Use the table to find the percentile of the following data values. Explain the meaning of the percentile in each case.

a. A data value 

11 

standard 

deviationdeviation 

belowbelow 

the mean.

b. A data value 

2.82.8 

standard 

deviationsdeviations 

belowbelow 

the mean.

c. A data value 

0.60.6 

standard 

deviationdeviation 

aboveabove 

the mean.

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Part 1

a. The percentile is 

enter your response here. 

Approximately 

enter your response here​% 

of values in a normal distribution are 

▼
 
less than

less than or equal to
 
this data value.

​(Type integers or​ decimals.)

To find the percentiles, we need to use the **z-scores** (standard deviations from the mean) and match them to their corresponding percentiles in a **standard normal distribution**.

For reference:
- A z-score of 0 corresponds to the 50th percentile (the mean of the distribution).
- Negative z-scores indicate values below the mean, and positive z-scores indicate values above the mean.

Let's compute each part:

---

**a. A data value 1 standard deviation below the mean.**

- **Z-score**: \(-1\)
- Using a standard normal distribution table (z-score table), a z-score of \(-1\) corresponds to approximately the **15.87th percentile**.
- This means that approximately **15.87%** of values in a normal distribution are **less than** this data value.

**b. A data value 2.8 standard deviations below the mean.**

- **Z-score**: \(-2.8\)
- Looking up a z-score of \(-2.8\) gives approximately the **0.26th percentile**.
- This means that about **0.26%** of values in a normal distribution are **less than** this data value.

**c. A data value 0.6 standard deviations above the mean.**

- **Z-score**: \(+0.6\)
- From the table, a z-score of \(0.6\) corresponds to approximately the **72.57th percentile**.
- This means that about **72.57%** of values in a normal distribution are **less than** this data value.

---

**Summary of Percentiles:**

a. For a z-score of \(-1\), the percentile is approximately **15.87%**.  
b. For a z-score of \(-2.8\), the percentile is approximately **0.26%**.  
c. For a z-score of \(+0.6\), the percentile is approximately **72.57%**.

Let me know if you'd like more details or if you have any questions!

### Related Questions:
1. What is the percentile for a z-score of 1.5?
2. How does the percentile change as the z-score increases?
3. Can a z-score have a percentile greater than 99%? Explain.
4. What is the meaning of the 50th percentile in a normal distribution?
5. How do you calculate the z-score if you are given the percentile?

**Tip**: Percentiles show the position of a data value within a distribution relative to others, and they are especially useful for comparing different data sets.

Use the table to find the percentile of the following data values. Explain the meaning of the percentile in each case.

 a. A data value 1 standard deviation below the mean.
 b. A data value 2.8 standard deviations below the mean.
 c. A data value 0.6 standard deviation above the mean.

Learn how to calculate the percentiles for data values with Z-scores of -1, -2.8, and 0.6 in a standard normal distribution. This problem covers key statistical concepts like normal distribution and Z-scores. Suitable for students in Grades 10-12.

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