The mean of a set of normally distributed data is 600 with a standard deviation of 20.  What percent of the data is between 580 and 620?

To find the percentage of data between 580 and 620 in a normally distributed dataset with a mean of 600 and a standard deviation of 20, we can use the properties of the normal distribution.

1. **Calculate the Z-scores**:
   - The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
   - For \( X = 580 \):  
     \( Z = \frac{580 - 600}{20} = \frac{-20}{20} = -1 \)
   - For \( X = 620 \):  
     \( Z = \frac{620 - 600}{20} = \frac{20}{20} = 1 \)

2. **Find the percentage corresponding to these Z-scores**:
   - A Z-score of -1 corresponds to the 15.87th percentile of the data.
   - A Z-score of 1 corresponds to the 84.13th percentile of the data.

3. **Calculate the percentage between Z = -1 and Z = 1**:
   - The percentage of data between these two Z-scores is \( 84.13\% - 15.87\% = 68.26\% \).

So, **68.26% of the data** lies between 580 and 620.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How much data falls between 560 and 640 for this distribution?
2. What percentage of data is less than 580?
3. What is the probability of a value greater than 620 occurring in this distribution?
4. How would the percentage change if the standard deviation were 10 instead of 20?
5. How do we interpret Z-scores in a real-world context?

**Tip:** Z-scores help standardize data, making it easier to compare across different datasets or distributions.

Learn how to calculate the percentage of data between 580 and 620 in a normally distributed dataset with mean 600 and standard deviation 20. Use Z-scores to determine the percentile of data within this range. Ideal for students and professionals exploring statistics and probability.

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