CP MathCP MathLesson 12-6Lesson 12-6

Measures of SpreadMeasures of Spread

Quiz 11-1Quiz 11-1 1. 1. Find the Find the meanmean of the following data. of the following data.

{ 11, 3, 6, 2, 9, 4}{ 11, 3, 6, 2, 9, 4} 52

64

median

2. 2. Find the Find the medianmedian of of the following data.the following data.

{ 10, 15, 6, 27, 9}Ums{ 10, 15, 6, 27, 9}Ums 10median

3. 3. Find the Find the rangerange of the following data. of the following data.

{ 16, 12, 14, 17, 14}{ 16, 12, 14, 17, 14} 5range

4. 4. Find the Find the modemode of the following data. of the following data.

{ 16, 12, 14, 17, 14, 17, 16, 14}{ 16, 12, 14, 17, 14, 17, 16, 14} 14mode

Frequency Distribution curvesFrequency Distribution curvesStandardStandard deviationdeviation: a measurement of spread of the data : a measurement of spread of the data from the mean. from the mean.

Smaller standard Smaller standard deviation deviation higher peak in the middlehigher peak in the middle less spreadless spread. .

Larger standard Larger standard deviationdeviation wider the spread of the datawider the spread of the data Greater spreadGreater spread

Data DistributionData Distribution

34%34%34%34%

13.5%13.5%13.5%13.5%

2.25%2.25%2.25%2.25%

)22( xxxP

= 13.5 + 13.5 + 34 + 34 = 95%= 13.5 + 13.5 + 34 + 34 = 95%

Standard deviationStandard deviation a number that describes the spread of the data. a number that describes the spread of the data.

Standard deviationStandard deviation 68% of the data will be within one standard 68% of the data will be within one standard deviation of the mean.deviation of the mean.

mean

probabilityprobability of a data point of a data point being within two standard being within two standard deviations of the mean.deviations of the mean.

probabilityprobability of a data point of a data point being within three standard being within three standard deviations of the mean.deviations of the mean.

)33( xxxP = 68 + 27 + 4.5 = 99.5 %= 68 + 27 + 4.5 = 99.5 %

Your turn: Your turn: 1. 1. What is the name of the variable that combinesWhat is the name of the variable that combines(1) The height of the peak and (2) spread of the data.:(1) The height of the peak and (2) spread of the data.:

Go to: http://www.shodor.org/interactivate/activities/NormalDistribution

““standard standard deviation”deviation”

2. 2. What do we call the What do we call the number that is the center number that is the center of the peak?of the peak?

““mean”mean” x

http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html

Probability that an event is Probability that an event is more than 2 standard more than 2 standard deviations ABOVE the mean?deviations ABOVE the mean?

%5.2)2( xxP

34%34%34%34%

13.5%13.5%13.5%13.5%

2.25%2.25%2.25%2.25%

%5.2)2( xxP

Probability that an event is Probability that an event is between the mean and 2 std between the mean and 2 std dev dev ABOVEABOVE the mean? the mean?

34%34%34%34%

13.5%13.5%13.5%13.5%

2.25%2.25%2.25%2.25%

%5.47%5.13%34)2( xxP

Your turn:Your turn:3. 3. What is the probability that a data point will be between What is the probability that a data point will be between one standard deviation and 2 standard deviations below the one standard deviation and 2 standard deviations below the mean?mean?

4. 4. What is the probability that a data point will be within 2 What is the probability that a data point will be within 2 standard deviations of the mean?standard deviations of the mean?

34%34%34%34%

13.5%13.5%13.5%13.5%

2.25%2.25%2.25%2.25%

Probabilities and Standard Probabilities and Standard DeviationDeviationThe standard deviation for some data is 10.The standard deviation for some data is 10.

The mean of the data is 50The mean of the data is 50

What is the probability that the data falls between 40 and 60?What is the probability that the data falls between 40 and 60?

34%34%34%34%

13.5%13.5%13.5%13.5%

2.25%2.25%2.25%2.25%

30 40 50 60 70

68%68%

What is the probability What is the probability that the data falls that the data falls between 50 and 70?between 50 and 70?

= 34% + 13.5%= 34% + 13.5%

= 47.5% = 47.5% Is it very likely that a Is it very likely that a data point falls above data point falls above 90?90?

Your turn:Your turn:5. 5. The standard deviation for some data is 7. The mean for The standard deviation for some data is 7. The mean for this data is 42. Draw a bell curve and label the x-axis up to 3 this data is 42. Draw a bell curve and label the x-axis up to 3 standard deviations above and below the mean.standard deviations above and below the mean.

6. 6. What is the probability that a data point will be in the What is the probability that a data point will be in the range between 28 and 42?range between 28 and 42?

7. 7. What is the probability that a data point will be in the What is the probability that a data point will be in the range between 21 and 28?range between 21 and 28?

Standard DeviationStandard Deviation

n

xxxxxx n22

22

1 )(...)()(

StandardStandard deviationdeviation: a measurement of spread of the data : a measurement of spread of the data from the mean. from the mean.

Test Scores

107 95 93 93 86 79 43

Step oneStep one: calculate the mean ( ). : calculate the mean ( ). xn

xxxx n

...21

857

427986939395107

x

Step twoStep two: subtract the mean from each data point. : subtract the mean from each data point.

xxn 22 10 8 8 1 6 42

Step threeStep three: square each of the terms : square each of the terms xxn

Standard DeviationStandard DeviationStandardStandard deviationdeviation: a measurement of spread of the data : a measurement of spread of the data from the mean. from the mean.

Test Scores

107 95 93 93 86 76 43

22 10 8 8 1 -6 -42xxn 2)( xxn 484 100 64 64 1 36 1764

Step fourStep four: add the last row togetther.: add the last row togetther. 2513

Step fiveStep five: divide by the number of data points: : divide by the number of data points: 35972513

Step sixStep six: take the square root of the number. : take the square root of the number. 9.18359

9.18

n

xxxxxx n22

22

1 )(...)()(

Your turn:Your turn: n

xxxxxx n22

22

1 )(...)()(

88. . Find the standard deviation for the test scores:Find the standard deviation for the test scores:

Test Scores

97 92 90 85 77 62 62 55

xxn 2)( xxn

0.15

Two different Standard Two different Standard DeviationsDeviations

1. Data set is 1. Data set is all of the dataall of the data (the “population”) (the “population”)

2. Data set is just a 2. Data set is just a samplesample of a of a HUGEHUGE data data

ExampleExample: test scores of a single class: test scores of a single class

ExampleExample: the opinions of 1000 individuals randomly selected: the opinions of 1000 individuals randomly selected

:deviation standard

S :Deviation Standard

Let’s use the calculatorLet’s use the calculatorEnter the data into a list Enter the data into a list “ “Stat” then “edit” option.Stat” then “edit” option.

Scroll until the cursor has “L1” highlighted,Scroll until the cursor has “L1” highlighted, then then clearclear the list (don’t delete it). the list (don’t delete it).

Enter the data into list “L1”. Enter the data into list “L1”.

““stat” then scroll stat” then scroll over to “calc”over to “calc”

Option 1: 1-var stats Option 1: 1-var stats then “enter”then “enter”

Tell the calculator whereTell the calculator where the data is :“2the data is :“2ndnd” “1” (List 1)” “1” (List 1)

Let’s use the calculatorLet’s use the calculatorMean:Mean:

Standard Deviation:Standard Deviation:

Number of data points:Number of data points:

Minimum data point:Minimum data point:

Q1 (we’ll talk aboutQ1 (we’ll talk about this later)this later)

Q3 (we’ll talk aboutQ3 (we’ll talk about this later)this later)

Median:Median:

Maximum data point:Maximum data point:

Let’s use the calculatorLet’s use the calculator

Find “s” (“ln” button)Find “s” (“ln” button)

Go to the catalog (“2Go to the catalog (“2ndnd then “0” buttons) then “0” buttons)

Then scroll down to “stDev(“Then scroll down to “stDev(“

Hit “enter” buttonHit “enter” button

Left bracket (“2Left bracket (“2ndnd” and “(“ buttons),” and “(“ buttons),enter the data with commas, enter the data with commas, (“2(“2ndnd” and “)” buttons)” and “)” buttons)

0.16S

This gives the standard deviation of the This gives the standard deviation of the “ “population” if the data is just a “population” if the data is just a “samplesample”.”.

Your turn:Your turn:9. 9. Find the standard deviation using the “stDev” functionFind the standard deviation using the “stDev” function on your calculator for the following data. (this data is the on your calculator for the following data. (this data is the entire population of the data set, it not just a sample) entire population of the data set, it not just a sample)

Test Scores

105 100 95 80 79 75 60 35

5.21

Quartiles: Quartiles: are medians are medians 105100100100959085808075757575757065555050504535

75 = Quartile 275 = Quartile 2

90 = Quartile 390 = Quartile 3

55 = Quartile 155 = Quartile 1

First QuartileFirst Quartile: the : the medianmedian of the of the lower halflower half of the data. of the data.

Second QuartileSecond Quartile: the median of the data : the median of the data

Third QuartileThird Quartile: the : the medianmedian of the of the upper halfupper half of the data. of the data.

1Q

2Q

3Q

VocabularyVocabulary105100100100959085808075757575757065555050504535

75 = Quartile 275 = Quartile 2

90 = Quartile 390 = Quartile 3

55 = Quartile 155 = Quartile 1

Inter-Quartile rangeInter-Quartile range: the distance : the distance between quartile 1 and quartile 3. between quartile 1 and quartile 3.

13 QQ 35559013 QQ

Inter-quartile Inter-quartile range = 35range = 35

VocabularyVocabulary

First QuartileFirst Quartile: the : the medianmedian of the of the lower halflower half of the data. of the data.

Second QuartileSecond Quartile: the median of the data : the median of the data

Third QuartileThird Quartile: the median : the median upper halfupper half of the data of the data

Inter-Quartile rangeInter-Quartile range: :

1Q

2Q

3Q

13 QQ

QuartilesQuartiles

1 1052 1003 1004 1004 956 907 858 809 80

10 7511 7512 7513 7514 7515 7016 6517 5518 5019 5020 5021 4522 35

22 terms22 terms

Median = quartile 2Median = quartile 2

75 = Quartile 275 = Quartile 2

EvenEven number of data points: number of data points: Must average the middle two points.Must average the middle two points.

Top half will be an odd # of pointsTop half will be an odd # of pointsBottom half will be an odd # of pointsBottom half will be an odd # of points

105100100100959085808075757575757065555050504535

1 1052 1003 1004 1005 956 907 858 809 80

10 7511 7512 7513 7514 7515 7016 6517 5518 5019 5020 5021 4522 35

QuartilesQuartiles

11 terms11 terms

Quartile 1 = median of the upper halfQuartile 1 = median of the upper half of the data points.of the data points.

How do you find the median of a data setHow do you find the median of a data set with an odd number of data points?with an odd number of data points?

90 = Quartile 390 = Quartile 3

Find the Find the center pointcenter point of the data of the data

1 1052 1003 1004 1004 956 907 858 809 80

10 7511 7512 7513 7514 7515 7016 6517 5518 5019 5020 5021 4522 35

QuartilesQuartiles

11 terms11 terms

Quartile 3 = median of the lower halfQuartile 3 = median of the lower half of the data points.of the data points.

How do you find the median of a data setHow do you find the median of a data set with an odd number of data points?with an odd number of data points?

55 = Quartile 155 = Quartile 1

Find the Find the center pointcenter point of the data of the data

Your turn:Your turn:1 1072 1073 1074 1075 976 957 938 939 93

10 9311 9312 9313 8614 8615 7916 6517 5818 5719 4320 36

For the following data set, find:For the following data set, find:

10. 10. First QuartileFirst Quartile

11. 11. Second Quartile Second Quartile

12. 12. Third QuartileThird Quartile

13. 13. Inter-Quartile range Inter-Quartile range

963 Q

932 Q

721 Q

2413 QQ

Use the calculatorUse the calculatorNumber of data points:Number of data points:

Minimum data point:Minimum data point:

Q1 (we’ll talk aboutQ1 (we’ll talk about this later)this later)

Q3 (we’ll talk aboutQ3 (we’ll talk about this later)this later)

Median:Median:

Maximum data point:Maximum data point:

Your turn:Your turn:1 1172 1573 1234 1155 1016 977 968 999 92

10 9111 9012 8713 8514 8615 7916 6917 5218 4619 4020 31

Use the “power of the calculator” to Use the “power of the calculator” to find the following statistics for the data find the following statistics for the data set given.set given.

14. 14. First QuartileFirst Quartile

15. 15. Second Quartile Second Quartile

16. 16. Third QuartileThird Quartile

17. 17. Inter-Quartile range Inter-Quartile range

1003 Q

741 Q

2213 QQ

5.902 Q

QuartilesQuartiles105100100100979085808075757575757065555050504535

110110

100100

9090

8080

7070

6060

5050

4040

3030

2020

1010

00

Box PlotBox Plot: visually shows:: visually shows:

PointSmallest

PointLargest

3Q

2Q

1Q

QuartilesQuartiles105100100100959085808075757575757065555050504535

Inter-QuartileInter-Quartile range: range:

110110

100100

9090

8080

7070

6060

5050

4040

3030

2020

1010

00

Range Range

Box PlotBox Plot: visually shows:: visually shows:

PointSmallest

PointLargest

13 QQ

Your turn:Your turn:1071071071079793939393939393868679645757433614

Using the data from problems 3 – 5,Using the data from problems 3 – 5, build a box plot next to the number linebuild a box plot next to the number line that shows: that shows:

18. 18. Largest data point.Largest data point.

20. 20. Second Quartile Second Quartile

19. 19. Third QuartileThird Quartile

21. 21. First QuartileFirst Quartile

110110

100100

9090

8080

7070

6060

5050

4040

3030

2020

1010

00

22. 22. Smallest data pointSmallest data point

23. 23. Are there any “outliers” ?Are there any “outliers” ? If so, what are their values?If so, what are their values?

Box Plots: Allow Box Plots: Allow visualvisual comparisons of the data.comparisons of the data.

110110

100100

90908080707060605050404030302020101000

24. 24. What is the range of the data? What is the range of the data?

25. 25. Lower quartile = ?Lower quartile = ?

26. 26. Upper quartile = ?Upper quartile = ?

27. 27. Median = ?Median = ?

28. 28. Interquartile range = ?Interquartile range = ?

29. 29. Are there any outlier(s). Are there any outlier(s). If so, what are they ?If so, what are they ?

46 – 10 46 – 10 = 36= 36

Lower quartile: 14Lower quartile: 14

Upper quartile: 22 Upper quartile: 22

Median: 20 Median: 20

Interquartile range: 8Interquartile range: 8

Yes; 46 is an outlier Yes; 46 is an outlier

Skewed Skewed leftleft Skewed Skewed rightright

symetricalsymetrical

Dive Larry’s Score

Leslie’s Score

1 28 272 22 273 21 234 26 65 18 276 21 287 25 238 20 209 24 24

10 21 2311 12 2712 26 26

Mean 22 ~23.4Median 21.5 25Range 16 22

Lower Quartile 20.5 23Upper Quartile 25.5 27

5 4

HOMEWORKHOMEWORKSection 11-1Section 11-1

12, 14, 26 12, 14, 26

(find standard deviation only)(find standard deviation only)

Section 11-3Section 11-3

4, 6, 12, 14, 32a, 32b4, 6, 12, 14, 32a, 32b

Page 1009: problems: Page 1009: problems:

77, 8, 14 – 19 (all) 18 problems, 8, 14 – 19 (all) 18 problems

the 65th percentile can be defined as the lowest score that is greater than 65% of the scores. This is the way we defined it above and we will call this "Definition 1."

The 65th percentile can also be defined as the smallest score that is greater than or equal to 65% of the scores. This we will call "Definition 2."

Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding

A third way to compute percentiles (presented below), is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median (discussed in Chapter 3) to be defined conveniently as the 50th percentile.

HOMEWORKHOMEWORK Section 11-1Section 11-1

2-8, 12, 14, 18-22, 26, 32 2-8, 12, 14, 18-22, 26, 32

(for 12 & 14: find range only)(for 12 & 14: find range only)

(for 18-22: find the outlier only)(for 18-22: find the outlier only)

(for 26: don’t find standard deviation)(for 26: don’t find standard deviation)

Section 7-6: 18, 20, 42, 44Section 7-6: 18, 20, 42, 44

15 problems15 problems