Measures of Spread

Objective

Students will be able to:

• Calculate the mean, variance, and standard deviation of a data set.
• Interpret these measures to understand data distribution and variability.

Common Core State Standards Alignment

• CCSS.MATH.CONTENT.HSS-ID.A.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
• CCSS.MATH.CONTENT.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.
• CCSS.MATH.CONTENT.HSS-ID.A.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible outliers.

Materials

• Printed one-page graphic organizer (blank and completed versions)
• Whiteboard or projector
• Calculators

Lesson Duration: 10 minutes

Teacher Preparation:

Prepare two versions of the graphic organizer:

• Blank copy: Sections for vocabulary, formulas, examples, and space for student solutions.
• Answer key copy: Completed with examples and worked-out solutions.

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Lesson Procedure

1. Introduction (2 minutes)

• Begin by activating prior knowledge: ask students what they know about "average" and how spread in data can be described.
• Present the graphic organizer outline on the board/projector, briefly explaining its sections: Vocabulary, Formulas, Example Problems, and Solutions.

2. Vocabulary & Formulas (2 minutes)

• Review key vocabulary aloud with the class, writing terms on the board:
  • Mean
  • Variance
  • Standard Deviation
• Write and verbally explain the formulas clearly:

[ \text{Mean} (\bar{x}) = \frac{\sum x_i}{n} ]

[ \text{Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n-1} ]

[ \text{Standard Deviation} (s) = \sqrt{s^2} ]

Emphasize the sample variance and standard deviation formulas (using (n-1) denominator).

3. Guided Example (4 minutes)

• Walk students through an example data set together, filling the graphic organizer with them:

Example data: 4, 7, 8, 6, 5

• Calculate the mean:
  (\frac{4 + 7 + 8 + 6 + 5}{5} = 6)
• Compute variance steps:
  ((4-6)^2 = 4), ((7-6)^2 = 1), ((8-6)^2 = 4), ((6-6)^2 = 0), ((5-6)^2 = 1)
  Sum = 4 + 1 + 4 + 0 + 1 = 10
  Variance = (\frac{10}{5-1} = 2.5)
• Calculate standard deviation:
  (\sqrt{2.5} \approx 1.58)

Highlight interpretation: The average distance of values from the mean is about 1.58 units.

4. Independent Practice / Closing (2 minutes)

• Provide students with a blank copy of the graphic organizer and a new data set as practice: 3, 9, 7, 5, 6
• Ask them to fill in the mean, variance, and standard deviation individually or in pairs.
• Collect the graphic organizers or review together quickly as time permits, using the completed answer key.

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Assessment

• Informal assessment through observation during the guided practice and completion of the graphic organizer.
• Review student responses on the independent practice data set for accuracy.

Extensions (for teachers who want to go further)

• Discuss how outliers affect variance and standard deviation.
• Connect these concepts to real-world data (e.g., test scores, sports statistics).

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Graphic Organizer Layout (To be printed on one page)

Section	Content Description
Vocabulary	Mean, Variance, Standard Deviation with simple definitions.
Formulas	Mean, Variance (s^2), and Standard Deviation (s) formulas.
Example	Provided example (4, 7, 8, 6, 5) with all calculation steps shown.
Solutions	Step-by-step answers for the example.
Practice	Blank section with space to calculate mean, variance, std dev for new data set (e.g., 3, 9, 7, 5, 6).

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This lesson harnesses visual organization and active problem-solving within a compact timeframe tailored for juniors in high school, reinforcing critical CCSS quantitative reasoning skills with clear, direct assessment opportunities.