Chapter 2€¦ · 2 2/44 = 4.5% 2 45-49 7 7/44 = 15.9% 9 50-54 13 13/44 = 29.5% 22 55-59 12 12/44 = 34% 34 60-64 7 7/44 = 15.9% 41 65-69 3 3/44 = 6.8% 44 A cumulative relative frequency

Chapter 2

Data Analysis

Section 2.1Describing Location ina Distribution

Starnes/Tabor, The Practice of Statistics

By the end of this section, you should be able to:

LEARNING TARGETS

Describing Location in a Distribution

FIND and INTERPRET the percentile of an individual value within a distribution of data.

ESTIMATE percentiles and individual values using a cumulative relative frequency graph.

FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and variability of a distribution of data.


One way to describe the location of a value in a distribution is to tell what percent of observations are less than it.

Measuring Location: Percentiles

An individual’s percentile is the percent of values in a distribution that are less than the individual’s data value.




Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class?

6 | 7

7 | 2334

7 | 5777899

8 | 00123334

8 | 569

9 | 03






6 | 7

7 | 2334

7 | 5777899

8 | 00123334

8 | 569

9 | 03





6 | 7

7 | 2334

7 | 5777899

8 | 00123334

8 | 569

9 | 03

Jenny’s score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny’s score is at the 84th percentile in the distribution of test scores for the class.




Cumulative Relative Frequency Graphs

A cumulative relative frequency graph plots a point corresponding to the cumulative relative frequency in each interval at the smallest value of the next interval, starting with a point at a height of 0% at the smallest value of the first interval. Consecutive points are then connected with a line segment to form the graph.



Age of First 44 Presidents When They Were

Inaugurated

Age Frequency

40-

44

2

45-

49

7

50-

54

13

55-

59

12

60-

64

7

65-

69

3





Inaugurated

Age Frequency Relative

frequency

40-

44

2 2/44 =

4.5%

45-

49

7 7/44 =

15.9%

50-

54

13 13/44 =

29.5%

55-

59

12 12/44 =

34%

60-

64

7 7/44 =

15.9%

65-

69

3 3/44 =

6.8%





Inaugurated


frequency

Cumulative

frequency

40-

44

2 2/44 =

4.5%

2

45-

49

7 7/44 =

15.9%

9

50-

54

13 13/44 =

29.5%

22

55-

59

12 12/44 =

34%

34

60-

64

7 7/44 =

15.9%

41

65-

69

3 3/44 =

6.8%

44





Inaugurated


frequency

Cumulative

frequency

Cumulative

relative

frequency

40-

44

2 2/44 =

4.5%

2 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

7 7/44 =

15.9%

41 41/44 =

93.2%

65-

69

3 3/44 =

6.8%

44 44/44 =

100%





Inaugurated


frequency

Cumulative

frequency

Cumulative

relative

frequency

40-

44

2 2/44 =

4.5%

2 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

7 7/44 =

15.9%

41 41/44 =

93.2%

65-

69

3 3/44 =

6.8%

44 44/44 =

100%

0

20

40

60

80

100

40 45 50 55 60 65 70

Cu

mu

lati

ve r

ela

tive

fre

qu

en

cy (

%)

Age at inauguration



Measuring Position: z-Scores

A z-score tells us how many standard deviations from the mean an observation falls, and in what direction.

If x is an observation from a distribution that has known mean and standard deviation, the standardized score of x is:

A standardized score is often called a z-score.

𝑧 =𝑣𝑎𝑙𝑢𝑒 −𝑚𝑒𝑎𝑛

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛



Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. Find and interpret her standardized score.




𝑧 =𝑥 −𝑚𝑒𝑎𝑛

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛=86 − 80

6.07= 0.99






6.07= 0.99

Jenny’s test score is 0.99 standard deviations above the class mean of 80.






6.07= 0.99

Jenny’s test score is 0.99 standard deviations above the class mean of 80.


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.


Transforming Data


The Effect of Adding or Subtracting a Constant

Adding the same positive number a to (subtracting a from) each observation:


Transforming Data



Adding the same positive number a to (subtracting a from) each observation:• Adds a to (subtracts a from) measures of center and location

(mean, five-number summary, percentiles)


Transforming Data




(mean, five-number summary, percentiles)• Does not change measures of variability

(range, IQR, standard deviation)


Transforming Data




(mean, five-number summary, percentiles)• Does not change measures of variability

(range, IQR, standard deviation)• Does not change the shape of the distribution


Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.


Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?


Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?

a) The same shape as the original distribution of guesses: skewed to the right with two distinct peaks.


Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?b) Find the mean and the median of the distribution of error.



Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?b) Find the mean and the median of the distribution of error.


b) Mean: 16.02 – 13 = 3.02 meters; Median: 15 – 13 = 2 meters.


Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?b) Find the mean and the median of the distribution of error.c) Find the standard deviation and interquartile range (IQR) of the distribution of

error.




Transforming Data

Soon after the metric system was introduced in Australia, a group of students was asked to guess the width of their classroom to the nearest meter. The actual width of the room was 13 meters. We can examine the distribution of students’ errors by defining a new variable as follows: error = guess – 13.a) What shape does the distribution of error have?b) Find the mean and the median of the distribution of error.c) Find the standard deviation and interquartile range (IQR) of the distribution of

error.



c) Standard deviation = 7.14 meters; IQR = 6 meters


Transforming Data


Transforming Data



Transforming Data


The Effect of Multiplying or Dividing by a Constant

Multiplying (or dividing) each observation by the same positive number b:


Transforming Data



Multiplying (or dividing) each observation by the same positive number b:• Multiplies (divides) measures of center and location by b

(mean, five-number summary, percentiles) by b


Transforming Data




(mean, five-number summary, percentiles) by b• Multiplies (divides) measures of variability by b

(range, IQR, standard deviation)


Transforming Data




(mean, five-number summary, percentiles) by b• Multiplies (divides) measures of variability by b

(range, IQR, standard deviation)• Does not change the shape of the distribution


Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.


Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.(a) What shape does the resulting distribution of error have?


Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.(a) What shape does the resulting distribution of error have?



Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.(a) What shape does the resulting distribution of error have?(b) Find the median of the distribution of error in feet.



Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.(a) What shape does the resulting distribution of error have?(b) Find the median of the distribution of error in feet.


b) Median = 2 × 3.28 = 6.56 feet


Transforming Data

Because the students are having some difficulty with the metric system, it may not be helpful to tell them that their guesses tended to be about 2 meters too large. Let’s convert the error data to feet before we report back to them. To convert from meters to feet, we multiply each of the error values by 3.28.(a) What shape does the resulting distribution of error have?(b) Find the median of the distribution of error in feet.(c) Find the interquartile range (IQR) of the distribution of error in feet.


b) Median = 2 × 3.28 = 6.56 feet


Transforming Data



b) Median = 2 × 3.28 = 6.56 feetc) IQR = 6 × 3.28 = 19.68 feet


Transforming Data



b) Median = 2 × 3.28 = 6.56 feetc) IQR = 6 × 3.28 = 19.68 feet


Transforming Data


After this section, you should be able to:

LEARNING TARGETS

Section Summary

FIND and INTERPRET the percentile of an individual value within a distribution of data.

ESTIMATE percentiles and individual values using a cumulative relative frequency graph.

FIND and INTERPRET the standardized score (z-score) of an individual value within a distribution of data.

DESCRIBE the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and variability of a distribution of data.


Assignment

2.1 p. 104-109 #2-30 EOE (Every Other Even) and 33-39 all

(2, 6, 10, 14, 18, 22, 26, 30, 33-39)

If you are stuck on any of these, look at the odd before or after and the answer in the back of your book. If you are still not sure text a friend or me for help (before 8pm).

Tomorrow we will check homework and review for 2.1 Quiz.

Documents

Chapter 2€¦ · 2 2/44 = 4.5% 2 45-49 7 7/44 = 15.9% 9 50-54 13 13/44 = 29.5% 22 55-59 12 12/44 = 34% 34 60-64 7 7/44 = 15.9% 41 65-69 3 3/44 = 6.8% 44 A cumulative relative frequency