Topic 6: The Normal Distribution (Spreadsheets, Static)

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About

This activity covers working with normally distributed data, including the calculation of probabilities and percentiles, and introduces the notion of the z-score.

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Note. The button above resets multiple choice and checkbox questions. Currently, resetting answer entry cells must be done manually via hitting the Start Over button on each individual interactive cell.

Using a Spreadsheet Program

There are times throughout this activity where you’ll be asked to do work inside of your favorite spreadsheet program.

If you are a Google Sheets user, open a blank Google Sheet
If you are an Excel user, open an empty workbook on your computer or create a blank workbook in Excel online

The Normal Distribution

Throughout this activity we’ll investigate the probability distribution that is most central to our study of statistics: the normal distribution. If we are confident that our data are nearly normal, that opens the door for us to leverage many powerful statistical methods. This activity gives you practice working with normally distributed data.

Objectives

Workbook Objectives: After completing this workbook you should be able to:

Compute probabilities of events from populations well-modeled by a normal distribution.
Given a variable \(X\) which follows an assumed normal distribution, compute and interpret various percentile thresholds for \(X\).
Identify scenarios to which the normal or binomial distributions can be applied, and use the appropriate distribution to answer various probability-related questions.

The Normal Distribution

Definition: If a random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), we often write \(X\sim N\left(\mu, \sigma\right)\). Three different normal distributions appear below.

In blue is a normal distribution with \(\mu = 0\) and \(\sigma = 5\)
In red is a normal distribution with \(\mu = 0\) and \(\sigma = 0.5\)
In black is a normal distribution with \(\mu = 0\) and \(\sigma = 1\) (the so-called Standard Normal Distribution)

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 having mean \(\mu\) and standard deviation \(\sigma\).

Properties of 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, we write \(\mathbb{P}\left[X\leq \mu - k\right] = \mathbb{P}\left[X \geq \mu + k\right]\) for any \(k\). Geometrically, this means that corresponding lower-tail (left-tail) and upper-tail (right-tail) areas are equal.

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

Important

Unlike the binomial distribution, where the distinction between at least and more than required careful adjustment, the probability that a continuous variable \(X\) takes on any prescribed value exactly is \(0\) (that is, \(\mathbb{P}\left[X = k\right] = 0\)). This means there is no difference between \(\mathbb{P}\left[X \leq k\right]\) and \(\mathbb{P}\left[X < k\right]\). Similarly, there is no difference between \(\mathbb{P}\left[X \geq k\right]\) and \(\mathbb{P}\left[X > k\right]\).

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 set of rough estimates come from 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 \(X\sim N\left(\mu = 85, \sigma = 5\right)\).

Check Your Understanding: The Empirical Rule I

Use the Empirical Rule to approximate \(\mathbb{P}\left[80\leq X\leq 90\right]\).

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Check Your Understanding: The Empirical Rule II

According to the Empirical Rule, which of the following are boundary values for which we expect about 95% of observed values of \(X\) to fall between?

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Standardization and \(z\)-Scores

Scenario: Two students, Bob and Sally, are trying to compare how well they did on a college entrance exam. The difficulty comes in that Bob took the SAT which is known to follow an approximate normal distribution with a mean score of 1068 and a standard deviation of 210, while Sally took the ACT which also follows an approximately normal distribution but with a mean score of 20.8 and a standard deviation of 5.8. If Bob scored a 1400 on the SAT and Sally scored a 31 on the ACT, who scored relatively higher?

How do we answer this question? We’ll see two methods.

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

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

Try It! Use your spreadsheet to compute Bob and Sally’s \(z\)-scores and answer the questions that follow.

Computing a \(z\)-Score in a Spreadsheet

The \(z\)-score formula is simply arithmetic — no special spreadsheet function is required. In a blank cell, type = followed by the formula \((x - \mu) / \sigma\) using the appropriate values.

For example, to compute Bob’s \(z\)-score, find an empty cell, type =(1400-1068)/210 into it, and press Enter. Compute Sally’s \(z\)-score the same way using her ACT score (31), the ACT mean (20.8), and the ACT standard deviation (5.8).

Be sure to label each result so you remember which \(z\)-score belongs to which student.

Check Your Understanding: \(z\)-Scores I

Which of the following is the \(z\)-score corresponding to Bob? (round to two decimal places)

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Check Your Understanding: \(z\)-Scores II

Which of the following is the \(z\)-score corresponding to Sally? (round to two decimal places)

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Check Your Understanding: \(z\)-Scores III

Who did relatively better on their standardized exam, Bob or Sally? Why?

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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 its \(z\)-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.

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

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

In the images below, Bob’s percentile is shown as the shaded area in the distribution on the left, while Sally’s is shown in the distribution on the right.

Computing Percentiles in a Spreadsheet

In Excel and Google Sheets, the NORM.DIST() function computes the area to the left of a given boundary value — that is, the percentile. It takes four arguments:

The boundary value (\(q\))
The mean of the distribution (\(\mu\))
The standard deviation (\(\sigma\))
TRUE — this tells the function to return the cumulative probability/area rather than the height of the density curve (you’ll always use TRUE as the last argument to NORM.DIST())

So if \(X\sim N\left(\mu, \sigma\right)\), then \(\mathbb{P}\left[X\leq q\right] \approx\) =NORM.DIST(q, mean, sd, TRUE).

For example, Bob’s percentile is calculated by typing =NORM.DIST(1400, 1068, 210, TRUE) into an empty cell in your spreadsheet.

Try It! Use your spreadsheet to compute Bob’s and Sally’s percentiles. In a blank cell, type =NORM.DIST(1400, 1068, 210, TRUE) for Bob. Then adapt the formula to compute Sally’s percentile (her ACT score was 31, the mean was 20.8, and the standard deviation was 5.8). Label both results.

Check Your Understanding: Percentiles I

Which of the following is the percentile corresponding to Bob? (round to four decimal places)

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Check Your Understanding: Percentiles II

Which of the following is the percentile corresponding to Sally? (round to four decimal places)

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Check Your Understanding: Percentiles III

Who did relatively better on their standardized exam, Bob or Sally? Why?

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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 NORM.DIST() in your spreadsheet.

Computing Probability from a Normal Distribution

Through this section you’ll get practice finding probabilities by using NORM.DIST() to compute areas in your spreadsheet. Remember that NORM.DIST(q, mean, sd, TRUE) always returns the area to the left of the boundary value q in the distribution with the specified mean and standard deviation.

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

Question 1: Find \(\mathbb{P}\left[Z <\right.\) \(\left.\right]\) — remember that \(Z\sim N\left(\mu = 0, \sigma = 1\right)\).