Chapter 4 Displaying and Summarizing

Quantitative Data

Math2200

Example: Tsunamis and Earthquake

- The most disastrous tsunami

- Dec 26, 2004, in Sumatra

- Earthquake: magnitude 9.0

- Killed 225,000 people

Question

• Was the earthquake that caused it truly unusually big?

• US National Geophysical Data Center

• Data on the magnitude of underlying earthquakes for 1240 historical tsunamis

• How do we learn the data

Year Country Magnitude

-1300 GREECE 6

-479 GREECE 6.7

-426 GREECE 7.1

-373 GREECE 7.3

-330 GREECE 7

-227 GREECE 7.2

-57 ALBANIA 6.6

-26 CYPRUS ISLAND 7.3

2003 NEW ZEALAND 7.5

2003 JAPAN 8.1

2003 JAPAN 6.8

2003 USA 7.2

2004 INDONESIA 6.5

2004 JAPAN 7.2

2004 JAPAN 7.4

2004 INDONESIA 9

Histogram

• Display a quantitative variable by discretizing it into equal-width bins

• Counts for the bins give the distribution of the quantitative variable

• Make a bar chart based on these counts and align the bar according to the bin values, we get a histogram– Do not leave gaps between bars

Historgram of earthquake magnitude

Magnitude

Fre

qu

en

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3 4 5 6 7 8 9

05

01

00

15

02

00

25

03

00

35

0

With different number of bins

Magnitude

Fre

quen

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3 4 5 6 7 8 9

050

100

150

Magnitude

Fre

quen

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3 4 5 6 7 8 9

050

100

150

Magnitude

Fre

quen

cy

3 4 5 6 7 8 9

050

150

250

350

Magnitude

Fre

quen

cy

3 4 5 6 7 8 9

010

020

030

040

050

0

Summarize the histogram

• Magnitudes are typically around 7

• Most are between 5.5 and 8.5

• Minimum is around 3

• Maximum is around 9

• Why there is a sharp peak in the middle?

Stem-and-Leaf Plot• John W. Tukey

• Useful for small data sets

• Similar to histogram, but the bars give numerical values more than counts

5 | 6 6 | 0444 6 | 8888 7 | 2222 7 | 6666 8 | 000044 8 | 8Pulse-rates of 24 woman(8|8 means 88 beats/min)

Handwriting

• Handwriting may not give the same space for different digits. That violates the area principle

• When you make a stem-and-leaf plot, be sure to give each digit the same width.

Dotplot

• Replace digits in stem-and-leaf plot by dots

How to summarize the distribution of a quantitative variable?

• shape

mode, symmetry, outlier

• center

mean, median

• Spread

sd, IQR

Shape• Peak / Mode

– Is there a peak? If so, how many peaks?– For quantitative variables, the mode is where

the peak is at.– No peak: uniform– One peak: unimodal– Two peaks: bimodal– More than two peaks: multimodal

uniform

x

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

12

unimodal

x

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-2 -1 0 1 2

05

1015

20

bimodal

x

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-2 0 2 4 6

020

4060

8010

0

multimodal

x

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-5 0 5

020

4060

8010

0

Shape• Symmetry

– Tail: thinner ends of a distribution– Skewed: If one tail stretches out farther than the

other, we say the histogram is skewed to the side of the longer tail

skewed to the right

x1

Fre

quen

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

020

4060

8010

0

symmetric

x

Fre

quen

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-3 -2 -1 0 1 2 3

020

4060

8010

0skewed to the left

x2

Fre

quen

cy

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

020

4060

8010

0

Shape

• Outliers– Those that stand

away from the body of the distribution

– The judgment is vague sometimes

Fre

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-6 -4 -2 0 2

05

10

15

Center

• When a histogram is symmetric and unimodal, the center is obvious– The corresponding numerical value can be

taken as the sample average, or say the sample mean

– The sample mean is actually where the histogram balances

Center

• For skewed distribution– The sample mean is dragged to the side of

the longer tail– Usually, much more than 50% values will be

less or larger than the sample mean– Median is more appropriate

• Median is the value that splits the data in half

Finding the median

• Suppose that we have n numbers

• Order them first– If n is odd, the median is middle value. That

is, the value in the (n+1)/2 position– If n is even, we take median as the average of

the values in positionsn/2 and n/2+1

Mean versus median

• Extreme values / outliers:– Median only considers the order of the values,

so it is resistant to extreme values– Mean is very sensitive

• Skewed distribution– Median is preferred than mean

• Unimodal and symmetric distribution– Mean is preferred because it uses more

information from the data

Spread

To quantify the variation

• Range

• Interquartile range (IQR)

• Standard deviation

Range

• Range = max – min

• Very sensitive to extreme values

Interquartile Range

• Quartiles– Q1 (lower quartile or the 25th percentile): one quarter

of the data lies below Q1– Q2 (median or the 50th percentile)– Q3 (upper quartile or the 75th percentile): one quarter

of the data lies above Q3• IQR = Q3-Q1

– Not sensitive to extreme values• How to find Q1 and Q3?

– Split the order values into two halves using the median

– Q1 is the median of the first half– Q3 is the median of the second half

Standard deviation

• Sample variance = average of squared deviations

• Standard deviation (sd)– Sensitive to extreme values

σX in TI-83

How to obtain these numbers using TI-83?

• Press STAT• Move the cursor to CALC• Press 1• The screen shows 1-Var Stats• Put the list you want the statistics for. For examp

le, L1.• Press ENTER, then you will see

– Sample mean, sample sum, sample sum squares, sample standard deviation (Sx),σx (the same as except divided by n instead of n-1), sample size n, minimum, Q1, median, Q3, maximum

Summary

• Make a picture– Histogram, stem-and-leaf plot, dot plot

• Shape– How many modes?– Symmetric?– Outliers?

• If there are outliers, summarize once with the outliers and another time without the outliers

• Center and spread– Skewed distribution: median and IQR– Symmetric and unimodal distribution: mean and sd

What can go wrong?• Do not use what we learned in chapter 4 for a categorica

l variable– Do not make histogram of a categorical variable– Do not look for shape and center and spread of a bar chart– Do not use mean, sd, IQR, etc. for a categorical variable

• Graph with bars are not always histograms or bar charts• Choose a bin width appropriate to the data• Check the summary numbers. Do they make sense?• Do not worry about small differences when using differen

t methods– No need to use too many digits for the summary numbers– Using one or two more digits than data is enough

• Do not round in the middle of a calculation• Multiple modes, outliers (make a picture)