To determine the percentile rank of each data value in terms of standard deviations from the mean, we need to look up the corresponding $z$-scores in a standard normal distribution table, often called a $z$-table. Here’s how we interpret each case:

a. A data value 2 standard deviations below the mean

1. Calculation: A data value 2 standard deviations below the mean corresponds to a $z$-score of -2.
2. Percentile: Looking up $z = -2.00$ in the $z$-table, we find that the cumulative area to the left of $z = -2$ is approximately 0.0228 or 2.28%.
3. Interpretation: This means that a data value 2 standard deviations below the mean is at the 2.28th percentile. In other words, 2.28% of the data values are below this value, and approximately 97.72% of data values are above this point.

b. A data value 2.6 standard deviations below the mean

1. Calculation: A data value 2.6 standard deviations below the mean corresponds to a $z$-score of -2.6.
2. Percentile: Looking up $z = -2.60$ in the $z$-table, we find that the cumulative area to the left of $z = -2.6$ is approximately 0.0047 or 0.47%.
3. Interpretation: This means that a data value 2.6 standard deviations below the mean is at the 0.47th percentile. In other words, only 0.47% of data values fall below this point, and 99.53% of data values are above it.

c. A data value 2.8 standard deviations above the mean

1. Calculation: A data value 2.8 standard deviations above the mean corresponds to a $z$-score of 2.8.
2. Percentile: Looking up $z = 2.8$ in the $z$-table, we find that the cumulative area to the left of $z = 2.8$ is approximately 0.9974 or 99.74%.
3. Interpretation: This means that a data value 2.8 standard deviations above the mean is at the 99.74th percentile. In other words, 99.74% of data values are below this value, and only about 0.26% of data values are above it.

Summary

• 2 standard deviations below: 2.28th percentile
• 2.6 standard deviations below: 0.47th percentile
• 2.8 standard deviations above: 99.74th percentile

These percentiles give a measure of how extreme a data point is in relation to the mean. For instance, a data value at the 99.74th percentile is significantly higher than most values, whereas a value at the 0.47th percentile is much lower than most.

Would you like further details or have any questions?

Related Questions:

1. What is the percentile for a data value 1 standard deviation below the mean?
2. How do we interpret a data value at the 50th percentile?
3. What percentile corresponds to a data value 1.5 standard deviations above the mean?
4. How would we calculate the percentile for a data value exactly at the mean?
5. What are the steps to use a $z$-table to find percentiles?

Tip:

To quickly assess the extremity of a data point, remember that approximately 68% of data lies within 1 standard deviation, 95% within 2, and 99.7% within 3 in a normal distribution (the Empirical Rule).