1. Measuring location: percentiles
One way to describe your location within the distribution of data is to tell what percent of students are less than or equal to your observation.
How to tell:
Count the number of observations equal to or less than your observed value.
Put that value over the total number observations.
Divide
Round two places
Convert to percentage.

2. Z-Score
Z-score or Standardized value tells us how many standard deviations from the mean the original observation falls

3. Z-score

4. Example 1
Quiz scores are as follows:
If John scored a 10 what is his percentile?
If Jane scored an 18, what is her percentile?

5. Percentiles 6 | 7 7 | 2334 Describe Jenny’s
6 | 7
7 | Describe Jenny’s
7 | performance using
8 | percentiles.
8 | 569
9 | 03

6. Measuring location : z-score
Here are the heights for Mrs. Jones’ class. Lynette, who is 65 inches tall, would like to know if she is short or tall compared to the class.
Where does she fall relative to the mean of this distribution?

7. Remember Mean and Standard Deviation go hand and hand.
Standard deviation– the average distance of the observations from the mean.
We can describe Lynette’s location by telling how many standard deviations above or below the mean her height is.
Z-score or Standardizing— converting observations from original values to standard deviation units.

8. Comparing Scores Statistics Test: mean=80 , Std dev =6.07
Chemistry Test: mean=76, Std dev =4
Jenny got an 86 in Statistics and 82 in Chemistry
On which test did she perform better?

9. Exercise
On page 105 # 9

10. Density Curves
1. Always plot your data with usually stemplot or histogram
2. Look for overall pattern (shape, center, spread) and for outliers
3. Choose either 5 number summary or mean and standard deviation to describe.
4. Sometimes overall pattern is so regular we can describe it by a density curve

11. Density Curve

12. Density curves helps us describe the overall pattern of the data.
The total area under the curve is exactly 1

13. The median of a density curve is the equal-areas point, the point that divides the area under the curve in half
The mean of a density curve is the balance point.
The median and mean are the same for a symmetric density curve.

15. Exercise
Starting on page 110 #
3.7 and 3.12 together

16. Normal Density Curves
The normal curve is symmetric, bell- shaped curve that have these properties:
1. Described by mean and standard deviation
2. The mean determines center
3. The standard deviation determines spread of the curve

17. The 68-95-99.7 rule In any normal distribution, approximately
-68% of the observation fall within one standard deviation of the mean
-95% of the observations fall within two standard deviations of the mean
-99.7% of the observations fall within three standard deviations of the mean

18. The 68-95-99.7 Rule for Normal Distribution

19. Example
Jennie scored 600 on the critical reading section of the SAT. Is this a good Score? Given the mean=500 and S=100

21. The notation we use for the mean of a density curve is μ. “mu”
The standard deviation of a density curve as σ. “sigma”
The standardized score or z score fomula:

22. Excercise Starting on page 121 #3.21-3.23 together
# with your
partner

23. The Standard Normal distribution has a mean=0 and standard deviation =1

24. Finding proportions through table
Look in the back of the book for Table A:
Table A only gives you for less than not greater than
Must subtract by 1 to get greater than
Example:
Find were z is less than -1.25

25. Calculator Press 2nd Vars Choose 2:normalcdf(
normalcdf(lower, upper, µ, σ)
Press enter

26. Example 2 with calculator
Find were z is greater than .81
Find were z is between and .81

27. Working backwards We’ve been given z first to find percentile.
What if they give you percentile first to find z-score.
Example:
What is the 80th percentile for a standard normal distribution?

28. Working backwards on calculator
2nd vars
3: invNorm(
invNorm(percentile (in decimal), µ, σ)
What is the 80th percentile for a standard normal distribution?

29. Exercise
On Page 127 #

30. Review for Quiz Starting on page 135 #3.40, 3.42-3.46
Go back to page 106
3.5 and 3.6

31. Review for Test
Starting on page 137
# , ab