Download 2.1 Measures of Relative Standing and Density Curves

AP Stats
Chapter 2
2.1 Measures of Relative Standing and Density Curves
Here are the scores of all 25 students in Mr. Jones’ Algebra 2 class on their first test:
6
7
7
8
8
9
7
2234
5777899
00123334
569
03
Fred scored the 84.
How did Fred do compared to everyone else in the class?
Find mean, standard deviation, and 5 number summary for the data.
Where does Fred’s score fall relative to the center of the distribution?
Percentiles
𝑝th percentile: 𝑝 percent < the particular data value
Fred has the 20th highest score in the class, so 19 out of 25 scores are below his score. Fred scored in the
80th percentile. (19 ÷ 25 = .76 which is 76th percentile)
What percentile do Barney and Bam Bam fall into?
Barney scored 90, so 23 out of 25. 92nd percentile
Bam Bam scored 67, so he defaults to 1st percentile
What about the three people that scored 77? 6 out of 25 so 24th percentile
Cumulative Relative Frequency Graphs
Pg. 86-88 Take a few minutes and read these pages and examples. Talk through the check your
understanding on page 89 with a buddy.
z-scores
We can also describe Fred’s position in the class by determine how many standard deviations above or
below the mean it falls.
𝑥̅ = 80
𝑠. 𝑑. = 6.07
Since Fred scored 84, he scored about ½ of one standard deviation above the mean.
Standardizing: converting scores from original values to standard deviation units
Standardized value more commonly called a z -score
𝑧=
𝑥 − 𝑥̅
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
z-score tells us how many standard deviations away from the mean an original observation falls
if z-score is larger than the mean it is positive
if z-score is smaller than the mean it is negative
the mean for the z-scores is always 0 and the standard deviation is always 1
Fred’s z-score would be
84 − 80
= .657
6.07
So Fred scored .657 standard deviations above the mean.
𝑧=
Now calculate the z-score for Barney and Bam Bam:
Barney:
90 − 80
= 1.648
6.07
So Barney scored 1.648 standard deviations above the mean
𝑧=
Bam Bam
67 − 80
= −2.14
6.07
So Bam Bam scored 2.14 standard deviations below the mean
𝑧=
A z-score can be used to compare relative standing of individuals in different distributions.
Fred found out he got an 81 on Mr. Slates English test and was told the distribution was fairly symmetric
with 𝑥̅ = 76
𝑠. 𝑑. = 4 How did Fred do compared to the rest of his class?
𝑧=
81 − 76
= 1.25
4
So Fred scored better on his English test than on his Algebra 2 test relative to the of his classmates.
We often standardize observations from symmetric distributions to express them on a common scale.
Transformations
Transforming data converts from the original units of measurements to another scale
-can affect the shape, center, and spread
Suppose Mr. Jones decided to throw out a question that every student missed. This added 5 points to
everyone’s score. How does this change the mean and standard deviation for his class?
𝑥̅ = 85
𝑠. 𝑑. = 6.07
Adding or Subtracting a Constant (𝒂)
 adds/subtracts a to/from measures of center and location (location is percentiles)
 does not change the shape of distribution or measures of spread
Multiplying or Dividing a Constant (𝒃)
 multiplies/divides measures of center and location
 multiplies/divides measure of spread by |𝑏|
 does not change the shape of the distribution
Read the example on page 95-96.
Density Curves
This is a histogram of the vocabulary scores of all seventh-grade students in Gary, Indiana with a smooth
curve drawn through the tops of the bars.
Mathematical Model: idealized description for the distribution
-gives a compact picture of the overall pattern of the data but ignores minor irregularities
as well as any outliers
Density Curves: describes the overall pattern of a distribution
-Area underneath always equal to 1
-Is always on or above the 𝑥 − 𝑎𝑥𝑖𝑠
-Does not show outliers
These graphs represent the shape of a Normal curve. The area of the portion of the histogram on the left
is approximately equal to the area of the portion under the density curve at the same point.
Density curve is an approximation that is easy to use and accurate enough for practical use.
Measuring Center for Density Curve
-Median is the point with half the total area on each side (picture pg. 106)
-Mean is the balancing point (picture pg. 107)
-Mean is now represented as 𝜇 because “true” value or population value
-Standard Deviation is now represented as 𝜎 “true” value or population value
Homework: 2.1 pg. 99-103 #1,5,9-13 odd, 17-23 odd, 25-30 all
2.2 pg. 128-133 #35, 39