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Have out your calculator and your notes! The four C’s: Clear, Concise, Complete, Context

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Chapter 4 Displaying Quantitative Data

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Dealing With a Lot of Numbers... When looking at large sets of quantitative data, it can be difficult to get a sense of what the numbers are telling us without summarizing the numbers in some way. In this chapter, we will concentrate on graphical displays of quantitative data.

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Percent of Population over 65 per state (1996) 13.014.312.513.913.812.515.812.1 5.212.812.611.411.414.512.111.2 13.218.515.214.112.013.414.411.6 14.4 9.913.712.413.813.512.515.2 10.512.912.612.411.013.410.213.3 11.011.411.412.313.415.9 8.811.2 13.8 13.2 Put this in your calc!!! (L1)

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What do these data tell us? Make a picture Histogram Stem-and-Leaf Display Dot plot First three things to do with data Make a picture

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Displaying Quantitative Data Histogram Give each graph a title Give each one of the axes a label Make as neat as possible Computer Calculator Grid paper

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Displaying Quantitative Data Histogram Divide data values into equal-width piles (called bins) Count number of values in each bin Plot the bins on x-axis Plot the bin counts on y-axis

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Example – Population Over 65 Decide on bin values Low value is 5.2 and high value is 18.5 Bins are 5.0 up to 6.0, 6.0 up to 7.0, etc. Written as 5.0 ≤ X < 6.0, 6.0 ≤ X < 7.0 Count number of values in each bin Bin 5.0 ≤ X < 6.0 has 1 value Bin 6.0 ≤ X < 7.0 has 0 values Bin 7.0 ≤ X < 8.0 has 0 values Bin 8.0 ≤ X < 9.0 has 1 value Continue counting values in each bin “Up to but not including”

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Example – Population Over 65 Plot bins on x-axis Min: 5.2 and max: 18.5 14 bins from 5.0 ≤ X < 6.0 to 18.0 ≤ X < 19.0 Plot bin counts on y-axis Bin counts are: 1, 0, 0, 1, 1, 2, 9, 13, 13, 5, 4, 0, 0, 1

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Make one on your calc!!!

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Displaying Quantitative Data Stem and Leaf Display Picture of Distribution Generally used for smaller data sets Group data like histograms Still have original values (unlike histograms) Two columns Left column: Stem Right column: Leaf

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Displaying Quantitative Data Stem and Leaf Display Leaf Contains the last digit of the values Arranged in increasing order away from stem Stem Contains the rest of the values Arranged in increasing order from top to bottom **Always have a legend!**

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Example – Population Over 65 Leaf = tenths digit Stem = tens and ones digits Ex. 5 | 2 Ex. 10| 2 5 Ex. 14| 1 3 4 4 5

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Percent of Population over Age 65 (by state) in 1996 12 | 1 = 12.1%

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Same shape!

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Example – Frank Thomas Career Home Runs (1990- 2004) 4 7 15 18 24 28 29 32 35 38 40 40 41 42 43 1 | 8 = 18 home runs

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Displaying Quantitative Data Back-to-back Stem-and-Leaf Display Used to compare two variables Stems in center column Leafs for one variable – right side Leafs for other variable – left side Arrange leafs in increasing order, AWAY FROM STEM!

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Example – Compare Frank Thomas to Ryne Sandberg Career Home Runs for Ryne Sandberg (1981-1997) 0 5 7 8 9 12 14 16 19 19 25 26 26 26 30 40 SandbergThomas 1 | 8 = 18 home runs

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Displaying Quantitative Data If there are a large number of observations in only a few stems, we can split stems. Split the stems into two stems First stem is 0 – 4. Second stem is 5 – 9. If you choose to split one stem you MUST split them all!

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Example – Population Over 65 12 | 1 = 12.1%

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One Variable Statistics… Population Over 65: STAT > CALC > 1-Var Stats

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Looking at Distributions Always report 4 things when describing a distribution: 1. Shape 2. Unusual (Outliers and other notable features) 3. Center 4. Spread SUCS (Shape, Unusual, Center, Spread)

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Looking at Distributions Shape How many humps (called modes)? None = uniform One = unimodal Two = bimodal Three or more = multimodal

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Looking at Distributions Shape Is it symmetric? Symmetric = roughly equal on both sides Skewed = more values on one side Right = Tail stretches to large values Left = Tail stretches to small values Are there any outliers? Interesting observations in data Can impact statistical methods

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Examples of Skewness “Skewness to the fewness”

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Looking at Distributions Center A single number to describe the data Can calculate different numbers for center

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Looking at Distributions Spread Variation in the data values Range: Smallest observation to the largest observation May take into account any outliers Later, spread will be a single number

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SUCS (Shape, Unusual, Center, Spread)

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Example – Population Over 65 Shape Unimodal Symmetric Unusual Two Outliers (5% and 18%) Center: About 12% Spread: Almost all observations are between 8% and 16% SUCS (Shape, Unusual, Center, Spread)

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Now try… (Day one) Pg. 72 #5-12 “My father taught me that the only way you can make good at anything is to practice, and then practice some more.” -Pete Rose

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Example – Frank Thomas Shape –Unimodal –Skewed left –No outliers Center: Median = 32 home runs Spread: All values are between 4 and 43 1 | 8 = 18 home runs

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Example – Compare Frank Thomas to Ryne Sandberg Sandberg’s Home Runs Shape: –Unimodal –Skewed right**** –No Outliers Center: Median = 27.5 home runs Spread: All values are between 0 and 40 Both players have about the same spread (ranges are 40 and 39) Thomas has 50% of his home runs above 30, while Sandberg only has 6.25% above 30. 1 | 8 = 18 home runs

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What Do We Know? Histograms, Stem-and-Leaf Displays, Back-to- Back Stem-and-Leaf Displays When describing a display, always mention: Shape: number of modes, symmetric or skewed Unusual Features (Outliers, etc. Mention them if they exist; otherwise, say there are no outliers) Center Spread SUCS (Shape, Unusual, Center, Spread)

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What Do We Know? (cont.) A graph is either symmetric or skewed, not both! If a graph is skewed, be sure to specify the direction: Skewed left (negative) or skewed right (positive)… (“Skewness to the fewness”)

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Describing the Distribution Center Median (.5 quantile, 2nd quartile, 50th percentile) Mean Spread Range (max – min) Interquartile Range (Q3 – Q1) Standard Deviation

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Median Literally = middle number (data value) Has the same units as the data n (number of observations) is odd Order the data from smallest to largest Median is the middle number on the list (n+1)/2 number from the smallest value Ex: If n=11, median is the (11+1)/2 = 6 th number from the smallest value Ex: If n=37, median is the (37+1)/2 = 19 th number from the smallest value

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Example – Frank Thomas Career Home Runs 4 7 15 18 24 28 29 32 35 38 40 40 41 42 43 Remember to order the values, if they aren’t already in order! 15 observations (15+1)/2 = 8 th observation from bottom Median = 32 HRs

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Median n is even Order the data from smallest to largest Median is the average of the two middle numbers (n+1)/2 will be halfway between these two numbers Ex: If n=10, (10+1)/2 = 5.5, median is average of 5 th and 6 th numbers from smallest value

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Example – Ryne Sandberg Career Home Runs 0 5 7 8 9 12 14 16 19 19 25 26 26 26 30 40 Remember to order the values if they aren’t already in order! 16 observations (16 + 1)/2 = 8.5, average of 8 th and 9 th observations from bottom Median = average of 16 and 19 Median = 17.5 HRs

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Mean Everyday “average” that most people think of Add up all observations Divide by the number of observations Has the same units as the data Formula n observations y 1, y 2, y 3, …, y n are the values

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Mean

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Examples of mean… Thomas’ Career Home Runs: Sandberg’s Career Home Runs:

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Mean vs. Median Median = middle number Mean = value where histogram balances ***Mean and Median similar when Data are symmetric ***Mean and median different when Data are skewed There are outliers

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Mean vs. Median Mean influenced by unusually high or unusually low values Example: Income in a small town of 6 people $25,000 $27,000 $29,000 $35,000 $37,000 $38,000 **The mean income is $31,830 **The median income is $32,000 And then….

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Mean vs. Median Bill Gates moves to town $25,000 $27,000 $29,000 $35,000 $37,000 $38,000 $40,000,000 **The mean income is $5,741,571 **The median income is $35,000 Mean is pulled by the outlier Median is not Mean is not a good center of these data

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Mean vs. Median Skewness pulls the mean in the direction of the tail Skewed to the right = mean > median Skewed to the left = mean < median Outliers pull the mean in their direction Large outlier = mean > median Small outlier = mean < median

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Spread Range = maximum – minimum Thomas Min = 4, Max = 43, Range = 43 - 4 = 39 HRs Sandberg Min = 0, Max = 40, Range = 40 - 0 = 40 HRs

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Spread Range is a very basic measure of spread It is highly affected by outliers Makes spread appear larger than reality Ex. The annual numbers of deaths from tornadoes in the U.S. from 1990 to 2000: 53 39 39 33 69 30 25 67 130 94 40 Range with outlier: 130 – 25 = 105 tornadoes Range without outlier: 94 – 25 = 69 tornadoes

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Spread Interquartile Range (IQR) First Quartile (Q1) Larger than about 25% of the data Third Quartile (Q3) Larger than about 75% of the data IQR = Q3 – Q1 Center (Middle) 50% of the values The IQR is a single value, NOT AN INTERVAL!

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Finding Quartiles Order the data Split into two halves at the median When n is odd, include the median in both halves When n is even, do not include the median in either half Q1 = median of the lower half Q3 = median of the upper half

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Example – Frank Thomas Order the values (15 values) 4 7 15 18 24 28 29 32 35 38 40 40 41 42 43 Lower Half = 4 7 15 18 24 28 29 32 Q1 = Median of lower half = 21 HRs Upper Half = 32 35 38 40 40 41 42 43 Q3 = Median of upper half = 40 HRs IQR = 40 – 21 = 19 HRs

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Example – Ryne Sandberg Order the values (16 values) 0 5 7 8 9 12 14 16 19 19 25 26 26 26 30 40 Lower Half = 0 5 7 8 9 12 14 16 Q1 = Median of lower half = 8.5 HRs Upper Half =19 19 25 26 26 26 30 40 Q3 = Median of upper half = 26 HRs IQR = Q3 – Q1 = 26 – 8.5 = 17.5 HRs

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Five Number Summary Minimum Q1 Median Q3 Maximum We’ll use these to make boxplots next chapter!

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Examples Thomas Min = 4 HRs Q1 = 21 HRs Median = 32 HRs Q3 = 40 HRs Max = 43 HRs Sandberg Min = 0 HRs Q1 = 8.5 HRs Median = 17.5 HRs Q3 = 26 HRs Max = 40 HRs

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Spread Standard deviation “Average” spread from mean Most common measure of spread (Although it is influenced by skewness and outliers) Denoted by letter s Make a table when calculating by hand

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Standard Deviation

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Example – Deaths from Tornadoes 5353-56.27 =-3.2710.69 3939-56.27 = -17.27298.25 3939-56.27 = -17.27298.25 3333-56.27 = -23.27541.49 6969-56.27 = 12.73162.05 3030-56.27 = -26.27690.11 2525-56.27 = -31.27977.81 6767-56.27 = 10.73115.13 130130-56.27 = 73.735436.11 9494-56.27 = 37.731423.55 4040-56.27 = -16.27264.71

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Example – Frank Thomas Find the standard deviation of the number of home runs given the following statistic:

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Properties of s s = 0 only when all observations are equal; otherwise, s > 0 s has the same units as the data s is not resistant… Skewness and outliers affect s, just like mean Tornado Example: s with outlier: 31.97 tornadoes s without outlier: 21.70 tornadoes

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Which summaries should you use? What numbers are affected by outliers? Mean Standard deviation Range What numbers are not affected by outliers? Median IQR

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Which summaries should you use? Five Number Summary Skewed Data Data with outliers Mean and Standard Deviation Symmetric Data ALWAYS PLOT YOUR DATA!! Table…

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Now try… (Day two) Pg. 72 #13-23 odds

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Now try…do… (Day three) Pg. 72-78 #25, 29, 31, 37, 43, 45, 49

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