## Unit Overview

**Essential Questions: **

preferences of a population useful?

**Lesson Plans and Seeds**

Lesson Plan B.4: Use Measures of Center and Variability to Draw Inferences

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 7**

**Progressions from Common Core State Standards in Mathematics**

Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.

### Unit Overview

This unit depends on students’ prior knowledge of statistical variability, measures of variations and of central tendency, and ability to display numerical data on a number line, including dot plots, and on histograms and box plots. It focuses on visually analyzing the extent to which two numerical data distributions are similar or different; and computing measures of center and measures of variability for numerical data from random sampling to compare/contrast inferences about two populations.

**Teacher Notes:**

- Students should have a developed sense of statistical reasoning and be able to differentiate between a statistical question and a non-statistical question.
- Students should recognize that the answer to a statistical question is based on the distribution of values within the data set, as well as on the center, spread, and overall shape of the data.
- Students should be well-grounded in their ability to determine measures of center to determine the appropriateness of each measure for a given data set.
- Students should have prior understanding of plotting and analyzing numerical data on a dot plot.

**Enduring Understandings:**

At the completion of this unit on drawing informal comparative inferences about two populations, the student will understand that:

- Gathering statistics on an entire population is a difficult process because populations are often extremely large. Thus, a random sample of the population is intended to be representative of the total population. Random samples from more than one population can be gathered and used to compare specific characteristics of populations.
- A generalization about a given population is valid only if the sample is representative of that total population.
- Samples have variations because collected data is never completely accurate; steps can be taken to minimize the likelihood of inaccuracies.
- Measures of center include the mean, median, and mode for a group of numbers in a data set. These measures summarize the numerical values related to a random sample from a population (the data set) and indicate what is typical for the population.
- Measures of variability include range, standard deviation, and variance for a group of numbers in a data set. In general, these measures refer to the distances above or below the mean (or other measure of central tendency) each numerical value in a data set is located. Each numerical value in a data set can be measured in a variety of ways to obtain as representative a “picture” of the entire data set as possible.

**Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
**

- PARCC has not provided examples of opportunities for in-depth focus related to the cluster “Use Random Sampling to Draw Inferences about a Population.”

**Possible Student Outcomes:**

Students will be able to:

- Obtain defining information about two populations.
- Collect and use various data samples to draw conclusions about two populations in relation to one another.
- Create multiple samples for two populations to gauge and compare variations in potential estimates and predictions related to random sample data.

**Evidence of Student Learning:**

*The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.*

**Fluency Expectations and Examples of Culminating Standards:**

**Common Misconceptions:**

Students may

- Assume that a population sample is an exact “miniature” of the population, with characteristics that match exactly; random sampling means the sample is likely to be a good representation of the population.
- Confuse the terms “reliability” and “validity.”
- Believe that one random sample is enough to accurately represent the entire population. In reality, many samples must be taken in order to make a valid inference.

**Interdisciplinary Connections:**

Interdisciplinary connections fall into a number of related categories:

*Literacy standards within the Maryland Common Core State Curriculum*

*Science, Technology, Engineering, and Mathematics standards*

*Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.*

**Sample Assessment Items: ***The items included in this component will be aligned to the standards in the unit and will include:*

*Items purchased from vendors*

*PARCC prototype items*

*PARCC public release items*

*Maryland Public release items*

**Interventions/Enrichments/PD: ***(Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)*

**Vocabulary/Terminology/Concepts: ***This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.*

**
Part I – Focus Cluster
Use Random Sampling to Draw Inferences about a Population**

** population: **A population is whole set of individuals, items, or data from which information, or a statistical sample, is drawn.

** sample of the population: **The part or section of a whole set of individuals, items, or data about which information is wanted, or from which a statistical sample is drawn. Used as a verb, to sample means to get data from part of a population and use the data to provide information about the entire population.

** simulated sample: **Simulated samples refer to multiple sets of individuals, items, or data of the same size that are gathered from the same population in order to estimate or predict the accuracy of an experiment.

** random sample/sampling: **A random sample of consists of n individuals, items, or data from the population, chosen in such a way that every set of n individuals, items, or data has an equal chance to be the sample actually selected.

** invalid: **The results of a statistical experiment are invalid when they are based on faulty, incorrect, or null information; being without foundation in fact or truth.

** valid/validity: **Validity refers to the extent to which a concept, conclusion or measurement is well-founded and corresponds accurately to the real world. Validity of a measurement tool, for example a quiz or chapter test, is the degree to which the tool measures what it claims to measure.

** reliable/reliability: **In statistics, reliability is the consistency of a set of measurements or of a measuring instrument. Reliability refers to the degree to which results from different clinical trials or statistical experiments, using an identical set of measurements or measuring instrument, are the same or compatible with one another.

** variation/variability: **The spread of values that exists within an array of scores or other measures is referred to as variability. Measures of variability include range, standard deviation, and variance.

**Part II – Instructional Connections outside the Focus Cluster**

NOTE: |
None of the vocabulary, terminology, and concepts in this cluster are new, nor should they be particularly problematic for instruction of the related standards. |