.

Use probability to evaluate outcomes of decisions

HSS-MD.B.7 - Analyze decisions and strategies using probability concepts

CONTENT The students will know how to . . .

design and conduct an observational study/experiment.
effectively display and summarize data.
construct and draw conclusions from sampling distributions.
preform inference calculations and interpret the results in context.
verify the conditions for the appropriateness of statistical inference in context.
evaluate the results of reports based on inferential analysis.