Unit 1 Lesson 10 : The Normal Distribution – Introduction and Calculations

The Normal Distribution – Introduction and Calculations

Unit 1, Topic 1.10: The Normal Distribution – Introduction and Calculations

Overview

This lesson introduces the normal distribution, a common bell-shaped pattern in quantitative data (like heights or test scores). It’s symmetric and single-peaked, described by mean (μ, center) and standard deviation (σ, spread). Context, like who the data is from, matters because it explains why data follows a normal shape. For example, heights for adults might be normal around 170 cm with σ of 10 cm, but for kids, the center shifts.

The normal distribution helps calculate z-scores (standardized values) and probabilities (areas under the curve, like 68% within 1σ). We’ll use the empirical rule (68-95-99.7%) and tables for estimates.

z score: https: //

Assignment:

Part 1: Guided Practice Activity

Work on your own. Use the example below for a normal distribution of heights (μ = 170 cm, σ = 10 cm). Describe it and calculate z-scores.

Example Data Points: Heights: 148, 152, 155, 158, 160, 162, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 180, 182, 184, 185, 187, 189, 192, 195, 197, 200 (from a sample of adults, all in cm).

Tasks:

1. Describing the Normal Distribution:
   • Describe its shape and parameters .
   • Write 1-2 sentences about the empirical rule.
   • Extra Practice: For test scores (μ = 75, σ = 10), describe the shape and empirical rule range for 95%.
2. Calculating Z-Scores:
   • Calculate z-scores for 148, 152, 155, 158, 160, 162, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 180, 182, 184, 185, 187, 189, 192, 195, 197, 200 using z = (x - μ)/σ.
   • Write 1-2 sentences about relative positions (e.g., "z = -1 for 160 cm means it's 1σ below average.").
   • Extra Practice: For test scores, calculate z for 65 and 85, and explain positions.
3. Probabilities and Reflection:
   • Use the empirical rule or a table to find approximate probabilities (e.g., P(160 < height < 180) ≈ 68%).
   • Write 2-3 sentences interpreting in context (e.g., "68% of adults are 160-180 cm, useful for clothing sizes.").

Part 2: Independent Practice

Use this normal distribution for IQ scores (μ = 100, σ = 15). 55, 60, 65, 70, 75, 80, 85, 88, 90, 93, 95, 98, 100, 102, 104, 106, 108, 110, 112, 115, 118, 120, 123, 126, 128, 130, 133, 136, 140, 145

Tasks:

• Describe the shape, μ, and σ, with empirical rule ranges for 68% and 95%.
• Calculate z-scores for 85, 100, 115, 130, and explain relative positions (e.g., percentiles like z=1 is top 16%).
• Write 2-3 sentences justifying a claim with probabilities (e.g., "P(IQ > 115) ≈ 16%, meaning only 16% score above average in context of job requirements.").
• Extra Activity: Invent a dataset for weights (μ = 70 kg, σ = 10 kg). Calculate z for two points, estimate a probability, and interpret in context (e.g., fitness class).

Homework Assignment

• Choose a real normal dataset (e.g., heights from 15 people, estimate μ/σ). Describe the distribution, calculate 2 z-scores, estimate a probability with the empirical rule, and interpret with context to share next class.