Calculating Percentiles in Normal Distributions

Understanding Percentiles and Their Calculation

Introduction to Percentiles

• Definition: A percentile indicates the value below which a given percentage of observations in a group of observations falls. It typically describes how many people or values are smaller or less than a specific value, not how many are greater.

  • For example, if you are in the $80^{th}$ percentile, $80\%$ of people are shorter than you.

  • While one could say you are in the top $20\%$ (for the $80^{th}$ percentile), the standard definition of percentile focuses on the proportion of values less than yours.

• Relation to Probability: Percentiles are closely related to probability, as they represent the likelihood of a value falling below a certain point in a distribution.

Percentiles for Discrete Data

• Easier to Calculate: Finding percentiles is simpler for discrete data, which can be clearly categorized.

  • Example: Commute time to work in hours ($2$ hours, $3$ hours, $4$ hours, $5$ hours).

• Method: A frequency table can be used.

  • P

    • Frequency Count: The number of people in each group.

    • Relative Frequency: The percentage of people in each group.

    • Cumulative Relative Frequency: The sum of relative frequencies as you move down the table, representing the percentage of people at or below a certain category.

• Imperfect Example: For a commute time of $3.5$ hours, if $65\%$ of people take $3$ hours or less, one would be at least at the $65^{th}$ percentile. This illustrates that for ordered (ordinal) or clearly discrete datasets, counting values smaller than yours is straightforward.

Percentiles for Continuous Data

• More Complex: Calculating percentiles becomes more complicated with continuous values (e.g., exact time values, exam scores).

  • Example: "How high does a score need to be to be in the $90^{th}$ percentile?"

• Rephrasing the Question: This type of question can be rephrased in terms of probability for a standard normal distribution:

  • "At what value of $y$ is the statement P(Z < y) = 0.90 true?"

  • This means finding the Z-score ($y$) such that $90\%$ of the area under the standard normal curve (probability) is to the left of that Z-score.

• Tools to Answer: We can use established tools:

  • Standardization: Transforming any normal distribution into a standard normal distribution using the Z-score formula: $Z = \frac{(X - \mu)}{\sigma}$, where $Z$ is the Z-score, $X$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation.

  • Probability Tables: Using standard normal (Z-score) tables to find probabilities associated with Z-scores.

• Reverse Process: Instead of finding probability from a Z-score, we reverse the process:

  1. Start with the desired probability (percentile).

  2. Find the corresponding Z-score from the standard normal table.

  3. Use the Z-score equation to solve for the raw score ($X$).

Example 1: SAT Verbal Score at the $90^{th}$ Percentile

• Scenario: What is the SAT verbal score at the $90^{th}$ percentile?

  • Mean $\mu = 500$

  • Standard Deviation $\sigma = 100$

  • This is visually represented as finding the score ($X$) or Z-value ($Z$) where $90\%$ of the area under the curve is less than that value.

• Step 1: Find the Z-score for the $90^{th}$ percentile.

  • Locate the probability $\approx 0.9000$ in the middle of the standard normal table.

  • Table Reading: The values in the middle of the table are probabilities. The first column and first row represent the Z-score components.

  • Finding $0.9000$ (or the closest values, e.g., $0.8997$ or $0.9015$) corresponds to a Z-score derived from combining values from the left column (e.g., $1.2$) and the top row (e.g., $0.09$).

  • Result: The corresponding Z-score is $Z = 1.29$. This means $90\%$ of the probability is less than a Z-score of $1.29$.

• Step 2: Plug the Z-score into the Z-score equation and solve for $X$.

  • Equation: $Z = \frac{(X - \mu)}{\sigma}$

  • Substitute known values: $1.29 = \frac{(X - 500)}{100}$

  • Rearrange and solve for $X$ (algebra):

    • $1.29 \times 100 = X - 500$

    • $129 = X - 500$

    • $X = 129 + 500$

    • $X = 629$

  • Conclusion: An SAT verbal score of $629$ is at the $90^{th}$ percentile (higher than $90\%$ of other scores).

Steps to Finding Percentiles (Summary)

1. Use the percentile (probability) to find the corresponding Z-score on your standard normal table.

2. Plug that Z-score, the mean ($\mu$), and the standard deviation ($\sigma$) into your Z-score equation: $Z = \frac{(X - \mu)}{\sigma}$.

3. Rearrange the equation to solve for the actual value ($X$) that represents that percentile.

Example 2: SAT Verbal Score at the $20^{th}$ Percentile

• Scenario: What is the SAT verbal score at the $20^{th}$ percentile?

  • Mean $\mu = 500$

  • Standard Deviation $\sigma = 100$

• Step 1: Find the Z-score for the $20^{th}$ percentile.

  • Since $20\%$ is less than $50\%$ (the mean/center), we will be looking at the negative side of the Z-score table.

  • Locate the probability $\approx 0.2000$ in the middle of the negative Z-score table.

  • Result: The corresponding Z-score is $Z = -0.84$. This indicates that $20\%$ of the information is less than this Z-score, and $80\%$ is greater.

• Step 2: Plug the Z-score into the Z-score equation and solve for $X$.

  • Equation: $Z = \frac{(X - \mu)}{\sigma}$

  • Substitute known values: $-0.84 = \frac{(X - 500)}{100}$

  • Rearrange and solve for $X$ (algebra):

    • $-0.84 \times 100 = X - 500$

    • $-84 = X - 500$

    • $X = -84 + 500$

    • $X = 416$

  • Conclusion: An SAT verbal score of $416$ is at the $20^{th}$ percentile.

Tips and Tricks for Z-scores and Percentiles (Sanity Checks)

• Normal Distributions Are Symmetrical: This is a convenient property.

  • The negative Z-score for the $P^{th}$ percentile will be the same magnitude as the positive Z-score for the $(100 - P)^{th}$ percentile.

  • Example: The Z-score for the $20^{th}$ percentile ($-0.84$) is the negative equivalent of the Z-score for the $80^{th}$ percentile ($+0.84$). This means the area below $-0.84$ ($20\%$) is equal to the area above $+0.84$ ($20\%$).

  • You can use this to work with just the positive or negative Z-table if preferred, by finding the 'mirror' percentile.

• Mean as Center (Median, Mode):

  • A Z-score of $0$ is at the center of the graph, representing the mean and the $50^{th}$ percentile.

  • If you're finding a percentile less than $50^{th}$ (e.g., $20^{th}$ percentile), your calculated raw score ($X$) must be less than the mean ($\mu$), and your Z-score must be negative.

  • If you're finding a percentile greater than $50^{th}$ (e.g., $90^{th}$ percentile), your calculated raw score ($X$) must be greater than the mean ($\mu$), and your Z-score must be positive.

  • These checks help ensure you are looking in the correct general region of the distribution.

Outliers on a Normal Distribution

• Definition: An outlier is an extreme value that is highly unlikely or atypical of the distribution, though it could still be a real observation.

• Z-score Table Limits: Standard normal tables typically only extend to a certain range (e.g., Z-scores from $-3.99$ to $+3.99$).

• Extreme Cases: A Z-score greater than $+4$ or less than $-4$ is possible but extremely rare. Absence from the table does not mean these values don't exist; it simply means their probability of occurrence is very, very low. This is important to remember for future assignments where such extreme Z-scores might appear.