We are given the following information about the distribution of math scores on a standardized test:

• The scores follow a bell-shaped (normal) distribution.
• The mean ($\mu$) is 525.
• The standard deviation ($\sigma$) is 110.

We are asked to determine the percentage of test scores that are less than 415 or greater than 635.

Step 1: Convert the raw scores into z-scores.

A z-score tells us how many standard deviations a raw score is from the mean. The formula for the z-score is:

$z = \frac{x - \mu}{\sigma}$

Where:

• $x$ is the raw score,
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

For the score 415:

$z_{415} = \frac{415 - 525}{110} = \frac{-110}{110} = -1$

For the score 635:

$z_{635} = \frac{635 - 525}{110} = \frac{110}{110} = 1$

So, the z-scores for 415 and 635 are -1 and +1, respectively.

Step 2: Find the cumulative probabilities corresponding to these z-scores.

Using the standard normal distribution table or a calculator:

• The cumulative probability for $z = -1$ is approximately 0.1587.
• The cumulative probability for $z = 1$ is approximately 0.8413.

This means that:

• About 15.87% of the scores are below 415 (because $z = -1$).
• About 84.13% of the scores are below 635 (because $z = 1$).

Step 3: Calculate the percentage of scores less than 415 or greater than 635.

To find the percentage of scores less than 415 or greater than 635, we combine the probabilities:

• The percentage less than 415 is 15.87%.
• The percentage greater than 635 is the complement of the percentage less than 635, so: $100\% - 84.13\% = 15.87\%$

Now, add these two percentages together:

$15.87\% + 15.87\% = 31.74\%$

Final Answer:

The percentage of standardized test scores that is less than 415 or greater than 635 is approximately 31.74%.

Would you like further details or clarifications?

Here are some related questions you might find interesting:

1. What is the percentage of test scores between 415 and 635?
2. How can we calculate the probability of a score falling within two standard deviations of the mean?
3. What does a z-score of 0 represent in a normal distribution?
4. How do you find the probability of a score being between two given values?
5. How do you interpret the standard deviation in the context of a normal distribution?

Tip: For quick calculations, most statistical calculators and software can instantly provide cumulative probabilities for z-scores, making these steps faster!