What Percentile Is Z = − 3 Z = -3 Z = − 3 ?State Your Answer To The Nearest Tenth Of A Percent.

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Understanding the Concept of Percentiles and Z-Scores

In statistics, percentiles are used to express the relative standing of a value within a data set. A percentile represents the percentage of values in the data set that are less than or equal to the given value. On the other hand, a z-score is a measure of how many standard deviations an element is from the mean. In this article, we will explore the relationship between z-scores and percentiles, and specifically, we will determine the percentile corresponding to a z-score of -3.

The Relationship Between Z-Scores and Percentiles

The z-score formula is given by:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. The z-score tells us how many standard deviations away from the mean our value is. A z-score of 0 indicates that the value is equal to the mean, while a positive z-score indicates that the value is above the mean, and a negative z-score indicates that the value is below the mean.

To find the percentile corresponding to a z-score, we need to use a standard normal distribution (also known as a z-table). The z-table shows the area under the standard normal curve to the left of a given z-score. This area represents the proportion of values in the data set that are less than or equal to the given z-score.

Finding the Percentile Corresponding to $z = -3$

To find the percentile corresponding to a z-score of -3, we need to look up the z-score in a z-table. The z-table shows that the area to the left of a z-score of -3 is approximately 0.0013. This means that 0.13% of the values in the data set are less than or equal to -3 standard deviations away from the mean.

Calculating the Percentile

To calculate the percentile, we need to multiply the proportion of values by 100:

$$\text{Percentile} = 0.13 \times 100 = 13\%$$

Therefore, the percentile corresponding to a z-score of -3 is approximately 13%.

Conclusion

In this article, we explored the relationship between z-scores and percentiles, and specifically, we determined the percentile corresponding to a z-score of -3. We used a z-table to find the area under the standard normal curve to the left of a z-score of -3, and then calculated the percentile by multiplying the proportion of values by 100. The result is that the percentile corresponding to a z-score of -3 is approximately 13%.

Frequently Asked Questions

• What is a z-score? A z-score is a measure of how many standard deviations an element is from the mean.
• How do I find the percentile corresponding to a z-score? You can use a z-table to find the area under the standard normal curve to the left of a given z-score, and then calculate the percentile by multiplying the proportion of values by 100.
• What is the relationship between z-scores and percentiles? A z-score tells us how many standard deviations away from the mean our value is, while a percentile represents the percentage of values in the data set that are less than or equal to the given value.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Additional Resources

• [1] Khan Academy: Z-scores and percentiles
• [2] Stat Trek: Z-scores and percentiles

Note: The references and additional resources provided are for informational purposes only and are not an exhaustive list of resources on the topic.

Understanding Z-Scores and Percentiles

In our previous article, we explored the relationship between z-scores and percentiles, and specifically, we determined the percentile corresponding to a z-score of -3. In this article, we will answer some frequently asked questions about z-scores and percentiles.

Q&A

Q: What is a z-score?

A: A z-score is a measure of how many standard deviations an element is from the mean. It tells us how many standard deviations away from the mean our value is.

Q: How do I calculate a z-score?

A: To calculate a z-score, you need to use the following formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Q: What is the difference between a z-score and a percentile?

A: A z-score tells us how many standard deviations away from the mean our value is, while a percentile represents the percentage of values in the data set that are less than or equal to the given value.

Q: How do I find the percentile corresponding to a z-score?

A: You can use a z-table to find the area under the standard normal curve to the left of a given z-score, and then calculate the percentile by multiplying the proportion of values by 100.

Q: What is the relationship between z-scores and percentiles?

A: A z-score tells us how many standard deviations away from the mean our value is, while a percentile represents the percentage of values in the data set that are less than or equal to the given value.

Q: Can I use a z-score to determine the percentile of a value?

A: Yes, you can use a z-score to determine the percentile of a value. However, you need to use a z-table to find the area under the standard normal curve to the left of a given z-score, and then calculate the percentile by multiplying the proportion of values by 100.

Q: What is the significance of a z-score of 0?

A: A z-score of 0 indicates that the value is equal to the mean.

Q: What is the significance of a z-score of 1?

A: A z-score of 1 indicates that the value is 1 standard deviation above the mean.

Q: What is the significance of a z-score of -1?

A: A z-score of -1 indicates that the value is 1 standard deviation below the mean.

Q: Can I use a z-score to compare values from different data sets?

A: No, you cannot use a z-score to compare values from different data sets. Z-scores are only meaningful within a single data set.

Q: Can I use a z-score to determine the median of a data set?

A: No, you cannot use a z-score to determine the median of a data set. The median is a measure of central tendency that is not related to z-scores.

Conclusion

In this article, we answered some frequently asked questions about z-scores and percentiles. We hope that this article has provided you with a better understanding of the relationship between z-scores and percentiles, and how to use z-scores to determine the percentile of a value.

Frequently Asked Questions

• What is a z-score? A z-score is a measure of how many standard deviations an element is from the mean.
• How do I calculate a z-score? To calculate a z-score, you need to use the following formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

• What is the difference between a z-score and a percentile? A z-score tells us how many standard deviations away from the mean our value is, while a percentile represents the percentage of values in the data set that are less than or equal to the given value.

References

• [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.

Additional Resources

• [1] Khan Academy: Z-scores and percentiles
• [2] Stat Trek: Z-scores and percentiles