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 =...

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.

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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

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

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

Starnes/Tabor, The Practice of Statistics

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

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

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

Starnes/Tabor, The Practice of Statistics

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

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.

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

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

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

Cumulative Relative Frequency Graphs

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

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.

Starnes/Tabor, The Practice of Statistics

Cumulative Relative Frequency Graphs

Age of First 44 Presidents When They Were

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%

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.

Starnes/Tabor, The Practice of Statistics

Cumulative Relative Frequency Graphs

Age of First 44 Presidents When They Were

Inaugurated

Age Frequency Relative

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

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.

Starnes/Tabor, The Practice of Statistics

Cumulative Relative Frequency Graphs

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 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%

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.

Starnes/Tabor, The Practice of Statistics

Cumulative Relative Frequency Graphs

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 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

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.

Starnes/Tabor, The Practice of Statistics

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

Starnes/Tabor, The Practice of Statistics

Measuring Position: z-Scores

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.

Starnes/Tabor, The Practice of Statistics

Measuring Position: z-Scores

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

Starnes/Tabor, The Practice of Statistics

Measuring Position: z-Scores

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

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

Starnes/Tabor, The Practice of Statistics

Measuring Position: z-Scores

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

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

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Adding or Subtracting a Constant

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

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Adding or Subtracting a Constant

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)

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Adding or Subtracting a Constant

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)• Does not change measures of variability

(range, IQR, standard deviation)

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Adding or Subtracting a Constant

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)• Does not change measures of variability

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

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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?

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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.

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

Starnes/Tabor, The Practice of Statistics

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.

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

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

Starnes/Tabor, The Practice of Statistics

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.

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

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

Starnes/Tabor, The Practice of Statistics

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.

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

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

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

Starnes/Tabor, The Practice of Statistics

Transforming Data

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Multiplying or Dividing by a Constant

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

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Multiplying or Dividing by a Constant

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

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Multiplying or Dividing by a Constant

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• Multiplies (divides) measures of variability by b

(range, IQR, standard deviation)

Starnes/Tabor, The Practice of Statistics

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.

The Effect of Multiplying or Dividing by a Constant

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• Multiplies (divides) measures of variability by b

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

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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?

Starnes/Tabor, The Practice of Statistics

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?

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

Starnes/Tabor, The Practice of Statistics

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.

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

Starnes/Tabor, The Practice of Statistics

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.

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

b) Median = 2 × 3.28 = 6.56 feet

Starnes/Tabor, The Practice of Statistics

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.

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

b) Median = 2 × 3.28 = 6.56 feet

Starnes/Tabor, The Practice of Statistics

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.

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

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

Starnes/Tabor, The Practice of Statistics

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.

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

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

Starnes/Tabor, The Practice of Statistics

Transforming Data

Starnes/Tabor, The Practice of Statistics

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.

Starnes/Tabor, The Practice of Statistics

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.