Determine the following percentiles for the standard normal distribution:
a. 91st
b. 9th
c. 22nd
d. 99.9th
The Correct Answer and Explanation is:
Here are the requested percentiles for the standard normal distribution:
Answers:
a. 91st percentile ≈ 1.34
b. 9th percentile ≈ -1.34
c. 22nd percentile ≈ -0.77
d. 99.9th percentile ≈ 3.09
Explanation
The standard normal distribution is a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. It is symmetric around the mean and is used extensively in statistics to standardize values and determine probabilities using z-scores.
A percentile represents the value below which a certain percentage of data falls. For example, the 91st percentile is the value (z-score) below which 91% of the values in the standard normal distribution lie.
To determine percentiles for the standard normal distribution, we use the inverse cumulative distribution function (also called the quantile function). This can be found in Z-tables, calculated using statistical software, or determined using a calculator with statistical functions (like a TI-84 or online tools).
Here’s how the provided values are interpreted:
- a. 91st percentile (z ≈ 1.34): 91% of the values lie below z = 1.34. This is on the right side of the mean.
- b. 9th percentile (z ≈ -1.34): 9% of the values lie below z = -1.34. This is symmetrical to the 91st percentile but on the left side.
- c. 22nd percentile (z ≈ -0.77): 22% of the values lie below z = -0.77. This also lies to the left of the mean but closer than -1.34.
- d. 99.9th percentile (z ≈ 3.09): 99.9% of values lie below z = 3.09. This is very far to the right tail, indicating a rare, extreme value.
Understanding percentiles in the standard normal distribution is fundamental in many statistical applications, including hypothesis testing, grading systems, and quality control. These values allow analysts to make judgments about how typical or atypical a value is within a distribution.
