Notice that all three distributions are bell-shaped and are centered at their mean (\(\mu = 0\)). The larger the standard deviation, the shorter and wider the curve, while the smaller the standard deviation, the taller and more narrow the curve.

Given that \(X\sim N\left(\mu, \sigma\right)\), we can compute probabilities associated with observed values of \(X\) by finding the corresponding area beneath the normal curve with mean \(\mu\) and standard deviation \(\sigma\).

Properties of the Normal Distribution: We have the following properties associated with the normal distribution. Consider \(X\sim N\left(\mu, \sigma\right)\).

• The area beneath the entire distribution is 1 (since this is equivalent to the probability that \(X\) takes on any of its possible values).

• \(\displaystyle{\mathbb{P}\left[X\leq \mu\right] = \mathbb{P}\left[X\geq \mu\right] = 0.5}\) (the area underneath a full half of the distribution is 0.5)

• The distribution is symmetric. In symbols, \(\mathbb{P}\left[X\leq \mu - k\right] = \mathbb{P}\left[X \geq \mu + k\right]\) for any \(k\).

• \(\displaystyle{\mathbb{P}\left[X = k\right] = 0}\) (the probability that \(X\) takes on any prescribed value exactly is \(0\))

Sometimes it is useful to be able to estimate probabilities or to estimate the proportion of a population that falls into a range as long as the population is nearly normal. A convenient rule of thumb is the Empirical Rule.

The Empirical Rule: If \(X\sim N\left(\mu, \sigma\right)\), then

• \(\mathbb{P}\left[\mu - \sigma \leq X\leq \mu + \sigma\right] \approx 0.67\) – that is, about 67% of observations lie within one standard deviation of the mean.
• \(\mathbb{P}\left[\mu - 2\sigma \leq X\leq \mu + 2\sigma\right] \approx 0.95\) – that is, about 95% of observations lie within two standard deviations of the mean.
• \(\mathbb{P}\left[\mu - 3\sigma \leq X\leq \mu + 3\sigma\right] \approx 0.997\) – that is, about 99.7% of observations lie within three standard deviations of the mean.

For each of the following, assume that \(X\sim N\left(\mu = 85, \sigma = 5\right)\)

Method 1: We can standardize the test scores so that they have comparable units.

• Definition: If an observation \(x\) comes from a nearly normal population with mean \(\mu\) and standard deviation \(\sigma\) then we compute \(z\)-score associated with \(x\) as follows:

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

An observation’s \(z\)-score is simply the number of standard deviations it falls above or below the mean.

A recap on \(z\)-scores: We can use \(z\)-scores as a common unit for comparing observations from completely different populations (such as SAT scores and ACT scores). Here’s a recap of the most important information so far:

• If an observation \(x\) comes from a nearly normal population with mean \(\mu\) and standard deviation \(\sigma\), we can compute it’s \(z\)-score using the formula: \(\displaystyle{z = \frac{x - \mu}{\sigma}}\).

• A \(z\)-score measures the number of standard deviations which an observation falls above or below the mean.

  • A positive \(z\)-score means that an observation was above the mean.
  • A negative \(z\)-score means that an observation was below the mean.
  • The larger a \(z\)-score is in absolute value, the further the corresponding observation falls from the mean. That is, the larger the magnitude of a \(z\)-score, the further into the tail of the distribution the corresponding observation falls.

Method 2: We can compute the percentile corresponding to Bob’s SAT score and the percentile corresponding to Sally’s ACT score.

• Definition: Given an observation \(x\) from a population – the percentile corresponding to \(x\) is the proportion of the population which falls below \(x\).

There are many ways to compute percentiles. Before the widespread availability of statistical software, people converted observed values to \(z\)-scores and then looked up the percentile in a table. Luckily R provides nice functionality for computing percentiles.

Computing Percentiles in R: If \(X\sim N\left(\mu, \sigma\right)\), then \[\mathbb{P}\left[X\leq q\right] \approx \tt{pnorm(q, mean = \mu, sd = \sigma)}\]

The block below is preset to compute the Bob’s percentile. Execute the code cell and then adapt the code to find Sally’s percentile. Use your results to answer the questions below.

pnorm(1400, 1068, 210)

We’ll make good use of this second method for a while, but don’t forget about standardization and \(z\)-scores. We’ll need that strategy quite often later in our course! For now, let’s move on to practicing with finding probabilities from a normal distribution using R’s pnorm() function.

For these first few questions I’ll draw pictures for you, but you should be prepared to draw your own shortly.

Through the last three problems you only worked with the standard normal distribution – that’s the \(Z\)-distribution, which is \(N\left(\mu = 0, \sigma = 1\right)\). We can find probabilities from arbitrary normal distributions (normal distributions with any mean and any standard deviation) using R’s pnorm() functionality – just supply the appropriate mean and sd arguments to pnorm() instead of the 0 and 1 that we passed earlier.

If \(X\sim N\left(\mu, \sigma\right)\), then to find the cutoff \(x^*\) for which \(\mathbb{P}\left[X < x^*\right] = p\), we can use R’s qnorm() function. Similar to pnorm(), this function takes three arguments. The first is the \(\tt{area~to~the~\underline{LEFT}}\) of the desired cutoff, the second is the \(\tt{mean}\) of the distribution, and the third is the \(\tt{standard~deviation}\) of the distribution.

Recall from earlier that SAT scores followed \(N\left(\mu = 1068, \sigma = 210\right)\) and ACT scores followed \(N\left(\mu = 20.8, \sigma = 5.8\right)\). The code block below is set up to find the minimum required SAT score to fall in the 95th percentile (to do better than 95% of other test-takers). Execute the code and note the required score. Adapt the code to find the minimum ACT score required to fall into the top 10% of all ACT test takers. Does your answer seem right? How can you judge?

qnorm(0.95, 1068, 210)