**Example:**

Suppose a teacher has collected all the final exam scores for all statistics classes she has ever taught. This dataset is normally distributed with a mean of 81 and a std dev of 3.5.

Using this information, **estimate the percentage** of students who will get the following scores using the **Empirical Rule** (also called the 95 – 68 – 34 Rule and the 50 – 34 – 14 Rule):

**a) Probability that a score is above 81?**

In this example, the mean of the dataset (the average score) is 81. Therefore, 50% of students are expected to score above this value and 50% below. The answer here is 50%

**b) Probability that a score is below 81?**

In this example, the mean of the dataset (the average score) is 81. Therefore, 50% of students are expected to score above this value and 50% below. The answer here is 50%

**c) ** **Probability that a score is** **between 81 (the mean) and 84.5**?

Here, 81 is the mean, so we know that 50% of the class is below this point. Next, the score of 84.5 is a one standard deviation above the mean. Why? Because each deviation in this question is “3.5” points. So, a score of 84.5 is 81 + 3.5 or one deviation above the mean.

Using the Empirical Rule, we can see that about 34% of scores are BETWEEN the mean and the first deviation. So there is 34% chance that a student will score between 81 and 84.5.

**d) ** **Probability that a score is** **between 81 (the mean) and 74?**

Here, 81 is the mean, so we know that 50% of the class is below this point. Next, the score of 74 is a two standard deviations BELOW the mean. Why? Because each deviation in this question is “3.5” points. So, a score of 74 is 81 – 3.5 – 3.5 = 74 or TWO deviations below the mean.

Using the Empirical Rule, we can see that about 34% + 14% of scores are BETWEEN the mean and the second deviation below it. So there is a 34% + 14% = 48% chance that a student will score between 81 and 74.

**e) ** **Probability that a score is between 74 and 88?**

Here, 74 is two deviation below the mean and 88 is two deviations above the mean. Using the Empirical Rule, we can see that about 14% + 34% + 34% + 14% of scores are BETWEEN 74 and 88 and to there is a **95% chance** that a score will be between 74 and 88.

**f) ** **Probability that a score is****above 88?**

Here, 88 is two deviations above the mean. To score ABOVE 88 there is only a 2.5% chance.

**NOTICE: These examples use the Empirical Rule to Estimate the Probability. However, the z value (also called z score) and z table can be used to get the exact probability for any score.**

**Using the z Table and z Value for Percentages and Probability**