1. Chapter 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution
2.2 Normal Distributions

2. Section 2.1 Describing Location in a Distribution
Learning Objectives
After this section, you should be able to…
MEASURE position using percentiles
INTERPRET cumulative relative frequency graphs
MEASURE position using z-scores
TRANSFORM data
DEFINE and DESCRIBE density curves

3. 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.
Definition:
The pth percentile of a distribution is the value with p percent of the observations less than it.
6 7
9 03
Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class?
Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is at the 84th percentile in the class’s test score distribution.
6 7
9 03

4. TIPs!!!!!!
Percentiles should be whole numbers.
If you get a decimal, round the answer to the nearest integer
If you get , assume 92nd percentile.
If two observations have the same value, they will be at the same percentile. To find the percentile, calculate the percent of the values in the distribution that are below both values
Unfortunately , there is no universally agreed upon definition of “percentiles”
On the AP exam, students will be allowed to use any reasonable definition of “percentile” on the free response section.

5. Your Turn: Wins in Major League Baseball
The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009.
Find the percentiles for the following teams:
(a) The Colorado Rockies, who won 92 games.
(b) The New York Yankees, who won 103 games.
(c) The Kansas City Royals and Cleveland Indians, who both won 65 games.

6. Solution:
(a) The Colorado Rockies, who won 92 games, are at the 80th percentile, since 24/30 teams had fewer wins than they did.
(b) The New York Yankees, who won 103 games, are at about the 97th percentile since 29/30 teams had fewer wins than they did.
(c) The two teams with 65 wins, the Kansas City Royals and the Cleveland Indians, are at the 10th percentile since only 3/30 teams had fewer wins than they did.

7. Cumulative Relative Frequency Graphs
A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each class of a frequency distribution.
Age of First 44 Presidents When They Were Inaugurated
Age
Frequency
Relative frequency
Cumulative frequency
Cumulative relative frequency
40-44 2
2/44 = 4.5%
2/44 =
4.5%
45-49 7
7/44 = 15.9% 9
9/44 = 20.5%
50-54 13
13/44 = 29.5% 22
22/44 = 50.0%
55-59 12
12/44 = 34% 34
34/44 = 77.3%
60-64 41
41/44 = 93.2%
65-69 3
3/44 = 6.8% 44
44/44 = 100%

8. Interpreting Cumulative Relative Frequency Graphs
Use the graph from page 88 to answer the following questions.
Was Barack Obama, who was inaugurated at age 47, unusually young?
Estimate and interpret the 65th percentile of the distribution 65 11 58 47

9. Your turn: State Median Household Incomes Here is a table showing the distribution of median household incomes for the 50 states and the District of Columbia.
Median
Income
($1000s)
Frequency
Relative
Cumulative
35 to < 40 1
40 to < 45 10
45 to < 50 14
50 to < 55 12
55 to < 60 5
60 to < 65 6
65 to < 70 3

10. Median
Income
($1000s)
Frequency
Relative
Cumulative
35 to < 40 1
1/51 = 0.020
40 to < 45 10
10/51 = 0.196 11
11/51 = 0.216
45 to < 50 14
14/51 = 0.275 25
25/51 = 0.490
50 to < 55 12
12/51 = 0.236 37
37/51 = 0.725
55 to < 60 5
5/51 = 0.098 42
42/51 = 0.824
60 to < 65 6
6/51 = 0.118 48
48/51 = 0.941
65 to < 70 3
3/51 = 0.059 51
51/51 = 1.000

11. Make a Cumulative Frequency Graph: OGIVE
What does the point ( 50, 0.49) and
( 50,0.725) mean?
The point at (50,0.49) means 49% of
the states had median household
incomes less than $50,000.
The point at (55, 0.725) means that 72.5% of the states had median household incomes less than $55,000.
Thus, 72.5% - 49% = 23.5% of the states had median household incomes between $50,000 and $55,000 since the cumulative relative frequency increased by
Due to rounding error, this value is slightly different than the relative frequency for the 50 to <55 category.

12. Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.
Definition:
If x is an observation from a distribution that has known mean and standard deviation, the standardized value of x is:
A standardized value is often called a z-score.

13. Measuring Position: z-Scores
Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is What is her standardized score?

14. We can see that Jenny’s score of 86 is “above average” but how far above is it?
Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is What is her standardized score?
This means that Jenny’s score was almost 1 full standard deviation above the mean.

15. Calculating z-scores 6 | 7 7 | 2334 7 | 5777899 8 | 00123334 8 | 56979 81 80 77 73 83 74 93 78 75 67 86 90 85 89 84 82 72
Jenny: z = (86-80)/6.07
z = {above average = +z}
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Kevin: z = (72-80)/6.07
z = {below average = -z}
Kim: z = (80-80)/6.07
z = {average z = 0}

16. Using z-scores for Comparison
Standardized values can be used to compare scores from two different distributions.
Jenny’s 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?
Although she had a lower score, she performed relatively better in Chemistry.

17. Percentiles
Another measure of relative standing is a percentile rank.
pth percentile: Value with p% of observations below it.
median = 50th percentile {mean=50th %ile if symmetric}
Q1 = 25th percentile
Q3 = 75th percentile
6 | 7
7 | 2334
7 |
8 |
8 | 569
9 | 03
Jenny got an 86.
22 of the 25 scores are ≤ 86.
Jenny is in the 22/25 = 88th %ile.

18. Transforming Data
Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution.
Effect of Adding (or Subracting) a Constant
Adding the same number x (either positive, zero, or negative) to each observation:
adds x to measures of center and location (mean, median, quartiles, percentiles), but
Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation).
Example, p. 93 n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Guess(m) 44
16.02
7.14 8 11 15 17 40 6 32
Error (m)
3.02 -5 -2 2 4 27

19. Transforming Data Effect of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each observation by the same number b (positive, negative, or zero):
multiplies (divides) measures of center and location by b
multiplies (divides) measures of spread by |b|, but
does not change the shape of the distribution n
Mean sx
Min Q1 M Q3
Max
IQR
Range
Error(ft) 44
9.91
23.43
-16.4
-6.56
6.56
13.12
88.56
19.68
104.96
Error (m)
3.02
7.14 -5 -2 2 4 27 6 32
Example, p. 95

20. Density Curves
In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.
Exploring Quantitative Data
Always plot your data: make a graph.
Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers.
Calculate a numerical summary to briefly describe center and spread.
4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

21. Density Curve Definition: A density curve is a curve that
is always on or above the horizontal axis, and
has area underneath is exactly 1 it. ( 100% of observations)
A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.
The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

22. Density Curves
Density Curves come in many different shapes; symmetric, skewed, uniform, etc.
The area of a region of a density curve represents the % of observations that fall in that region.
The median of a density curve cuts the area in half.
The mean of a density curve is its “balance point.”

23. Area under the curve

24. Describing Density Curves
Our measures of center and spread apply to density curves as well as to actual sets of observations.
Distinguishing the Median and Mean of a Density Curve
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, at which the curve would balance if made of solid material.
The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

25. Section 2.1 Describing Location in a Distribution
Summary
In this section, we learned that…
There are two ways of describing an individual’s location within a distribution – the percentile and z-score.
A cumulative relative frequency graph allows us to examine location within a distribution.
It is common to transform data, especially when changing units of measurement. Transforming data can affect the shape, center, and spread of a distribution.
We can sometimes describe the overall pattern of a distribution by a density curve (an idealized description of a distribution that smooths out the irregularities in the actual data).

26. Lets Practice:
# 1, 4, 6,
9,10, 14,
15,16, 27

29. 2.4:
Larry’s wife should gently break the news that being in the 90th percentile is not good news in this situation. About 90% of men similar to Larry have lower blood pressures. The doctor was suggesting that Larry take action to lower his blood pressure.

30. 2.6. Peter’s time was slower than 80% of his previous race times that season, but it was slower than only 50% of the racers at the league championship meet.

32. DO NOW: In 2010, Taxi Cabs in New York City charged an initial fee of $2.50 plus $2 per mile. In equation form, fare = (miles). At the end of a month a businessman collects all of his taxi cab receipts and calculates some numerical summaries. The mean fare he paid was $15.45 with a standard deviation of $ What are the mean and standard deviation of the lengths of his cab rides in miles? Answer:

33. Looking Ahead… In the next Section…
We’ll learn about one particularly important class of density curves – the Normal Distributions
We’ll learn
The Rule
The Standard Normal Distribution
Normal Distribution Calculations, and
Assessing Normality
In the next Section…

34. Chapter 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution
2.2 Normal Distributions

35. Section 2.2 Normal Distributions
Learning Objectives
After this section, you should be able to…
DESCRIBE and APPLY the Rule
DESCRIBE the standard Normal Distribution
PERFORM Normal distribution calculations
ASSESS Normality

36. Because density curves are idealized descriptions, we need to distinguish between the mean and standard deviation of the density curve ( sample) vs. the mean and standard deviation from the actual observations. ( Population)
Ideal
Actual
Parameters
(Population) s
Statistics
( Sample)
mean
st. dev

37. Normal Distributions Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
All Normal curves are symmetric, single-peaked, and bell- shaped
A Specific Normal curve is described by giving its mean µ and standard deviation σ.
Normal Distributions
Two Normal curves, showing the mean µ and standard deviation σ.

38. Normal Distributions
Normal Distributions
Definition:
A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ.
The mean of a Normal distribution is the center of the symmetric Normal curve.
The standard deviation is the distance from the center to the change-of-curvature points on either side.
We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

39. Why Normal Distributions?
1) Normal distributions describe many sets of real data. For example...
Scores on tests taken by many people (SAT's, psychological tests).
Repeated careful measurements of the same quantity.
Characteristics of biological populations (such as yields of corn and lengths of animal pregnancies). 39

40. Why Normal Distributions?
2) Normal distributions are good approximations to the results of many kinds of chance outcomes, such as tossing a coin many times.
3) Third, and most important, we will see that many statistical inference procedures based on Normal distributions work well for other roughly symmetric distributions. 40

41. Be aware...
Even though many sets of data follow a Normal distribution, many do not.
Income distributions are skewed right.
Some symmetric distributions are NOT Normal.
Don't assume a distribution is Normal just because it looks like it. 41

42. The Rule
Definition: The Rule (“The Empirical Rule”)
In the Normal distribution with mean µ and standard deviation σ:
Approximately 68% of the observations fall within σ of µ.
Approximately 95% of the observations fall within 2σ of µ.
Approximately 99.7% of the observations fall within 3σ of µ.

43. The Rule

44. Suppose that the distribution with = 0.261 , = 0.034
Example: The batting averages for Major League Baseball players in 2009, the mean of the 432 batting averages was with a standard deviation of
Suppose that the distribution with
= , = 0.034
(a) Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean.
(b) What percent of the batting averages are above 0.329? Show your work.
(c) What percent of the batting averages are between and .295? Show your work.

46. (b) Here is a curve showing the proportion of batting averages above 0
(b) Here is a curve showing the proportion of batting averages above Since is exactly two standard deviations above the mean, we know that about 95% of batting averages will be between and Since the curve is symmetric, half of the remaining 5%, or 2.5% should be above

47. (c) Here is a curve showing the proportion of batting averages between and We know that about 68% of batting averages are between and Also, we know that half of the difference between 95% and 68% should be between and Therefore, about 13.5% + 68% = 81.5% of batting averages will be between and

48. The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean µ as center.
Normal Distributions
Definition:
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable
has the standard Normal distribution, N(0,1).

49. Recall an area under a density curve is a proportion of the observations in a distribution.
Because all Normal distributions are the same once we standardize, you can now find percentages under Normal curves without Calculus, using the standard Normal table.

51. 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

52. Be aware...
A common mistake is to look up a z-value in Table A and report the entry corresponding to that z-value, regardless of whether the problem asks for the area to the left or to the right of that z-value.
Always sketch the standard Normal curve, mark the z- value, shade the area of interest, and make sure your answer is reasonable in the context of the problem.

53. Normal Distributions Finding Areas Under the Standard Normal Curve
Example, p. 117
Normal Distributions
Find the proportion of observations from the standard Normal distribution that are between and 0.81.
Can you find the same proportion using a different approach?
1 - ( ) = 1 – =

54. Using the Standard Normal Table
Find the proportion between and 2.11
(Area left of 2.11) – (Area right of -1.23)
– =

55. Cholesterol in Young Boys
For 14-year-old boys, the mean is µ=170 milligrams of cholesterol per deciliter of blood (mg/dl) and the standard deviation is σ = 30 mg/dl.
Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol?
Draw a picture Use the table

56. Cholesterol in Young Boys
What percent of 14-year-old boys have blood cholesterol between 170 and 240 mg/dl
Area between 0 and 2.33 =
area below 2.33-area below 0.00
= = .4901
Calculator :
normalcdf ( 170, 240 , 170, 30) =
lb ub mean sd

57. Normal Distribution Calculations
Normal Distributions
How to Solve Problems Involving Normal Distributions
State: Express the problem in terms of the observed variable x.
Plan: Draw a picture of the distribution and shade the area of interest under the curve.
Do: Perform calculations.
Standardize x to restate the problem in terms of a standard Normal variable z.
Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve.
Conclude: Write your conclusion in the context of the problem.

58. Normal Distribution Calculations
Normal Distributions
When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?
Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13.
– = About 44% of Tiger’s drives travel between 305 and 325 yards.

59. FINDING A VALUE WHEN GIVEN A PROPORTION
To find the observed value that corresponds to a given percentile
Rule :
Use Table A backwards.
Find the given proportion in the body of the table and read the corresponding z value from the left column and top row.

60. Cholesterol in boys problem Mean = 170 mg/dl , Sd = 30 mg/dl
Find the First Quartile of the distribution of blood cholesterol.
Answer: First quartile means area from the left is 0.25.
Look in the body of Table A for number closest to 0.25.
It is The z value for this is z = ( which is standardized).
To unstandardize: It helps to find the x value.
Solve for x  x =
The 1st Quartile of blood cholesterol is about 150 mg/dl.

61. Calculator: To get area from z value: 2nd Distribution ( Vars) normalcdf ( lower bound, upper bound, mean, Sd) To get z value from area (unstandardize) invNorm ( proportion, mean, sd)

62. IQ Scores
Based on N(100,16) what % of people’s IQ scores would you expect to be:
a) over 80?
b) under 90?
c) between 112 and 132?
d) What IQ represents the 15th percentile
e) What IQ represents the 98th percentile
f) What is the IQR of the IQ’s

63. AP EXAM COMMON ERROR You will not get full credit if you use calculator speak. Look at AP Exam tip P Label each number and also draw the sketch.

64. Assessing Normality Normal Distributions
The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. Consequently, we need a strategy for assessing Normality.
Normal Distributions
Plot the data.
Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped.
Check whether the data follow the rule.
Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution.

65. 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.

66. Do Check your understanding: P- 119.

67. P- 119

68. P- 119

69. P- 119

70. P- 119

71. How to use calculator for normality plot.
Make list L1 ready
Then do 2nd Stat plot: Choose : type 6th one.

72. Do: Page : 133 # 63, 66, 69-74( all)

73. Section 2.2 Normal Distributions
Summary
In this section, we learned that…
The Normal Distributions are described by a special family of bell- shaped, symmetric density curves called Normal curves. The mean µ and standard deviation σ completely specify a Normal distribution N(µ,σ). The mean is the center of the curve, and σ is the distance from µ to the change-of-curvature points on either side.
All Normal distributions obey the Rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.

74. Section 2.2 Normal Distributions
Summary
In this section, we learned that…
All Normal distributions are the same when measurements are standardized. The standard Normal distribution has mean µ=0 and standard deviation σ=1.
Table A gives percentiles for the standard Normal curve. By standardizing, we can use Table A to determine the percentile for a given z-score or the z-score corresponding to a given percentile in any Normal distribution.
To assess Normality for a given set of data, we first observe its shape. We then check how well the data fits the rule. We can also construct and interpret a Normal probability plot.