Lesson 8: The Normal Distribution

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.title[
# Lesson 8: The Normal Distribution
]
.author[
### Fei Ye
]
.date[
### May, 2024
]

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## Unit 6C: The Normal Distribution

*Primary Source:* PPT for the book "Using & Understanding Mathematics".

---

## The Normal Distribution

The normal distribution is a symmetric, bell-shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution. Its variation is characterized by the standard deviation of the distribution.

---

## Example: The Normal Shape (1 of 2)

The following figure shows two distributions:  
(a) a famous data set of the chest sizes of 5,738 Scottish militiamen collected in about 1846 and  
(b) the distribution of the population densities of the 50 states.

Is either distribution a normal distribution? Explain.
![image-20200818150148457](data:image/png;base64,#img/image-20200818150148457.png)

---

## Example: The Normal Shape (2 of 2)

**Solution:**

The distribution of chest sizes in Figure (a) is nearly symmetric, with a mean between 39 and 40 inches. Values far from the mean are less common, giving it the bell shape of a normal distribution.

The distribution in Figure (b) shows that most states have low population densities, but a few have much higher densities. This fact makes the distribution right-skewed, so it is not a normal distribution.

---

## Conditions for a Normal Distribution

A data set satisfying the following criteria is likely to have a nearly normal distribution.

1. *Most data values are clustered near the mean*, giving the distribution a well-defined single peak.
2. Data values are *spread evenly* around the mean, making the distribution symmetric.
3. *Larger deviations* from the mean are increasingly *rare*, producing the tapering tails of the distribution.
4. Individual data values result from a combination of many different factors.

---

## Example: Is It a Normal Distribution (1 of 2)

Which of the following variables would you expect to have a normal or nearly normal distribution?

a. Scores on a very easy test

b. Foot length of a random sample of adult women

**Solution:**

**a.** Tests have a maximum score (100%) that limits the size of the data values.

If a test is very easy, the mean will be high and many scores will be near the maximum. The few lower scores can be spread out well below the mean. We therefore expect the distribution of scores to be left-skewed and not normal.

---

## Example: Is It a Normal Distribution (2 of 2)

**b.** Foot length is a human trait determined by many genetic and environmental factors. We therefore expect lengths of women’s feet to cluster near a mean and become less common farther from the mean in both directions, giving the distribution the bell shape of a normal distribution.

---

## The 68-95-99.7 Rule for a Normal Distribution (1 of 2)

![image-20200818150203471](data:image/png;base64,#img/image-20200818150203471.png)

---

## The 68-95-99.7 Rule for a Normal Distribution (2 of 2)

- About 68% (more precisely, 68.3%), or just over two-thirds, of the data points fall within 1 standard deviation of the mean.

- About 95% (more precisely, 95.4%) of the data points fall within 2 standard deviations of the mean.

- About 99.7% of the data points fall within 3 standard deviations of the mean.

The 68-95-99.7 rule applies to data values that are exactly 1, 2, or 3 standard deviations from the mean.

---

## Example: Detecting Counterfeits (1 of2)

Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal U.S. quarters are normally distributed with a mean of 5.67 grams and a standard deviation of 0.0700 gram.

If a vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what percentage of legal quarters will be rejected by the machine?

---

## Example: Detecting Counterfeits (2 of2)

**Solution:** A weight of 5.81 is 0.14 gram, or 2 standard deviations, above the mean.

A weight of 5.53 is 0.14 gram, or 2 standard deviations, below the mean.

Therefore, by accepting only quarters within the weight range 5.53 to 5.81 grams, the machine accepts quarters that are within 2 standard deviations of the mean and rejects those that are more than 2 standard deviations from the mean.

By the 68-95-99.7 rule, about 95% of legal quarters will be accepted and about 5% of legal quarters will be rejected.

---

## Standard Scores

The number of standard deviations a data value lies above or below the mean is called its standard score (or z-score ), defined by

$$
𝑧=\text{standard score}=\frac{\text{data value−mean}}{\text{standard deviation}}
$$

.center[
|Data value||Standard Score|
|:---:|:---:|:---:|
|Above the mean| `$\longrightarrow$` |positive|
|Below the mean| `$\longrightarrow$` |negative|
]

---

## Example: Standard IQ Scores

The Stanford-Binet IQ test is scaled so that scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the standard scores for IQ scores of 85, 100, and 125.

**Solution:** We calculate the standard scores for these IQs by using the standard score formula with a mean of 100 and a standard deviation of 15.

$$
\text{Standard score for 85:}\quad z=\dfrac{85-100}{15}=-1.00
$$
$$
\text{Standard score for 100:}\quad z=\dfrac{100-100}{15}=0.00
$$
$$
\text{Standard score for 125:}\quad z=\dfrac{125-100}{15}\approx1.67
$$

---

## Standard Scores and Percentiles (1 of 3)

- A **percentile** is a data value below which a given percentage of data fall.

- A **percentile rank** of a data value is the percentage of data below the given data value.

**Example:** In the previous example, the 50-th percentile is 100. The percentile rank of 100 is 50 which means that 50% of scores are at most 100.

**Example:** The quartiles Q1, Q2, and Q3 have percentile rank 25, 50 and 75 respectively.

---

## Standard Scores and Percentiles (2 of3)

TABLE 6.4: Standard Scores and Percentile Ranks

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> -3.5 </td>
   <td style="text-align:left;"> 0.02 </td>
   <td style="text-align:left;"> -1.0 </td>
   <td style="text-align:left;"> 15.87 </td>
   <td style="text-align:left;"> 0.0 </td>
   <td style="text-align:left;"> 50.00 </td>
   <td style="text-align:left;"> 1.1 </td>
   <td style="text-align:left;"> 86.43 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -3.0 </td>
   <td style="text-align:left;"> 0.13 </td>
   <td style="text-align:left;"> -0.95 </td>
   <td style="text-align:left;"> 17.11 </td>
   <td style="text-align:left;"> 0.05 </td>
   <td style="text-align:left;"> 51.99 </td>
   <td style="text-align:left;"> 1.2 </td>
   <td style="text-align:left;"> 88.49 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.9 </td>
   <td style="text-align:left;"> 0.19 </td>
   <td style="text-align:left;"> -0.90 </td>
   <td style="text-align:left;"> 18.41 </td>
   <td style="text-align:left;"> 0.10 </td>
   <td style="text-align:left;"> 53.98 </td>
   <td style="text-align:left;"> 1.3 </td>
   <td style="text-align:left;"> 90.32 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.8 </td>
   <td style="text-align:left;"> 0.26 </td>
   <td style="text-align:left;"> -0.85 </td>
   <td style="text-align:left;"> 19.77 </td>
   <td style="text-align:left;"> 0.15 </td>
   <td style="text-align:left;"> 55.96 </td>
   <td style="text-align:left;"> 1.4 </td>
   <td style="text-align:left;"> 91.92 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.7 </td>
   <td style="text-align:left;"> 0.35 </td>
   <td style="text-align:left;"> -0.80 </td>
   <td style="text-align:left;"> 21.19 </td>
   <td style="text-align:left;"> 0.20 </td>
   <td style="text-align:left;"> 57.93 </td>
   <td style="text-align:left;"> 1.5 </td>
   <td style="text-align:left;"> 93.32 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.6 </td>
   <td style="text-align:left;"> 0.47 </td>
   <td style="text-align:left;"> -0.75 </td>
   <td style="text-align:left;"> 22.66 </td>
   <td style="text-align:left;"> 0.25 </td>
   <td style="text-align:left;"> 59.87 </td>
   <td style="text-align:left;"> 1.6 </td>
   <td style="text-align:left;"> 94.52 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.5 </td>
   <td style="text-align:left;"> 0.62 </td>
   <td style="text-align:left;"> -0.70 </td>
   <td style="text-align:left;"> 24.20 </td>
   <td style="text-align:left;"> 0.30 </td>
   <td style="text-align:left;"> 61.79 </td>
   <td style="text-align:left;"> 1.7 </td>
   <td style="text-align:left;"> 95.54 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.4 </td>
   <td style="text-align:left;"> 0.82 </td>
   <td style="text-align:left;"> -0.65 </td>
   <td style="text-align:left;"> 25.78 </td>
   <td style="text-align:left;"> 0.35 </td>
   <td style="text-align:left;"> 63.68 </td>
   <td style="text-align:left;"> 1.8 </td>
   <td style="text-align:left;"> 96.41 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.3 </td>
   <td style="text-align:left;"> 1.07 </td>
   <td style="text-align:left;"> -0.60 </td>
   <td style="text-align:left;"> 27.43 </td>
   <td style="text-align:left;"> 0.40 </td>
   <td style="text-align:left;"> 65.54 </td>
   <td style="text-align:left;"> 1.9 </td>
   <td style="text-align:left;"> 97.13 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.2 </td>
   <td style="text-align:left;"> 1.39 </td>
   <td style="text-align:left;"> -0.55 </td>
   <td style="text-align:left;"> 29.12 </td>
   <td style="text-align:left;"> 0.45 </td>
   <td style="text-align:left;"> 67.36 </td>
   <td style="text-align:left;"> 2.0 </td>
   <td style="text-align:left;"> 97.72 </td>
  </tr>
</tbody>
</table>

---

## Standard Scores and Percentiles (3 of3)

TABLE 6.4: Standard Scores and Percentile Ranks (continued)

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
   <th style="text-align:left;"> z-score </th>
   <th style="text-align:left;"> Rank </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> -2.1 </td>
   <td style="text-align:left;"> 1.79 </td>
   <td style="text-align:left;"> -0.50 </td>
   <td style="text-align:left;"> 30.85 </td>
   <td style="text-align:left;"> 0.50 </td>
   <td style="text-align:left;"> 69.15 </td>
   <td style="text-align:left;"> 2.1 </td>
   <td style="text-align:left;"> 98.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -2.0 </td>
   <td style="text-align:left;"> 2.28 </td>
   <td style="text-align:left;"> -0.45 </td>
   <td style="text-align:left;"> 32.64 </td>
   <td style="text-align:left;"> 0.55 </td>
   <td style="text-align:left;"> 70.88 </td>
   <td style="text-align:left;"> 2.2 </td>
   <td style="text-align:left;"> 98.61 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.9 </td>
   <td style="text-align:left;"> 2.87 </td>
   <td style="text-align:left;"> -0.40 </td>
   <td style="text-align:left;"> 34.46 </td>
   <td style="text-align:left;"> 0.60 </td>
   <td style="text-align:left;"> 72.57 </td>
   <td style="text-align:left;"> 2.3 </td>
   <td style="text-align:left;"> 98.93 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.8 </td>
   <td style="text-align:left;"> 3.59 </td>
   <td style="text-align:left;"> -0.35 </td>
   <td style="text-align:left;"> 36.32 </td>
   <td style="text-align:left;"> 0.65 </td>
   <td style="text-align:left;"> 74.22 </td>
   <td style="text-align:left;"> 2.4 </td>
   <td style="text-align:left;"> 99.18 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.7 </td>
   <td style="text-align:left;"> 4.46 </td>
   <td style="text-align:left;"> -0.30 </td>
   <td style="text-align:left;"> 38.21 </td>
   <td style="text-align:left;"> 0.70 </td>
   <td style="text-align:left;"> 75.80 </td>
   <td style="text-align:left;"> 2.5 </td>
   <td style="text-align:left;"> 99.38 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.6 </td>
   <td style="text-align:left;"> 5.48 </td>
   <td style="text-align:left;"> -0.25 </td>
   <td style="text-align:left;"> 40.13 </td>
   <td style="text-align:left;"> 0.75 </td>
   <td style="text-align:left;"> 77.34 </td>
   <td style="text-align:left;"> 2.6 </td>
   <td style="text-align:left;"> 99.53 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.5 </td>
   <td style="text-align:left;"> 6.68 </td>
   <td style="text-align:left;"> -0.20 </td>
   <td style="text-align:left;"> 42.07 </td>
   <td style="text-align:left;"> 0.80 </td>
   <td style="text-align:left;"> 78.81 </td>
   <td style="text-align:left;"> 2.7 </td>
   <td style="text-align:left;"> 99.65 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.4 </td>
   <td style="text-align:left;"> 8.08 </td>
   <td style="text-align:left;"> -0.15 </td>
   <td style="text-align:left;"> 44.04 </td>
   <td style="text-align:left;"> 0.85 </td>
   <td style="text-align:left;"> 80.23 </td>
   <td style="text-align:left;"> 2.8 </td>
   <td style="text-align:left;"> 99.74 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.3 </td>
   <td style="text-align:left;"> 9.68 </td>
   <td style="text-align:left;"> -0.10 </td>
   <td style="text-align:left;"> 46.02 </td>
   <td style="text-align:left;"> 0.90 </td>
   <td style="text-align:left;"> 81.59 </td>
   <td style="text-align:left;"> 2.9 </td>
   <td style="text-align:left;"> 99.81 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.2 </td>
   <td style="text-align:left;"> 11.51 </td>
   <td style="text-align:left;"> -0.05 </td>
   <td style="text-align:left;"> 48.01 </td>
   <td style="text-align:left;"> 0.95 </td>
   <td style="text-align:left;"> 82.89 </td>
   <td style="text-align:left;"> 3.0 </td>
   <td style="text-align:left;"> 99.87 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> -1.1 </td>
   <td style="text-align:left;"> 13.57 </td>
   <td style="text-align:left;"> -0.0 </td>
   <td style="text-align:left;"> 50.00 </td>
   <td style="text-align:left;"> 1.0 </td>
   <td style="text-align:left;"> 84.13 </td>
   <td style="text-align:left;"> 3.5 </td>
   <td style="text-align:left;"> 99.98 </td>
  </tr>
</tbody>
</table>

---

## Example: Cholesterol Levels (1 of 2)

Cholesterol levels in men 18 to 24 years of age are normally distributed with a mean of 178 and a standard deviation of 41.

**a.** In what percentile rank is a 20-year-old man with a cholesterol level of 190?

**b.** What cholesterol level corresponds to the 90th percentile, the level at which treatment may be necessary?

**Solution:**  
**a.** The standard score for a cholesterol level of 190 is
$$
z=\dfrac{\text{data value}-\text{mean}}{\text{standard deviation}}=\dfrac{190-178}{41}\approx 0.29.
$$

Table 6.4 shows that a standard score of 0.29 corresponds to almost the 62nd percentile.

---

## Example: Cholesterol Levels (2 of 2)

**b.** Table 6.4 shows that 90.32% of all data values have a standard score less than 1.3.

That is, the 90th percentile is about 1.3 standard deviations above the mean.

The standard deviation is 41, so 1.3 standard deviations is `$1.3 \times 41 = 53.3$`.

We find that the 90th percentile begins at about `$178 + 53 = 231$`.

A person with a cholesterol level above 231 may need treatment.

---

## Practice: Calculate the standard score `$z$`

.iframecontainer[
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---

## Practice: Using the 68-95-99.7 rule to find range

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---

## Practice: Using the 68-95-99.7 rule to find percentage

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  <iframe src="https://www.myopenmath.com/embedq2.php?id=19757&amp;seed=2021&amp;showansafter" width="100%" height="400px" data-external="1"></iframe>
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---

## Practice: Percentile and percentile rank of a test

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  <iframe src="https://www.myopenmath.com/embedq2.php?id=677031&amp;seed=2021&amp;showansafter" width="100%" height="400px" data-external="1"></iframe>
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.footmark[
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