1. Relative Cumulative Frequency Graph
Notes Ogive
Relative Cumulative Frequency Graph

2. I.Relative Cumulative Frequency Graph
Tells relative standing of an individual observation.

3. II. How to construct an ogive:
Step 1:
Decide on class intervals and make a frequency table, just as in making a histogram.
Add three columns to your frequency table:
relative frequency
cumulative frequency
relative cumulative frequency.

4. Step 2: Label and scale your axes and title your graph.
Label the horizontal axis “Age at inauguration”
Label the vertical axis “Relative cumulative frequency.”
Scale the horizontal axis according to your choice of class intervals
Scale the vertical axis from 0% to 100%.

5. Step 3:
Plot a point corresponding to the relative cumulative frequency in each class interval at the left endpoint of the next class interval.
Begin your ogive with a point at a height of 0% at the left endpoint of the lowest class interval.
Connect consecutive points with a line segment to form the ogive. The last point you plot should be at a height of 100%.

6. Sample Ogive

7. How to locate an individual within the distribution:
What about Bill Clinton? He was age 46 when he took office.

8. How to locate a value corresponding to a percentile:
What inauguration age corresponds to the 60th percentile?

9. Example: 1.14
Glucose levels: People with diabetes must monitor and control their blood glucose level. The goal is to maintain “fasting plasma glucose” between about 90 and 130 milligrams per deciliter (mg/dl) of blood. Here are the fasting plasma glucose levels for 18 diabetics enrolled in a diabetes control class, five months after the end of the class.
Construct a relative cumulative frequency graph (ogive) for these data.
Use the graph to answer the following questions:
What percent of blood glucose levels were between 90 and 130?
What is the center of this distribution?
What relative cumulative frequency is associated with a blood glucose level of 130?

10. With your group, complete the activities on the ogive worksheet.
Check for Understanding
Assessment
Prompt 1
With your group, complete the activities on the ogive worksheet.
#1.12, and 1.19

11. Standardized Scores

12. I. Standardizing Scores
Statistics Notes ~ Chapter 6 part 2
I. Standardizing Scores
Standard deviations are used to compare different-looking values.
Find a z-score or standardized value by:
Finding the difference between the observation and the mean.
Divide the difference by the standard deviation.
Measures distance from mean.
Have no units.

13. When standardizing data:
The further the value is from the mean, the more unusual it is considered.
When standardizing data:
Shift data by subtracting mean.
Rescale values by dividing by their standard deviation.

14. II. Measures of Position
Outliers
Quartiles
Percentiles
Z-scores (standard scores)
III. A number that is more than two standard deviations from the mean are very unusual.

15. Example:
The college your sister is considering has a combined SAT scores between 1530 and 1850 for the middle 50% of students accepted. However, your sister took the ACT and needs to compare her score with the SAT scores. If the average combined SAT score is about 1500 with a standard deviation of 250 points and the ACT average is 20.8 with a standard deviation of 4.8, what score does you sister need to be in the top 25% of those applying for this college?

16. THINK Quantitative data What: ACT & SAT scores
Who: students scores (unknown)
Why: to find out ACT score that corresponds to the upper 25% SAT score
Where: USA
When: present
How: official scores from college

17. SHOW

18. TELL
To be in the top 25% of applicants in terms of combined SAT score, my sister will need to have a minimum ACT score of at least

19. POPULATION SAMPLE Parameter Statistic Population mean Sample mean
Population SD
s = Sample SD
N = Population size
n = sample size
Or p = population percentage
= sample percentage
Actual values
Represent actual value
Models
Approximate distributions

20. Transforming Data
Adding a constant to every value adds the same constant to measures of position, but does not change measures of spread.
Multiplying by a constant to every value multiplies the same constant to measures of position and measures of spread.
Standardizing scores changes the center to a mean of zero and the spread to a standard deviation of one.

21. Activity – Transforming Data
Check for Understanding:
Why can’t variability be negative?

22. Density curve review

23. Statistics Notes ~ Chapter 6 part 2
III. Empirical Rule
Rule
Approximately 68% of data is within one standard deviation of the mean.
Approximately 95% of data is within two standard deviations of the mean.
Approximately 99.7% of data is within three standard deviations of the mean.
Another measure of spread
Can use when data is “approximately” normal.

24. IV. Normal Models MAKE A PICTURE Bell shaped Roughly Symmetric
Unimodal
Standard Normal Model
Mean = zero
Standard deviation = one

25. V. Calculator Tips To find the area under the normal curve:
2nd DISTR select normalcdf
Normalcdf(zLeft,zRight)
Normalcdf(left,right,mean,sd)
To find the z-score for a percentile:
2nd DISTR select invNorm(
invNorm(percentile as decimal)

26. The Standard Normal Table
Normal Distributions
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.
Definition: The Standard Normal Table
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.
Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81.
We can use Table A:
P(z < 0.81) =
.7910 Z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212

27. Assessment prompt 1. P(z > 0.68) 2. P(z < -1.38)
3. P(z > k) =
4. P(-1.38 < z < 0.68)
5. P(z < -3.81)
6. P(z < k) =

28. Examples – on notes page
7. Sketch Normal models using the Rule:
Birthweights of babies,
N(7.6 lb, 1.3 lb)
8. Sketch Normal models using the Rule
ACT scores at a certain college, N(21.2, 4.4)

29. 9. Suppose the class took a 40-point quiz
9. Suppose the class took a 40-point quiz. Results show a mean score of 30, median 32, IQR 8, SD 6, min 12 and Q (Suppose you received a 35.) What happens to each of the statistics if…
I decide to weight the quiz as 50 points, and will add 10 points to every score. Your score is now 45.
I decide to weight the quiz as 80 points, and double each score. Your score is now 70.
I decide to count the quiz as 100 points; I’ll double each score and add 20 points. Your score is now 90.

30. 9. Suppose the class took a 40-point quiz
9. Suppose the class took a 40-point quiz. Results show a mean score of 30, median 32, IQR 8, SD 6, min 12 and Q (Suppose you received a 35.) What happens to each of the statistics if…
a) I decide to weight the quiz as 50 points, and will add 10 points to every score. Your score is now 45.
b) I decide to weight the quiz as 80 points, and double each score. Your score is now 70.
c) I decide to count the quiz as 100 points; I’ll double each score and add 20 points. Your score is now 90.

31. 10. Suppose a Normal model describes the fuel efficiency of cars currently registered in your state. The mean is 24 mpg, with a standard deviation of 6 mpg.
a. Sketch the normal model.
b. What percent of all cars get less than 15 mpg?
c. What percent of all cars get between 20 and 30 mpg?
What percent of cars get more than 40 mpg?
Describe the fuel efficiency of the worst 20% of all cars?
What gas mileage represents the third quartile?
Describe the gas mileage of the most efficient 5% of all cars.

32. V. Applications
A patient recently diagnosed with Alzheimer's disease takes a cognitive abilities test. The patient scores a 45 on the test (µ = 52, σ = 5). What is this patient's percentile rank?
Another patient with Parkinson's disease takes the same cognitive abilities test as in question 1 and scores a 54. What percent of individuals would receive a higher score?

33. VI. Normal Probability Plot
If it is a straight diagonal line, then the data is approximately normal.
To get the plot
Type data into STAT EDIT
Turn on STATPLOT
Select graph #6
Select ZOOMSTAT

34. Normal Probability Plots
Most software packages can construct Normal probability plots. These plots are constructed by plotting each observation in a data set against its corresponding percentile’s z-score.
Normal Distributions
Interpreting Normal Probability Plots
If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot.

35. CAUTIONS Think-Pair-Group-Share: Activity 2-2 Assessing Normality
Don’t use a Normal model is the distribution is not unimodal and symmetric
Don’t use the mean and standard deviation when outliers are present
Don’t round off too soon…keep 4 decimal places to be safe.
Don’t round in the middle of a calculation.
Think-Pair-Group-Share:
Activity 2-2 Assessing Normality

36. PAGE 129 Data Exploration Activity
Lesson Task
PAGE 129
Data Exploration Activity

37. Ticket out the door
What is the Rule?