Chapter 3 - Distributions of Random Variables - 3.1 Normal Distribution

Sep 14, 2026

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Chapter 3 explains how random variables are modeled through probability distributions, focusing on the normal distribution as the most common real-world pattern. It introduces the normal distribution’s key properties, then models it using two parameters: mean and standard deviation. The chapter also shows how standardization via Z-scores converts observations to a common scale, enabling comparisons across nearly normal data and supporting interpretation of how unusual values are relative to the mean.

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Chapter 3: Distributions of Random Variables

3.1 Normal distribution

3.1.1 Normal distribution model

3.1.2 Standardizing with Z-scores

3.1.3 Normal probability table

Chapter 3 Distributions of random variables 3.1 Normal distribution Among all the distributions we see in practice, one is overwhelmingly the most common. The symmetric, unimodal, bell curve is ubiquitous throughout statistics. Indeed it is so common, that people often know it as the normal curve or normal distribution, 1 shown in Figure 3.1. Variables such as SAT scores and heights of US adult males closely follow the normal distribution. | Normal distribution facts
Many variables are nearly normal, but none are exactly normal. Thus the normal distribution, while not perfect for any single problem, is very useful for a variety of problems. We will use it in data exploration and to solve important problems in statistics. | | | --- | --- | | | | 1 It is also introduced as the Gaussian distribution after Frederic Gauss, the ﬁrst person to formalize its mathematical expression. Figure 3.1: A normal curve. N (μ, σ) Normal dist. with mean μ& [st. dev](st. dev). σ 128 CHAPTER 3 . DISTRIBUTIONS OF RANDOM VARIABLES 3.1.1 Normal distribution model The normal distribution model always describes a symmetric, unimodal, bell-shaped curve. However, these curves can look diﬀerent depending on the details of the model. Speciﬁcally, the normal distribution model can be adjusted using two parameters: mean and standard deviation. As you can probably guess, changing the mean shifts the bell curve to the left or right, while changing the standard deviation stretches or constricts the curve. Figure 3.2 shows the normal distribution with mean 0 and standard deviation 1 in the left panel and the normal distributions with mean 19 and standard deviation 4 in the right panel. Figure 3.3 shows these distributions on the same axis. −3 −2 −1 0 1 2 3 7 11 15 19 23 27 31 Figure 3.2: Both curves represent the normal distribution, however, they diﬀer in their center and spread. The normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution. 0 10 20 30 Figure 3.3: The normal models shown in Figure 3.2 but plotted together and on the same scale. If a normal distribution has mean μ and standard deviation σ, we may write the distribution as N (μ, σ) . The two distributions in Figure 3.3 can be written as N (μ = 0,σ = 1) and N (μ = 19,σ = 4) Because the mean and standard deviation describe a normal distribution exactly, they are called the distribution’s parameters. 􀀂 Guided Practice 3.1 Write down the short-hand for a normal distribution with2 (a) mean 5 and standard deviation 3, (b) mean-100 and standard deviation 10, and (c) mean 2 and standard deviation 9 . 2 (a) N (μ = 5,σ = 3) . (b) N (μ = −100,σ = 10) . (c) N (μ = 2,σ = 9) . 3.1. NORMAL DISTRIBUTION 129 | | SAT | ACT | | --- | --- | --- | | Mean | 1500 | 21 | | SD | 300 | 5 | Table 3 .4: Mean and standard deviation for the SAT and ACT. 11 16 21 26 31 Figure 3.5: Ann’s and Tom’s scores shown with the distributions of SAT and ACT scores. 3.1.2 Standardizing with Z-scores 􀀂 Example 3.2 Table 3.4 shows the mean and standard deviation for total scores on the SAT and ACT. The distribution of SAT and ACT scores are both nearly normal. Suppose Ann scored 1800 on her SAT and Tom scored 24 on his ACT. Who performed better? We use the standard deviation as a guide. Ann is 1 standard deviation above average on the SAT: 1500 + 300 = 1800 . Tom is 0 .6 standard deviations above the mean on the ACT: 21 + 0 .6 × 5 = 24 . In Figure 3.5, we can see that Ann tends to do better with respect to everyone else than Tom did, so her score was better. Example 3.2 used a standardization technique called a Z-score, a method most commonly employed for nearly normal observations but that may be used with any distribution. The Z-score of an observation is deﬁned as the number of standard deviations it falls above or below the mean. If the observation is one standard deviation above the mean, its Z-score is 1 . If it is 1 .5 stan

FAQs

What makes the normal distribution so widely used in statistics?

It has a symmetric, unimodal, bell-shaped form and appears frequently in practice. Many variables are close to normal even if none are exactly normal.

How do mean and standard deviation change a normal curve?

Changing the mean shifts the bell curve left or right, while changing the standard deviation stretches or constricts the curve.

What is a Z-score and how is it computed?

A Z-score is the number of standard deviations an observation lies above or below the mean. For X ~ N(μ, σ), it is computed as Z = (x − μ)/σ.

Sep 14, 2026

Uploaded by Oliver