Use The Information Given In The Table To Complete Each Of The Following Statements.Brenda Is 50 Inches Tall. Her $z$-score Is $\square$.$\[ \begin{tabular}{|c|c|c|} \hline \text{Age} & \text{Mean} & \text{Standard Deviation}
Introduction
In mathematics, particularly in statistics and probability, the z-score is a crucial concept used to measure the number of standard deviations an element is from the mean. It provides a way to compare the performance of different individuals or data points relative to a standard. In this article, we will delve into the concept of z-scores, how to calculate them, and how to use the information given in a table to complete a statement about Brenda's height.
What is a Z-Score?
A z-score, also known as a standard score, is a measure of how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. The z-score formula is:
z = (X - μ) / σ
Where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Calculating Z-Scores
To calculate a z-score, you need to know the value of the element, the mean of the dataset, and the standard deviation of the dataset. Let's use an example to illustrate this:
Suppose we have a dataset of heights of students in a class, with a mean height of 60 inches and a standard deviation of 5 inches. If a student's height is 65 inches, we can calculate their z-score as follows:
z = (65 - 60) / 5 z = 5 / 5 z = 1
This means that the student's height is 1 standard deviation above the mean.
Using a Table to Complete a Statement
Now, let's use the information given in the table to complete a statement about Brenda's height.
| Age | Mean | Standard Deviation |
|---|---|---|
| 50 | 60 | 5 |
Brenda is 50 inches tall. Her z-score is .
To complete this statement, we need to calculate Brenda's z-score using the formula:
z = (X - μ) / σ
Where:
- X is Brenda's height (50 inches)
- μ is the mean height (60 inches)
- σ is the standard deviation (5 inches)
Plugging in the values, we get:
z = (50 - 60) / 5 z = -10 / 5 z = -2
Therefore, Brenda's z-score is -2.
Interpretation of Z-Scores
Z-scores can be interpreted in the following ways:
- A z-score of 0 means that the element is equal to the mean.
- A positive z-score means that the element is above the mean.
- A negative z-score means that the element is below the mean.
- The magnitude of the z-score indicates how many standard deviations the element is from the mean.
Real-World Applications of Z-Scores
Z-scores have numerous real-world applications, including:
- Quality control: Z-scores can be used to detect outliers in a dataset, which can indicate a problem with the manufacturing process.
- Finance: Z-scores can be used to evaluate the performance of a stock or a portfolio of stocks.
- Medicine: Z-scores can be used to evaluate the performance of a patient's treatment.
Conclusion
In conclusion, z-scores are a powerful tool in mathematics that can be used to measure the number of standard deviations an element is from the mean. By understanding how to calculate z-scores and how to interpret them, we can gain valuable insights into the performance of different individuals or data points relative to a standard. In this article, we used a table to complete a statement about Brenda's height and learned how to calculate her z-score.
Frequently Asked Questions
Q: What is a z-score?
A: A z-score, also known as a standard score, is a measure of how many standard deviations an element is from the mean.
Q: How is a z-score calculated?
A: A z-score is calculated by subtracting the mean from the value and then dividing by the standard deviation.
Q: What is the formula for calculating a z-score?
A: The formula for calculating a z-score is:
z = (X - μ) / σ
Where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Q: How do I interpret a z-score?
A: A z-score can be interpreted in the following ways:
- A z-score of 0 means that the element is equal to the mean.
- A positive z-score means that the element is above the mean.
- A negative z-score means that the element is below the mean.
- The magnitude of the z-score indicates how many standard deviations the element is from the mean.
Q: What are some real-world applications of z-scores?
Q: What is a z-score?
A: A z-score, also known as a standard score, is a measure of how many standard deviations an element is from the mean. It provides a way to compare the performance of different individuals or data points relative to a standard.
Q: How is a z-score calculated?
A: A z-score is calculated by subtracting the mean from the value and then dividing by the standard deviation. The formula for calculating a z-score is:
z = (X - μ) / σ
Where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Q: What is the significance of a z-score?
A: A z-score is significant because it provides a way to compare the performance of different individuals or data points relative to a standard. It helps to identify outliers, which can indicate a problem with the data or the process.
Q: How do I interpret a z-score?
A: A z-score can be interpreted in the following ways:
- A z-score of 0 means that the element is equal to the mean.
- A positive z-score means that the element is above the mean.
- A negative z-score means that the element is below the mean.
- The magnitude of the z-score indicates how many standard deviations the element is from the mean.
Q: What is the difference between a z-score and a standard deviation?
A: A z-score is a measure of how many standard deviations an element is from the mean, while a standard deviation is a measure of the spread of the data. A z-score is a relative measure, while a standard deviation is an absolute measure.
Q: Can a z-score be negative?
A: Yes, a z-score can be negative. A negative z-score means that the element is below the mean.
Q: Can a z-score be zero?
A: Yes, a z-score can be zero. A z-score of zero means that the element is equal to the mean.
Q: What is the relationship between a z-score and a percentile?
A: A z-score can be used to determine a percentile. For example, a z-score of 1.5 corresponds to the 95th percentile.
Q: Can a z-score be used to compare data from different distributions?
A: No, a z-score cannot be used to compare data from different distributions. A z-score is specific to a particular distribution and cannot be used to compare data from different distributions.
Q: What are some real-world applications of z-scores?
A: Some real-world applications of z-scores include:
- Quality control: Z-scores can be used to detect outliers in a dataset, which can indicate a problem with the manufacturing process.
- Finance: Z-scores can be used to evaluate the performance of a stock or a portfolio of stocks.
- Medicine: Z-scores can be used to evaluate the performance of a patient's treatment.
Q: How do I calculate a z-score using a calculator?
A: To calculate a z-score using a calculator, you need to enter the value of the element, the mean, and the standard deviation. The calculator will then calculate the z-score for you.
Q: Can I use a z-score to determine the probability of an event?
A: Yes, a z-score can be used to determine the probability of an event. By using a z-score, you can determine the probability of an event occurring within a certain number of standard deviations from the mean.
Q: What is the relationship between a z-score and a normal distribution?
A: A z-score is a measure of how many standard deviations an element is from the mean in a normal distribution. The normal distribution is a probability distribution that is symmetric about the mean and has a bell-shaped curve.
Q: Can a z-score be used to compare data from different populations?
A: No, a z-score cannot be used to compare data from different populations. A z-score is specific to a particular population and cannot be used to compare data from different populations.
Q: What are some common mistakes to avoid when calculating a z-score?
A: Some common mistakes to avoid when calculating a z-score include:
- Rounding errors: Rounding errors can occur when calculating a z-score, especially when working with large numbers.
- Sign errors: Sign errors can occur when calculating a z-score, especially when working with negative numbers.
- Unit errors: Unit errors can occur when calculating a z-score, especially when working with different units of measurement.
Q: Can a z-score be used to determine the median of a dataset?
A: No, a z-score cannot be used to determine the median of a dataset. A z-score is a measure of how many standard deviations an element is from the mean, while the median is a measure of the middle value of a dataset.