What Is The $z$-score If $\mu=122$, $\sigma=52.3$, And $x=264.3$?A. $-1.74$ B. 1.74 C. $-2.72$ D. 2.72
In statistics, the z-score is a measure that indicates how many standard deviations an element is from the mean. It is a crucial concept in understanding the distribution of data and is widely used in various fields such as finance, engineering, and social sciences. In this article, we will explore the concept of z-score and learn how to calculate it using a given formula.
What is a Z-Score?
A z-score is a numerical value that represents how many standard deviations an element is from the mean. It is calculated using the following formula:
z = (x - ╬╝) / ╧Г
where:
- z is the z-score
- x is the value of the element
- ╬╝ is the mean of the dataset
- ╧Г is the standard deviation of the dataset
Calculating the Z-Score
To calculate the z-score, we need to substitute the given values into the formula. Let's assume we have the following values:
- ╬╝ = 122 (mean)
- ╧Г = 52.3 (standard deviation)
- x = 264.3 (value of the element)
Now, let's substitute these values into the formula:
z = (264.3 - 122) / 52.3
Step-by-Step Calculation
To calculate the z-score, we need to follow the order of operations (PEMDAS):
- Subtract 122 from 264.3: 264.3 - 122 = 142.3
- Divide 142.3 by 52.3: 142.3 / 52.3 = 2.72
Conclusion
Therefore, the z-score is 2.72. This means that the value of 264.3 is 2.72 standard deviations above the mean.
Interpretation of Z-Score
A z-score of 2.72 indicates that the value of 264.3 is significantly higher than the mean. In a normal distribution, about 95% of the data points fall within 2 standard deviations of the mean. Therefore, a z-score of 2.72 indicates that the value of 264.3 is in the upper 2.5% of the distribution.
Comparison with Other Options
Let's compare the calculated z-score with the given options:
A. -1.74 B. 1.74 C. -2.72 D. 2.72
The calculated z-score (2.72) matches option D.
Conclusion
In the previous article, we explored the concept of z-scores and learned how to calculate them using a given formula. In this article, we will answer some frequently asked questions (FAQs) about z-scores to help you better understand this concept.
Q: What is the purpose of calculating a z-score?
A: The purpose of calculating a z-score is to determine how many standard deviations an element is from the mean. This helps to understand the distribution of data and identify outliers.
Q: What is the difference between a z-score and a standard deviation?
A: A standard deviation is a measure of the amount of variation or dispersion of a set of values. A z-score, on the other hand, is a measure of how many standard deviations an element is from the mean.
Q: Can a z-score be negative?
A: Yes, a z-score can be negative. A negative z-score indicates that the element is below the mean.
Q: What does a z-score of 0 mean?
A: A z-score of 0 means that the element is equal to the mean.
Q: Can a z-score be greater than 3?
A: Yes, a z-score can be greater than 3. However, this is relatively rare and indicates that the element is significantly above the mean.
Q: How do I interpret a z-score?
A: To interpret a z-score, you need to understand the distribution of the data. A z-score of 1 or 2 indicates that the element is within 1 or 2 standard deviations of the mean, respectively. A z-score of 3 or more indicates that the element is significantly above the mean.
Q: Can I use a z-score to compare data from different distributions?
A: No, a z-score is specific to a particular distribution. You cannot use a z-score to compare data from different distributions.
Q: How do I calculate a z-score for a dataset with a non-normal distribution?
A: If the dataset is not normally distributed, you cannot use the z-score formula. Instead, you can use other statistical methods, such as the Wilcoxon rank-sum test or the Kruskal-Wallis test, to analyze the data.
Q: Can I use a z-score to determine the probability of an event?
A: Yes, you can use a z-score to determine the probability of an event. By looking up the z-score in a standard normal distribution table, you can find the probability of the event occurring.
Conclusion
In conclusion, z-scores are a powerful tool for understanding the distribution of data and identifying outliers. By answering these frequently asked questions, we hope to have provided you with a better understanding of z-scores and how to use them in your analysis.
Additional Resources
If you want to learn more about z-scores and statistical analysis, here are some additional resources:
We hope this article has been helpful in answering your questions about z-scores. If you have any further questions, please don't hesitate to ask.