for a normal distribution with a mean of 49 and a standard deviation of 7, is a value of 28 rare or expected?

In a normal distribution with a **mean of 49** and a **standard deviation of 7**, we can determine whether a value of **28** is rare or expected by calculating the **z-score**. The z-score helps measure how many standard deviations a value is from the mean. The formula for the z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x = 28 \) (the value you're interested in)
- \( \mu = 49 \) (the mean)
- \( \sigma = 7 \) (the standard deviation)

Substituting the values into the formula:

\[
z = \frac{28 - 49}{7} = \frac{-21}{7} = -3
\]

A z-score of **-3** means that 28 is 3 standard deviations below the mean.

### Interpretation:
In a normal distribution:
- About **68%** of the values lie within 1 standard deviation of the mean.
- About **95%** of the values lie within 2 standard deviations.
- About **99.7%** of the values lie within 3 standard deviations.

Since a z-score of **-3** is quite far from the mean and lies on the boundary of the 99.7% interval, a value of **28** is **rare**, occurring in only about 0.15% of cases on the lower tail of the distribution.

Would you like more details on this? 

Here are 5 related questions to explore:
1. How do you calculate probabilities from z-scores?
2. What is the empirical rule in a normal distribution?
3. What is the significance of values that are multiple standard deviations from the mean?
4. How do we interpret the rarity of extreme z-scores?
5. How does the standard deviation influence the spread of data in a normal distribution?

**Tip**: When z-scores exceed ±2, the values become relatively rare in a normal distribution.

For a normal distribution with a mean of 49 and a standard deviation of 7, is a value of 28 rare or expected?

Explore the rarity of a value of 28 in a normal distribution with a mean of 49 and a standard deviation of 7. Learn about z-scores, standard deviations, and the empirical rule in this detailed statistical breakdown. Suitable for students in Grades 10-12.

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