A normal distribution is a bell-shaped curve that is symmetrical about the mean, depicting how values are distributed around a central point. It is often referred to as a Gaussian distribution. The key parameters for a normal distribution are:

• Mean (μ): The center of the distribution. It represents the average value.
• Standard deviation (σ): It measures the spread of values. A larger standard deviation means a wider spread, while a smaller one means a narrower spread.

Properties of the Normal Distribution:

1. Symmetry: The distribution is symmetric around the mean.
2. 68-95-99.7 Rule:
   • 68% of data points fall within 1 standard deviation of the mean.
   • 95% fall within 2 standard deviations.
   • 99.7% fall within 3 standard deviations.
3. The area under the curve: The total area under the normal distribution curve is 1 (or 100%).

Example Calculation:

Suppose you have a normal distribution with:

• Mean (μ) = 100
• Standard deviation (σ) = 15

Now, if you want to find the probability of a value X falling within a specific range (say 85 to 115), you can calculate the Z-scores for 85 and 115:

• Z-score formula: $Z = \frac{X - \mu}{\sigma}$

1. For $X = 85$: $Z = \frac{85 - 100}{15} = \frac{-15}{15} = -1$

2. For $X = 115$: $Z = \frac{115 - 100}{15} = \frac{15}{15} = 1$

Next, you would use standard normal distribution tables or software to find the probability between these Z-scores.

• The probability for $Z = -1$ is approximately 0.1587, and for $Z = 1$ is approximately 0.8413.
• Thus, the probability of a value between 85 and 115 is the difference between these two: $P(85 \leq X \leq 115) = 0.8413 - 0.1587 = 0.6826$ So, there is a 68.26% chance that a value will fall between 85 and 115.

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If you have specific values you want to explore within a normal distribution, or a specific problem, feel free to share it! Would you like me to help with that?

Additional questions you might explore:

1. How do you calculate the probability for values outside of a given range?
2. What does the Z-score represent in real-world terms?
3. How does a normal distribution relate to sampling and the Central Limit Theorem?
4. How do you use normal distribution to find percentiles or quantiles?
5. What is the relationship between normal distribution and standard deviation in practical data analysis?

Tip: When calculating probabilities with a normal distribution, always remember to check if the problem is asking for values to the left, right, or between two points, as this will affect which areas under the curve you need to calculate.