Performance Task: Understanding Data through Range, Standard Deviation, and Z-Score

Teacher Guide

Standards and Skills

Common Core Standard:

  • Calculate the range and the standard deviation of a set of data.

  • Calculate a z-score for a given dataset.

Skills Taught:

  • Understanding range and standard deviation as measures of data variability.

  • Computing z-scores to interpret data in relation to the mean.

Task Description

Students will engage in a real-world data analysis task where they will calculate the range, standard deviation, and z-scores for a dataset representing the heights of a group of students. The task will deepen their understanding of these statistical concepts and develop their reasoning and problem-solving skills.

Task Delivery

Introduction (10 minutes)

  • Introduce statistical concepts: range, standard deviation, and z-scores.

  • Discuss their importance in understanding data variability and comparison.

Data Exploration (20 minutes)

  • Provide students with a dataset of heights (in inches) of 10 students: 65, 62, 68, 70, 72, 62, 75, 67, 64, 66.

  • Ask students to calculate the range and the standard deviation of the dataset.

Calculating Z-Scores (20 minutes)

  • Students will compute the z-score for each height using the formula:

    [ z = \frac{(x - \mu)}{\sigma} ]

    where (x) is each individual data point, (\mu) is the mean, and (\sigma) is the standard deviation.

  • Instruct students to interpret the meaning of the z-scores calculated, discussing their significance in understanding how each student's height compares to the average.

Review and Reflection (15 minutes)

  • Have a classroom discussion where students can share their findings and interpretations of their calculations.

  • Address any misconceptions and reinforce the importance of these calculations in real-world contexts, such as health or sports statistics.

Materials Needed

  • Dataset for student heights

  • Calculators

  • Worksheets for calculations and reflections

  • Whiteboard for instructor demonstrations

Student Directions

Performance Task: Analyzing Student Heights

Objective: To calculate the range, standard deviation, and z-scores of a dataset representing student heights.

Steps:

  1. Calculate the Range:

    • Determine the highest and lowest values in the dataset (65, 62, 68, 70, 72, 62, 75, 67, 64, 66).


    • Calculate the range using the formula:[ \text{Range} = \text{Maximum} - \text{Minimum} ]

    • Record your answer.

  2. Calculate the Standard Deviation:

    • Find the mean of the dataset.

    • Using the mean, calculate the standard deviation.

    • Record your answer.

  3. Calculate Z-Scores:

    • Use the formula ( z = \frac{(x - \mu)}{\sigma} ) to find the z-score for each student's height.

    • Record the z-scores for each height in a table.

  4. Interpret Your Results:

    • Describe what the range and standard deviation tell you about the dataset.

    • Explain the significance of the calculated z-scores, and how they help you understand the heights relative to the mean.

Remember:

  • Accuracy in calculations is key! Double-check your work.

  • Think critically about how these statistical measures offer insights into the data you analyzed.

Assessment Guide

Distractor Analysis

  • Correct Calculation: Accurately calculating the range, standard deviation, and z-scores for each height.

  • Partial Calculation: Making errors in some calculations but demonstrating an understanding of the process.

  • Incorrect Calculation: Misunderstanding the calculations or applying wrong formulas.

  • No Calculation: Failing to attempt the calculations.

Grading Rubric (4 Points)

  • 4 Points: Accurately calculates all range, standard deviation, and z-scores, providing clear interpretations of results.

  • 3 Points: Makes small errors in calculations or interpretations but shows understanding of the concepts.

  • 2 Points: Demonstrates some understanding but makes significant errors in calculations or interpretations.

  • 1 Point: Fails to calculate any values accurately and provides no meaningful interpretations.

Exemplar Solution

After calculating, a student determines:

  • Range: 13 (75 - 62)

  • Standard Deviation: 4.44

  • Z-scores: individual heights with calculated z-scores are shared; students interpret a z-score of +1.5 indicates a height significantly above average, enhancing their understanding of population data analysis.