A $z$-score Of +1.6 Represents A Value Which Is How Many Standard Deviations Above The Mean?A. $-1.6$ B. 0.6 C. 1.6 D. $-0.6$

Understanding $z$-scores

In statistics, a $z$-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:

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

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

Interpreting $z$-scores

A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean. This is because the $z$-score is positive, indicating that the value is above the mean.

Calculating the Number of Standard Deviations

To calculate the number of standard deviations, we can use the $z$-score formula and rearrange it to solve for $X$:

$$X = \mu + (z \times \sigma)$$

Substituting the given $z$-score of +1.6, we get:

$$X = \mu + (1.6 \times \sigma)$$

This means that the value is $1.6$ standard deviations above the mean.

Conclusion

A $z$-score of +1.6 represents a value which is 1.6 standard deviations above the mean.

Key Takeaways

• A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean.
• The $z$-score formula is $z = \frac{(X - \mu)}{\sigma}$.
• To calculate the number of standard deviations, we can use the formula $X = \mu + (z \times \sigma)$.

Frequently Asked Questions

• What does a $z$-score of +1.6 mean? A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean.
• How is the $z$-score formula used to calculate the number of standard deviations? The $z$-score formula is rearranged to solve for $X$, which gives us the number of standard deviations.

Example Use Case

Suppose we have a dataset with a mean of 10 and a standard deviation of 2. If we want to find the value that is 1.6 standard deviations above the mean, we can use the $z$-score formula:

$$X = \mu + (z \times \sigma)$$

Substituting the values, we get:

$$X = 10 + (1.6 \times 2)$$

$$X = 10 + 3.2$$

$$X = 13.2$$

This means that the value is 13.2, which is 1.6 standard deviations above the mean.

Conclusion

In conclusion, a $z$-score of +1.6 represents a value which is 1.6 standard deviations above the mean. This can be calculated using the $z$-score formula and rearranging it to solve for $X$.

Understanding $z$-scores

In statistics, a $z$-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:

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

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

Interpreting $z$-scores

A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean. This is because the $z$-score is positive, indicating that the value is above the mean.

Calculating the Number of Standard Deviations

To calculate the number of standard deviations, we can use the $z$-score formula and rearrange it to solve for $X$:

$$X = \mu + (z \times \sigma)$$

Substituting the given $z$-score of +1.6, we get:

$$X = \mu + (1.6 \times \sigma)$$

This means that the value is $1.6$ standard deviations above the mean.

Conclusion

A $z$-score of +1.6 represents a value which is 1.6 standard deviations above the mean.

Key Takeaways

• A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean.
• The $z$-score formula is $z = \frac{(X - \mu)}{\sigma}$.
• To calculate the number of standard deviations, we can use the formula $X = \mu + (z \times \sigma)$.

Frequently Asked Questions

Q: What does a $z$-score of +1.6 mean?

A: A $z$-score of +1.6 means that the value is 1.6 standard deviations above the mean.

Q: How is the $z$-score formula used to calculate the number of standard deviations?

A: The $z$-score formula is rearranged to solve for $X$, which gives us the number of standard deviations.

Q: What is the formula for calculating the $z$-score?

A: The formula for calculating the $z$-score is $z = \frac{(X - \mu)}{\sigma}$.

Q: How do I calculate the number of standard deviations from a given $z$-score?

A: To calculate the number of standard deviations, use the formula $X = \mu + (z \times \sigma)$.

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.

Q: Can I use the $z$-score formula to calculate the mean?

A: No, the $z$-score formula is used to calculate the number of standard deviations from the mean, not the mean itself.

Example Use Cases

Example 1: Calculating the Number of Standard Deviations

Suppose we have a dataset with a mean of 10 and a standard deviation of 2. If we want to find the value that is 1.6 standard deviations above the mean, we can use the $z$-score formula:

$$X = \mu + (z \times \sigma)$$

Substituting the values, we get:

$$X = 10 + (1.6 \times 2)$$

$$X = 10 + 3.2$$

$$X = 13.2$$

This means that the value is 13.2, which is 1.6 standard deviations above the mean.

Example 2: Interpreting $z$-scores

Suppose we have a dataset with a mean of 20 and a standard deviation of 5. If we have a $z$-score of -0.8, we can interpret it as follows:

• The value is 0.8 standard deviations below the mean.
• The value is 20 - (0.8 x 5) = 20 - 4 = 16.

This means that the value is 16, which is 0.8 standard deviations below the mean.

Conclusion

In conclusion, a $z$-score of +1.6 represents a value which is 1.6 standard deviations above the mean. This can be calculated using the $z$-score formula and rearranging it to solve for $X$.