Activity - 11: Normal Distributions - Normal (Gaussian) distribution

Q: What are the main learning objectives for this activity on normal distributions?

The activity focuses on sketching a normal distribution given μ and σ, converting it to a standard normal distribution using a z-score, and calculating probabilities and percentiles under N(μ, σ). It also asks students to compare histogram shapes and decide which variables could be modeled by normal distributions.

Q: How do you identify the shape and key points of a normal curve?

All normal distributions share the same overall shape; they differ by center μ and spread σ. When sketching, center the curve at μ and place inflection points at (μ − σ) and (μ + σ).

Q: How can probabilities under a normal distribution be computed for continuous variables like IQ scores?

Because the normal pdf does not require perfect fitting and the required integrals involve the error function, the activity suggests using technology tools to estimate probabilities such as P(X < 90) for X ~ N(100, 16). Methods mentioned include StatKey, Rossman/Chance applets, Wolfram Alpha, and R functions like pnorm and qnorm.

Oct 3, 2026

Upload by Riley

Activity #11 investigates the family of normal (Gaussian) distributions and how they model measurement error and approximate other distributions. The activity targets practical skills: sketching a normal curve from μ and σ, converting to standard normal using a z-score, and computing probabilities and percentiles. It also includes conceptual checks on identifying variables that may be adequately modeled by normal distributions, plus guidance on estimating IQ-based questions using technology rather than direct integration.

LanguageEnglish

CategoryExam

FormatPDF

Pages7

Reading Time18 min

AccessFree to download

Normal distributions overview

Learning objectives

Sketching and standardizing

Probabilities and percentiles

Normal pdf and notation

Inflection points at μ ± σ

Modeling random variables with normal distributions

IQ example and estimating P(X < 90)

Error function and technology-based methods

Tools: StatKey, Rossman/Chance, Wolfram Alpha, and R

Activity \#11: Normal Distributions Name: So far, we’ve investigated the binomial, hypergeometric, geometric, negative binomial, Poisson, uniform, exponential, and gamma distributions. In this activity, we’re going to investigate the family of normal (or Gaussian) distributions. Normal distributions can be useful in modeling measurement error or physical measurements of organisms, but they’re also extremely useful for approximating other distributions (such as the binomial and Poisson distributions) . Even when we have a population that is not well approximated by a normal distribution, averages of samples from the population can approximate a normal distribution (as we’ll see when we learn about the Central Limit Theorem) . By the end of this activity, we’ll need to be able to do the following: 1. Sketch a normal distribution with a given μ (mean) and σ (standard deviation) 2. Convert a normal distribution to a standard normal distribution (calculating a z-score) 3. Calculate probabilities and percentiles under a normal distribution with given μ and σ . 1. Identify similarities and differences among the following three histograms: These unimodal, symmetric distributions can be approximated by normal distributions. A continuous variable has a normal distribution with parameters μ and σ if its probability density function is: 1 "x−µ %2 pdf: f (x) = 1 e − 2 $\# σ &' for −∞ < x < ∞ , where µ = E [ X ] and σ = var (X) σ 2π If variable X follows a normal distribution, we can write X ~ N(μ, σ) Below, I’ve sketched normal distributions on top of the histograms. To do this, I only needed to know μ and σ . Notice that the normal pdf does not perfectly ﬁt any of the datasets; it ﬁts some better than others. 20 40 60 80 100 Level of Academic Challenge (2012 Freshmen) µ = 57.89 σ = 13.75 µ = 3156.30 σ = 570.44 2. Normal distributions are theoretical, so no data will follow a normal distribution perfectly. That doesn’t mean a normal distribution can’t be useful, though. Identify which of the following random variables might be adequately modeled by a normal distribution. Explain. a) The weights of randomly selected American adults. b) The income of randomly selected American families. c) The shoe size of Olympic athletes. 3. All normal distributions have the same shape; they differ only in terms of where they are centered (μ) and how spread out they are (σ) . When you want to sketch a normal curve, you should center it at μ and put the inﬂection points at (μ — σ) and (μ + σ) . IQ scores follow a normal distribution with μ =100 and σ = 16. That means IQ scores are distributed as f (x) = 16 1 "x−100%2 1 − 2 $\# 16 &' e 2π To the right, sketch the distribution of IQ scores. How could we demonstrate that the inﬂection points on a normal distribution are located at µ ± σ ? 4. If we randomly choose an individual, what’s the probability we choose someone with an IQ less than 90? Estimate the answer using the distribution you sketched above. Estimated probability = 1 "x−100%2 Because IQ scores are continuous, we need to ﬁnd: P (x < 90) = ∫090 ~~ ~~16~~ ~~12π~~ ~~ e − 2 $\# 16 &' dx That doesn’t look like something I want to integrate, so I’m going to hand [it off to ](it off to wolframalpha.com)[wolframalpha.com](it off to wolframalpha.com). Before I get the answer, let’s [have ](have wolframalpha.com)[wolframalpha.com](have wolframalpha.com) ﬁnd the indeﬁnite integral. Here’s the output: Even if you know what the error function is*, you won’t want to integrate to ﬁnd normal distribution probabilities. Instead, we’ll use technology. 5. Remember, we’re trying to ﬁnd P(X < 90) given X ~ N(100, 16) . Choose the method you prefer… 1 StatKey: [http://lock5stat.com/statkey/theoretical_distribution/theoretical_distribution.html\#normal](http://lock5stat.com/statkey/theoretical_distribution/theoretical_distribution.html#normal) a) Click to enter in the mean and [std.

FAQs

What are the main learning objectives for this activity on normal distributions?

The activity focuses on sketching a normal distribution given μ and σ, converting it to a standard normal distribution using a z-score, and calculating probabilities and percentiles under N(μ, σ). It also asks students to compare histogram shapes and decide which variables could be modeled by normal distributions.

How do you identify the shape and key points of a normal curve?

All normal distributions share the same overall shape; they differ by center μ and spread σ. When sketching, center the curve at μ and place inflection points at (μ − σ) and (μ + σ).

How can probabilities under a normal distribution be computed for continuous variables like IQ scores?

Because the normal pdf does not require perfect fitting and the required integrals involve the error function, the activity suggests using technology tools to estimate probabilities such as P(X < 90) for X ~ N(100, 16). Methods mentioned include StatKey, Rossman/Chance applets, Wolfram Alpha, and R functions like pnorm and qnorm.

Oct 3, 2026

Upload by Riley