ch04 - STA 2023 - Holbrook Displaying and Summarizing...

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Unformatted text preview: STA 2023 - Holbrook Displaying and Summarizing Quantitative Data Chapter 4 4-1 Data Presentation STA 2023 - Holbrook D a t a P r e s e n ta t io n C a t e g o r ic a l D a ta (C h a p te r 3 ) F r e q u e n c y T a b le B a r C h a rt 4-2 P ie C h a r t Q u a n t it a t iv e D a ta (C h a p te r 4 ) F r e q u e n c y D is t r ib u t io n T a b le H is t o r g r a m S t e m a n d L e a f D is p la y A n d D o tp lo ts STA 2023 - Holbrook Presenting Quantitative Data 4-3 Data Presentation STA 2023 - Holbrook D a t a P r e s e n t a t io n C a t e g o r ic a l D a t a (C h a p te r 3 ) F r e q u e n c y T a b le B a r C h a rt 4-4 P ie C h a r t Q u a n t it a t iv e D a t a (C h a p te r 4 ) F r e q u e n c y D is t r ib u t io n T a b le H is t o r g r a m S t e m a n d L e a f D is p la y A n d D o t p lo ts Stem­and­Leaf Display STA 2023 - Holbrook 1. Divide Each Divide Observation into Stem Value and Leaf Value Value Stem Value Defines Stem Class Class Leaf Value Defines Leaf Frequency (Count) Frequency 2 144677 3 028 26 41 2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 2. 26 4-5 Dotplot STA 2023 - Holbrook 1. Create a dotplot with ages in our class. 4-6 Data Presentation STA 2023 - Holbrook D a t a P r e s e n t a t io n C a t e g o r ic a l D a t a (C h a p te r 3 ) F r e q u e n c y T a b le B a r C h a rt 4-7 P ie C h a r t Q u a n t it a t iv e D a t a (C h a p te r 4 ) F r e q u e n c y D is t r ib u t io n T a b le H is t o r g r a m S t e m a n d L e a f D is p la y A n d D o t p lo t s STA 2023 - Holbrook Frequency Distribution Table Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Class 15 but < 25 3 25 but < 35 5 35 but < 45 4-8 Frequency 2 Relative Frequency & % Distribution Tables STA 2023 - Holbrook Relative Frequency Relative Distribution Distribution Percentage Percentage Distribution Distribution Class Prop. Class % 15 but < 25 .3 15 but < 25 30.0 25 but < 35 .5 25 but < 35 50.0 35 but < 45 .2 35 but < 45 20.0 4-9 Data Presentation STA 2023 - Holbrook D a t a P r e s e n t a t io n C a t e g o r ic a l D a ta (C h a p te r 3 ) F r e q u e n c y T a b le B a r C h a rt 4 - 10 10 P ie C h a r t Q u a n t it a t iv e D a ta (C h a p te r 4 ) F r e q u e n c y D is t r ib u t io n T a b le H is t o r g r a m S t e m a n d L e a f D is p la y A n d D o tp lo ts Histogram STA 2023 - Holbrook Class 15 but < 25 25 but < 35 35 but < 45 Count 5 Frequency Relative Relative Frequency Frequency Percent 4 3 Bars Bars Touch Touch 2 1 0 0 4 - 11 11 15 25 35 45 Lower Boundary 55 Freq. 3 5 2 STA 2023 - Holbrook Shape, Center, and Spread 4 - 12 12 Humps? STA 2023 - Holbrook 1. Humps in a histogram are called Humps “Modes” “Modes” 1. Unimodal - one main peak 2. Bimodal - two peaks 3. Multimodal - three or more peaks 4 - 13 13 Symmetry STA 2023 - Holbrook 1. A histogram is symmetric if you fold it histogram along a vertical line through the middle and have edges match closely. and 2. See page 54. (page 40 – First Edition) (page See 4 - 14 14 Shape STA 2023 - Holbrook 1. Describes How Data Are Distributed 2. Measures of Shape Skewness Left-Skewed Mean Median Mode Median Mode 4 - 15 15 Symmetric Mean = Median = Mode Median Mode Right-Skewed Mode Median Mean Median Mean STA 2023 - Holbrook Describing Distributions Numerically 4 - 16 16 Standard Notation STA 2023 - Holbrook Measure Mean Stand. Dev. Sample Population Y µ S σ 2 2 Variance S σ Size n N 4 - 17 17 Numerical Data Properties STA 2023 - Holbrook Central Tendency Central (Location) (Location) Variation Variation (Dispersion) (Dispersion) Shape 4 - 18 18 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 19 19 STA 2023 - Holbrook Central Tendency 4 - 20 20 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 21 21 Mean STA 2023 - Holbrook 1. 2. 3. 4. 5. 5. Measure of Central Tendency Most Common Measure Acts as ‘Balance Point’ Affected by Extreme Values (‘Outliers’) Formula (Sample Mean) Formula n Y= 4 - 22 22 ∑ Yi i= 1 Y1+ Y 2 + + Y n = n n Mean Example STA 2023 - Holbrook Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 n Y= ∑ Yi i= 1 Y1+ Y 2 + Y 3 + Y 4 + Y 5 + Y 6 = n 6 10.3 + 4.9 + 8.9 + 11.7 + 6.3 + 7.7 = 6 = 8.30 4 - 23 23 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 24 24 Median STA 2023 - Holbrook 1. Measure of Central Tendency 2. Middle Value In Ordered Sequence If Odd n, Middle Value of Sequence If Even n, Average of 2 Middle Values 3. Position of Median in Sequence 3. n+ 1 Positioning Point = Point 2 4. Not Affected by Extreme Values 4. 4 - 25 25 STA 2023 - Holbrook Median Example Odd­Sized Sample Raw Data: 24.1 22.6 21.5 23.7 22.6 Ordered: 21.5 22.6 22.6 23.7 24.1 Position: 1 2 3 4 5 n + 1 5+ 1 Positioning Point = = = 3.0 Point 2 2 Median = 22.6 4 - 26 26 STA 2023 - Holbrook Median Example Even­Sized Sample Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6 n + 1 6+ 1 Positioning Point = = = 3.5 Point 2 2 7.7 + 8.9 Median = = 8.30 2 4 - 27 27 Thinking Challenge STA 2023 - Holbrook You’re a financial analyst You’re for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. 21, Describe the stock prices iin terms of central n tendency. tendency 4 - 28 28 Central Tendency Solution* STA 2023 - Holbrook Mean n Y= ∑ Yi i= 1 Y1+ Y 2 + + Y 8 = n 8 17 + 16 + 21+ 18 + 13 + 16 + 12+ 11 = 8 = 15.5 4 - 29 29 STA 2023 - Holbrook Central Tendency Solution* Median Raw Data: 17 16 21 18 13 16 12 11 Ordered: 11 12 13 16 16 17 18 21 Position: 12345678 n + 1 8+ 1 Positioning Point = = = 4.5 Point 2 2 16 + 16 Median = = 16 2 4 - 30 30 STA 2023 - Holbrook Summary of Central Tendency Measures Measure Equation Mean Σ Yi / n Median (n+1) Position Position 2 4 - 31 31 Description Balance Point Middle Value When Ordered STA 2023 - Holbrook Quartiles 4 - 32 32 Quartiles STA 2023 - Holbrook 1. Measure of Noncentral Tendency Measure Noncentral 2. Split Ordered Data into 4 Quarters 25% 25% Q1 4 - 33 33 25% Q2 25% Q3 Quartiles STA 2023 - Holbrook 1. Note: We will be using the TI-83 to find the Note: quartiles. quartiles. 25% 25% Q1 25% Q2 25% Q3 2. The algorithm that the book uses is 2. different than the one that your TI-83 different uses (for odd sized data sets). uses 4 - 34 34 STA 2023 - Holbrook Q1 and Q3 Example Odd­Sized Sample Raw Data: 24.1 22.6 21.5 23.7 22.6 Ordered: 21.5 22.6 22.6 23.7 24.1 Position: 1 2 3 4 5 n + 1 5+ 1 Positioning Point = = = 3.0 Point 2 2 Median = 22.6 4 - 35 35 Using the TI­83 STA 2023 - Holbrook 1. Entering the data in a list: Hit “STAT” Hit Under “EDIT” select “Edit” Position cursor over “L1” and hit Position “CLEAR” and “ENTER” “CLEAR” Enter each number by typing the number Enter and hitting “ENTER”. and 4 - 36 36 “21.5” and “ENTER”, etc. Using the TI­83 STA 2023 - Holbrook 1. After the data has been entered in a After list (in any order) do the following: list Hit “STAT” and scroll to “CALC” Select “1-Var Stats” Hit “2nd” and “1” (for L1) Hit “ENTER” and scroll to the bottom to Hit view your quartiles. view 4 - 37 37 STA 2023 - Holbrook Q1 and Q3 Example Odd­Sized Sample Raw Data: 24.1 22.6 21.5 23.7 22.6 Ordered: 21.5 22.6 22.6 23.7 24.1 Position: 1 2 3 4 5 Q1=22.05 4 - 38 38 STA 2023 - Holbrook Q1 and Q3 Example Odd­Sized Sample Raw Data: 24.1 22.6 21.5 23.7 22.6 Ordered: 21.5 22.6 22.6 23.7 24.1 Position: 1 2 3 4 5 Q3=23.9 4 - 39 39 STA 2023 - Holbrook Q1 and Q3 Example Even­Sized Sample Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6 n + 1 6+ 1 Positioning Point = = = 3.5 Point 2 2 7.7 + 8.9 Median = = 8.30 2 4 - 40 40 Using the TI­83 STA 2023 - Holbrook 1. After the data has been entered in a After list (in any order) do the following: list Hit “STAT” and scroll to “CALC” Select “1-Var Stats” Hit “2nd” and “1” (for L1) Hit “ENTER” and scroll to the bottom to Hit view your quartiles. view 4 - 41 41 STA 2023 - Holbrook Q1 and Q3 Example Even­Sized Sample Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6 Q1=6.3 4 - 42 42 STA 2023 - Holbrook Q1 and Q3 Example Even­Sized Sample Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7 Ordered: 4.9 6.3 7.7 8.9 10.3 11.7 Position: 1 2 3 4 5 6 Q3=10.3 4 - 43 43 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 44 44 Interquartile Range STA 2023 - Holbrook 1. 2. Measure of Dispersion Difference Between Third & First Difference Quartiles Quartiles Interquartile Range = Q3 – Q1 3. 4. 5. Spread in Middle 50% Not Affected by Extreme Values “IQR” for short 4 - 45 45 Thinking Challenge STA 2023 - Holbrook You’re a financial analyst for You’re Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. 21, What are the quartiles, Q1 What quartiles, and Q3, and the interquartile and 3, interquartile range? range 4 - 46 46 Quartile Solution* STA 2023 - Holbrook Q1 Raw Data: 17 16 21 18 Ordered: 11 12 13 16 Position: 1234 Q1=12.5 4 - 47 47 13 16 5 16 17 6 12 18 7 11 21 8 Quartile Solution* STA 2023 - Holbrook Q3 Raw Data: 17 16 21 18 Ordered: 11 12 13 16 Position: 1234 13 16 5 16 17 6 12 18 7 11 21 8 Q3=17.5 4 - 48 48 STA 2023 - Holbrook Interquartile Range Solution* Interquartile Range Raw Data: 17 16 21 Ordered: 11 12 13 Position: 123 18 16 4 13 16 5 16 17 6 12 18 7 11 21 8 Interquartile Range = Q3 − Q1 = 17.5 − 12.5 = 5.0 Range 4 - 49 49 STA 2023 - Holbrook Variation 4 - 50 50 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 51 51 Range STA 2023 - Holbrook 1. Measure of Dispersion 2. Difference Between Largest & Smallest Difference Observations Observations Range = Y largest − Y smallest 3. Ignores How Data Are Distributed 4 - 52 52 7 8 9 10 7 8 9 10 STA 2023 - Holbrook Numerical Data Properties & Measures Numerical Data Properties Central Tendency Variation Shape Mean Range Median Interquartile Range Skew Variance Standard Deviation 4 - 53 53 STA 2023 - Holbrook Variance & Standard Deviation 1. Measures of Dispersion 2. Most Common Measures 3. Consider How Data Are Distributed 4. Show Variation About Mean (Y or µ) or Y = 8.3 8.3 46 4 - 54 54 8 10 12 Sample Variance Definitional Formula STA 2023 - Holbrook n 2 S= ∑( i= 1 (Y = 4 - 55 55 Yi − Y ) 2 n - 1 in denominator! (Use N if Population Population Variance) Variance) n −1 1 −Y ) + (Y 2 2 −Y ) + + (Y 2 n −1 n −Y ) 2 What does S2 Measure? STA 2023 - Holbrook The Sample Variance (s2) measures measures the average squared distance that all the observations are from the mean. the 4 - 56 56 STA 2023 - Holbrook S= Sample Standard Deviation Formula S 2 n = = 4 - 57 57 ∑ (Y i − Y ) i= 1 2 n −1 (Y )( 2 1 ) 2 ( − Y + Y2 − Y + + Yn − Y n −1 ) 2 Variance Example STA 2023 - Holbrook Raw Data: 2, 3, 3, 4, 8 yi yi − y ( y i − y) 2 2 2- 4=-2 (-2)2=4 3 3- 4=-1 (-1)2=1 3 3- 4=-1 (-1)2=1 4 4- 4=0 (0)2=0 8 8- 4=4 (4)2=16 20 4 - 58 58 22 Variance Example STA 2023 - Holbrook Raw Data: s 2 2, ∑( y = 3, 3, 4, − y) 22 = = 5.5 n −1 5 −1 2 i s = s = 5.5 = 2.345 2 4 - 59 59 8 STA 2023 - Holbrook Variance Computational Formula (∑ y) ∑y − n 2 s= n −1 2 4 - 60 60 2 Variance Example STA 2023 - Holbrook Raw Data: 2, 3, 3, 4, yi ( yi ) 2 2 (2)2=4 3 (3)2=9 3 (3)2=9 2 4 8 (8)2=64 20 4 - 61 61 (4) =16 102 8 Variance Example STA 2023 - Holbrook Raw Data: 2, 3, 3, 4, 8 2 (∑y) (20) 2 102 − ∑y − n 2 5 = 5.5 s= = n −1 5 −1 2 s = s = 5.5 = 2.345 2 4 - 62 62 Using the TI­83 STA 2023 - Holbrook 1. After the data has been entered in a list After (in any order) do the following: (in Hit “STAT” and scroll to “CALC” Select “1-Var Stats” Hit “2nd” and “1” (for L1) Hit “ENTER” 4 - 63 63 We use y instead of x, so when you see x on We your calculator, think y. your So, “Sx” is your sample standard deviation (s). End of Chapter Any blank slides that follow are blank intentionally. ...
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