1. Activity: Where do I stand?
What percent of students have heights less than yours? (Percentile)
Calculate the mean and standard deviation for the class.
Is your height above or below the mean? How far?
How many standard deviations above or below the mean is your height? (z-score)

2. CHAPTER 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution

3. Describing Location in a Distribution
FIND and INTERPRET the percentile of an individual value within a distribution of data.
FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

4. Measuring Position: Percentiles
One way to describe the location of a value in a distribution is to tell what percent of observations are less than it.
The pth percentile of a distribution is the value with p percent of the observations less than it.
6 7
9 03
Harry earned a score of 86 on his charms test. How did he perform relative to the rest of the class (25 students total)?
Example
His score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below his, Harry is at the 84th percentile in the class’s test score distribution.
6 7
9 03

5. Measuring Position: Percentiles
Use the scores below to find the percentiles for the following students:
Neville, who earned a 72.
Hermione, who scored a 93.
Ron and Draco, who earned scores of 80.
6 7
9 03
Practice
4th percentile
96th percentile
48th percentile
6 7
9 03

6. Some places percentiles are used ACT/SAT (and other standardized testing) Weight/Height Measurements Income

8. Measuring Position: z-Scores
Let’s revisit Harry’s test score from the earlier example.
Harry earned a score of 86 on his test. The mean score for the class was an 80. How much above the average was his score?
6 7
9 03
Harry is 6 points above the average, but does that tell us very much about how well he did?
Example
Depending on the spread of the distribution, Harry might be barely above the average or really far above the average. A measure of spread (like standard deviation) can help us better determine how far above the mean he really is.
6 7
9 03

9. Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
A z-score is found from a raw score, x, as follows:
A z-score is also known as a standardized score.
A z-score does not have units as it simply tells us how many standard deviations away from the mean an observation is.
Z-scores can be used to compare observations from different distributions.

10. Measuring Position: z-Scores Example #1
Harry earned a score of 86 on his charms test. The class mean is 80 and the standard deviation is What is his standardized score?
What does Harry’s z-score tell us?

11. Measuring Position: z-Scores Example #2
Remember that Hermione scored a 93 on her test, and Neville scored a 72. The class mean is 80 and the standard deviation is Find their z-scores.
Hermione: z = 2.14
Neville: z = -1.32
What do these scores tell us?

12. Using z-scores for comparisons
After earning an 86 on his charms test, Harry was disappointed that he earned an 82 on his potions test the next day. Professor Snape informed the class that the mean score was 76 with a standard deviation of 4. (Snape secretly wishes Hogwarts offered a statistics course so he could teach it.)
How did Harry’s score on his potions test compare to the rest of the class?
Relatively, on which test did Harry score better?

14. Work with a partner to answer these questions (pg. 91).
1) Z =
2) Z = 1.63
3) 2.35 inches

15. Describing Location in a Distribution
FIND and INTERPRET the percentile of an individual value within a distribution of data.
FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

16. Sec. 2.1 Assignment
Pg. 99 #1,3,5,11,13,15,25,27,31,32