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.

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

Z-score

Example 1 Quiz scores are as follows: 5 7 9 10 11 13 14 15 16 17 18 18 20 21 37 If John scored a 10 what is his percentile? If Jane scored an 18, what is her percentile?

Percentiles 6 | 7 7 | 2334 Describe Jenny’s 6 | 7 7 | 2334 Describe Jenny’s 7 | 5777899 performance using 8 | 00123334 percentiles. 8 | 569 9 | 03

Measuring location : z-score 60 62 62 63 63 63 63 64 64 65 66 66 66 67 68 68 68 69 69 71 72 72 74 75 75 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?

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.

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?

Exercise On page 105 # 3.1-3.6 9

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

Density Curve

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

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.

Exercise Starting on page 110 #3.7-3.12 3.7 and 3.12 together

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

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

The 68-95-99.7 Rule for Normal Distribution

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

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:

Excercise Starting on page 121 #3.21-3.23 together #3.24-3.26 with your partner

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

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

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

Example 2 with calculator Find were z is greater than .81 Find were z is between -1.25 and .81

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?

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

Exercise On Page 127 #3.27-3.32

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

Review for Test Starting on page 137 #3.47-3.51, 3.53-3.55ab