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5.2 The Normal Distribution, Revisited

5 min readdecember 31, 2022

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

The normal model is a continuous probability distribution that is characterized by a bell-shaped curve that is symmetrical around the mean. The mean, median, and of a are all equal, and the distribution is , meaning that it has only one peak or mound. 🐪

To describe a normal model, you can talk about the center, shape, and spread of the distribution. The center of a normal model is represented by the mean, which is the arithmetic average of the data. The shape of a normal model is characterized by the bell-shaped curve, which is symmetrical around the mean. The spread of a normal model is represented by the , which measures the dispersion or spread of the data around the mean.

In general, a model (such as a sampling distribution) is approximately normal if the following conditions are met for the following types of data:

Categorical (Proportions)

Quantitative (Means)

  • The number of successes and failures is at least 10 (Large Counts)

  • The sample size is at least 30 (Central Limit Theorem which you’ll learn about in the next post)

OR

  • Population is normally distributed

Empirical Rule Review

The area underneath any density curve (including the normal curve) is equal to 1, or 100% of the data. There is 50% of the data on each size of the mean.

  • 68% of the data is within 1 of the mean (34% on each side). 🔴

  • 95% of the data is within 2 standard deviations of the mean (an additional 13.5% on each side). 🟣

  • 99.7% of the data is within 3 standard deviations of the mean (an additional 2.35% on each side). 🟢

https://miro.medium.com/max/24000/1*IdGgdrY_n_9_YfkaCh-dag.png

Source: Medium

🎥 Watch: AP Stats - Normal Distributions

A z-score indicates how many standard deviations above or below the mean a piece of data is. If a z-score is -2, it is 2 standard deviations below the mean. If a z-score is 1, it is 1 above the mean.  When asked to interpret a z-score, it is imperative that you indicate the direction (positive or negative). The magnitude of the z-scores is also important, especially when making inferences.

Z-scores are often used to compare scores regarding two sets of data, like SAT and ACT scores. The z-scores are resistant to units. Therefore the units do not matter and the z-scores remain the same if we convert the units.  Think of z-scores as a way of comparing apples (🍎) to oranges (🍊).

Interpreting Percentiles in Normal Distributions

In a , you can determine the intervals associated with a given area by assigning appropriate inequalities to the boundaries of the intervals.

For example, if you want to find the interval associated with an area of 0.95 (or 95%) under the normal curve, you would use the z-score values that correspond to that area. The z-score is a measure of how many standard deviations a value is from the mean of the distribution.

To find the z-scores that correspond to an area of 0.95, you can use a table of the standard or a calculator or software program that can calculate them for you. The z-scores will be the boundaries of the interval, and you can use them to determine the values that fall within that interval by plugging them back into the equation for the .

For example, if the mean of the distribution is 100 and the is 15, the interval associated with an area of 0.95 would be between the values obtained by plugging the corresponding z-scores into the equation for the .

We can also use actual equalities, as shown in the visual below:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-31%20at%2012.44-mluYVoqzgT1d.png?alt=media&token=bb04cc07-f7e7-4751-928a-a2db5d47493b

Source: College Board

Practice Problems

(1) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a of 3 inches. The mean height of women is 65 inches, with a of 2.5 inches. Calculate the z-score for a man who is 72 inches tall.

(2) A study is conducted to compare the GPA of students at two different colleges. The mean GPA of students at College A is 3.0, with a of 0.5. The mean GPA of students at College B is 2.5, with a of 0.4. Calculate the z-score for a student at College A who has a GPA of 3.5.

(3) A study is conducted to compare the number of hours spent studying per week by students at two different universities. The mean number of hours spent studying per week at University A is 15 hours, with a of 3 hours. The mean number of hours spent studying per week at University B is 12 hours, with a of 2 hours. Calculate the z-score for a student at University A who studies 21 hours per week.

(4) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a of 3 inches. The mean height of women is 65 inches, with a of 2.5 inches. Calculate the z-score for a woman who is 62 inches tall.

(5) Going back to problem (2), calculate the z-score for a student at College B who has a GPA of 3.0.

Answers

(1) To calculate the z-score for a man who is 72 inches tall, we can use the following formula:

z = (x - mean) /

Plugging in the values, we get:

z = (72 - 70) / 3 = 2 / 3 = 0.67

Thus, the z-score for a man who is 72 inches tall is 0.67.

(2) Plugging in the values, we get:

z = (3.5 - 3.0) / 0.5 = 0.5 / 0.5 = 1

Thus, the z-score for a student at College A who has a GPA of 3.5 is 1.

(3) Plugging in the values, we get:

z = (21 - 15) / 3 = 3 / 3 = 2

Thus, the z-score for a student at University A who studies 18 hours per week is 2.

(4) Plugging in the values, we get:

z = (62 - 65) / 2.5 = -3 / 2.5 = -1.2

Thus, the z-score for a woman who is 62 inches tall is -1.2.

(5) Plugging in the values, we get:

z = (3.0 - 2.5) / 0.4 = 0.5 / 0.4 = 1.25

Thus, the z-score for a student at College B who has a GPA of 3.0 is 1.25.

Key Terms to Review (6)

Empirical Rule

: The empirical rule, also known as the 68-95-99.7 rule, is a statistical guideline that states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

Mode

: The mode is the value or values that appear most frequently in a dataset.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Percentiles

: Percentiles are values that divide a dataset into 100 equal parts, representing the percentage of data points below a certain value. For example, if you are in the 75th percentile on a test, it means you scored higher than 75% of the other students.

Standard Deviation

: The standard deviation measures the average amount of variation or dispersion in a set of data. It tells us how spread out the values are from the mean.

Unimodal

: Unimodal describes a dataset that has one clear mode, or one value that appears most frequently.

5.2 The Normal Distribution, Revisited

5 min readdecember 31, 2022

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

The normal model is a continuous probability distribution that is characterized by a bell-shaped curve that is symmetrical around the mean. The mean, median, and of a are all equal, and the distribution is , meaning that it has only one peak or mound. 🐪

To describe a normal model, you can talk about the center, shape, and spread of the distribution. The center of a normal model is represented by the mean, which is the arithmetic average of the data. The shape of a normal model is characterized by the bell-shaped curve, which is symmetrical around the mean. The spread of a normal model is represented by the , which measures the dispersion or spread of the data around the mean.

In general, a model (such as a sampling distribution) is approximately normal if the following conditions are met for the following types of data:

Categorical (Proportions)

Quantitative (Means)

  • The number of successes and failures is at least 10 (Large Counts)

  • The sample size is at least 30 (Central Limit Theorem which you’ll learn about in the next post)

OR

  • Population is normally distributed

Empirical Rule Review

The area underneath any density curve (including the normal curve) is equal to 1, or 100% of the data. There is 50% of the data on each size of the mean.

  • 68% of the data is within 1 of the mean (34% on each side). 🔴

  • 95% of the data is within 2 standard deviations of the mean (an additional 13.5% on each side). 🟣

  • 99.7% of the data is within 3 standard deviations of the mean (an additional 2.35% on each side). 🟢

https://miro.medium.com/max/24000/1*IdGgdrY_n_9_YfkaCh-dag.png

Source: Medium

🎥 Watch: AP Stats - Normal Distributions

A z-score indicates how many standard deviations above or below the mean a piece of data is. If a z-score is -2, it is 2 standard deviations below the mean. If a z-score is 1, it is 1 above the mean.  When asked to interpret a z-score, it is imperative that you indicate the direction (positive or negative). The magnitude of the z-scores is also important, especially when making inferences.

Z-scores are often used to compare scores regarding two sets of data, like SAT and ACT scores. The z-scores are resistant to units. Therefore the units do not matter and the z-scores remain the same if we convert the units.  Think of z-scores as a way of comparing apples (🍎) to oranges (🍊).

Interpreting Percentiles in Normal Distributions

In a , you can determine the intervals associated with a given area by assigning appropriate inequalities to the boundaries of the intervals.

For example, if you want to find the interval associated with an area of 0.95 (or 95%) under the normal curve, you would use the z-score values that correspond to that area. The z-score is a measure of how many standard deviations a value is from the mean of the distribution.

To find the z-scores that correspond to an area of 0.95, you can use a table of the standard or a calculator or software program that can calculate them for you. The z-scores will be the boundaries of the interval, and you can use them to determine the values that fall within that interval by plugging them back into the equation for the .

For example, if the mean of the distribution is 100 and the is 15, the interval associated with an area of 0.95 would be between the values obtained by plugging the corresponding z-scores into the equation for the .

We can also use actual equalities, as shown in the visual below:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-31%20at%2012.44-mluYVoqzgT1d.png?alt=media&token=bb04cc07-f7e7-4751-928a-a2db5d47493b

Source: College Board

Practice Problems

(1) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a of 3 inches. The mean height of women is 65 inches, with a of 2.5 inches. Calculate the z-score for a man who is 72 inches tall.

(2) A study is conducted to compare the GPA of students at two different colleges. The mean GPA of students at College A is 3.0, with a of 0.5. The mean GPA of students at College B is 2.5, with a of 0.4. Calculate the z-score for a student at College A who has a GPA of 3.5.

(3) A study is conducted to compare the number of hours spent studying per week by students at two different universities. The mean number of hours spent studying per week at University A is 15 hours, with a of 3 hours. The mean number of hours spent studying per week at University B is 12 hours, with a of 2 hours. Calculate the z-score for a student at University A who studies 21 hours per week.

(4) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a of 3 inches. The mean height of women is 65 inches, with a of 2.5 inches. Calculate the z-score for a woman who is 62 inches tall.

(5) Going back to problem (2), calculate the z-score for a student at College B who has a GPA of 3.0.

Answers

(1) To calculate the z-score for a man who is 72 inches tall, we can use the following formula:

z = (x - mean) /

Plugging in the values, we get:

z = (72 - 70) / 3 = 2 / 3 = 0.67

Thus, the z-score for a man who is 72 inches tall is 0.67.

(2) Plugging in the values, we get:

z = (3.5 - 3.0) / 0.5 = 0.5 / 0.5 = 1

Thus, the z-score for a student at College A who has a GPA of 3.5 is 1.

(3) Plugging in the values, we get:

z = (21 - 15) / 3 = 3 / 3 = 2

Thus, the z-score for a student at University A who studies 18 hours per week is 2.

(4) Plugging in the values, we get:

z = (62 - 65) / 2.5 = -3 / 2.5 = -1.2

Thus, the z-score for a woman who is 62 inches tall is -1.2.

(5) Plugging in the values, we get:

z = (3.0 - 2.5) / 0.4 = 0.5 / 0.4 = 1.25

Thus, the z-score for a student at College B who has a GPA of 3.0 is 1.25.

Key Terms to Review (6)

Empirical Rule

: The empirical rule, also known as the 68-95-99.7 rule, is a statistical guideline that states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

Mode

: The mode is the value or values that appear most frequently in a dataset.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Percentiles

: Percentiles are values that divide a dataset into 100 equal parts, representing the percentage of data points below a certain value. For example, if you are in the 75th percentile on a test, it means you scored higher than 75% of the other students.

Standard Deviation

: The standard deviation measures the average amount of variation or dispersion in a set of data. It tells us how spread out the values are from the mean.

Unimodal

: Unimodal describes a dataset that has one clear mode, or one value that appears most frequently.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.