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

Have out your calculator and your notes!

The four C’s:Clear, Concise, Complete, Context

Chapter 4

Displaying Quantitative Data

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.

Percent of Population over 65 per state (1996)13.0 14.3 12.5 13.9 13.8 12.5 15.8 12.1

5.2 12.8 12.6 11.4 11.4 14.5 12.1 11.2

13.2 18.5 15.2 14.1 12.0 13.4 14.4 11.6

14.4 9.9 13.7 12.4 13.8 13.5 12.5 15.2

10.5 12.9 12.6 12.4 11.0 13.4 10.2 13.3

11.0 11.4 11.4 12.3 13.4 15.9 8.8 11.2

13.8 13.2

Put this in your calc!!! (L1)

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 Make a picture Make a picture


Histogram Give each graph a title Give each one of the axes a label Make as neat as possible

• Computer• Calculator• Grid paper


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

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”


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

Make one on your

calc!!!


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


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!**


Leaf = tenths digit Stem = tens and ones digits Ex. 5 | 2 Ex. 10| 2 5 Ex. 14| 1 3 4 4 5

Percent of Population over Age 65 (by state) in 1996

5 2678 89 9

10 2 511 0 0 2 2 4 4 4 4 612 0 1 1 3 4 4 5 5 5 6 6 8 913 0 2 2 3 4 4 4 5 7 8 8 8 914 1 3 4 4 515 2 2 8 9161718 5

12 | 1 = 12.1%

Same shape!

5 2678 89 9

10 2 511 0 0 2 2 4 4 4 4 612 0 1 1 3 4 4 5 5 5 6 6 8 913 0 2 2 3 4 4 4 5 7 8 8 8 914 1 3 4 4 515 2 2 8 9161718 5

52

6 7 88

99

102

511

00

22

44

44

612

01

13

44

55

56

68

913

02

23

44

45

78

88

914

13

44

515

22

89

16 17 185

Example – Frank Thomas

Career Home Runs (1990-2004)

4 7 15 18 24 28 29 32 35 38 40 40 41 42 43

0 4 71 5 82 4 8 93 2 5 84 0 0 1 2 3

1 | 8 = 18 home runs


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!

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

9 8 7 5 0 0 4 79 9 6 4 2 1 5 86 6 6 5 2 4 8 9

0 3 2 5 80 4 0 1 2 3



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!


12 0 1 1 3 4 4 5 5 5 6 6 8 913 0 2 2 3 4 4 4 5 7 8 8 8 9

12 0 1 1 3 4 412 5 5 5 6 6 813 0 2 2 3 4 413 5 7 8 8 8 9 12 | 1 = 12.1%

One Variable Statistics… Population Over 65:

STAT > CALC > 1-Var Stats

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)


Shape How many humps (called modes)?

• None = uniform• One = unimodal• Two = bimodal• Three or more = multimodal

0.1 0.2 0.3 0.4

0

5

10

15

Size (carats)

Fre

qu

ency

Size of Diamonds (carats)

86 87 88 89 90 91 92 93 94 95 96

0

1

2

3

4

5

6

7

8

9

10

Octane

Fre

qu

ency

Histogram of Octane Rating

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

86 87 88 89 90 91 92 93 94 95 96

0

1

2

3

4

5

6

7

8

9

10

OctaneF

req

uen

cy

Histogram of Octane Rating

Examples of Skewness

“Skewness to the fewness”


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


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

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%


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


• Shape– Unimodal – Skewed left– No outliers

• Center: Median = 32 home runs

• Spread: All values are between 4 and 43

0 4 71 5 82 4 8 93 2 5 84 0 0 1 2 3


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.

9 8 7 5 0 0 4 79 9 6 4 2 1 5 86 6 6 5 2 4 8 9

0 3 2 5 80 4 0 1 2 3


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


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”)

Describing the Distribution

Center Median (.5 quantile, 2nd quartile, 50th percentile) Mean

Spread Range (max – min) Interquartile Range (Q3 – Q1) Standard Deviation

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 = 6th number from the smallest value

• Ex: If n=37, median is the (37+1)/2 = 19th number from the smallest value


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 = 8th

observation from bottom

• Median = 32 HRs

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 5th and 6th numbers from smallest value

Example – Ryne Sandberg

Career Home Runs0 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 8th and 9th observations from bottom

• Median = average of 16 and 19

• Median = 17.5 HRs

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 y1, y2, y3, …, yn are the values

Mean

y y1 y2 y3 L yn

n

yn

1

ny

Examples of mean…

Thomas’ Career Home Runs:

Sandberg’s Career Home Runs:

(4 7 15 18 ... 43)

1526.4HRs

(0 5 7 8 ... 40)

1617.625 HRs

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

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….

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

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

Spread

Range = maximum – minimum Thomas

Min = 4, Max = 43, Range = 43 - 4 = 39 HRs Sandberg

Min = 0, Max = 40, Range = 40 - 0 = 40 HRs

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

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!

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


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

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

Five Number Summary

Minimum Q1 Median Q3 Maximum

We’ll use these to make boxplots next chapter!

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

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

Standard Deviation

s (y1 y )2 (y2 y )2 K (yn y )2

n 1

y y 2n 1

1

n 1y y 2

Example – Deaths from Tornadoes

53 53-56.27 =-3.27 10.69

39 39-56.27 = -17.27 298.25

39 39-56.27 = -17.27 298.25

33 33-56.27 = -23.27 541.49

69 69-56.27 = 12.73 162.05

30 30-56.27 = -26.27 690.11

25 25-56.27 = -31.27 977.81

67 67-56.27 = 10.73 115.13

130 130-56.27 = 73.73 5436.11

94 94-56.27 = 37.73 1423.55

40 40-56.27 = -16.27 264.71

y )( yy 2)( yy

s 10.69 298.25 L 264.71

11 131.97 tornadoes

Example – Frank Thomas Find the standard deviation of the number of

home runs given the following statistic:

74.2329)( 2 yy

s (y y )2n 1

2329.74

15 112.9HRs

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

Which summaries should you use? What numbers are affected by outliers?

Mean Standard deviation Range

What numbers are not affected by outliers? Median IQR

Which summaries should you use? Five Number Summary

Skewed Data Data with outliers

Mean and Standard Deviation Symmetric Data

ALWAYS PLOT YOUR DATA!!

Table…

Now try… (Day two)

Pg. 72 #13-23 odds

Now try…do… (Day three)

Pg. 72-78 #25, 29, 31,

37, 43, 45, 49

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

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