Chapter 3

Describing Distributions Numerically

Describing the Distribution

• Center– Median– Mean

• Spread– Range– Interquartile Range– Standard Deviation

Median

• Literally = middle number (data value)• 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

Example – August Temps

• High Temperatures for Des Moines, Iowa taken from the first 13 days of August 2005.

71 76 81 81 85 86 90 90 91 93 93 96 96

Remember to order the values, if they aren’t already in order!

• 13 observations

– (13+1)/2 = 7th observation from the bottom

• Median = 90

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 – Yankees

•Scores of last 10 games2 3 3 5 5 5 6 7 7 10

•Remember to order the values if they aren’t already in order!

• 10 observations– (10 + 1)/2 = 5.5,

average of 5th and 6th observations from bottom

• Median = 5

Mean

• Ordinary average– Add up all observations– Divide by the number of observations

• Formula – n observations

– y1, y2, y3, …, yn are the values

Mean ( )

xn

n

xxxxn

xx

n

1

321

x

Example – Vikings (as of 1/9)

• Find the mean of the (17 values)

13 14 16 18 20 22 23 27

27 28 28 31 31 31 34 35 38

65.2517

)38...18161413(

x

Example – Colts as of (1/9)

• Find the mean of the scores (17 values)14 20 23 24 24 24 31 31 34 35 35 41 41 45 49 49 51

59.3317

)51...24232014(

x

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

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

Weighted Mean

• Used when values are not equally represented.

• Weighted mean =

n

nn

www

xwxwxw

w

wXX

...

...

21

2211

Example (weighted mean)

Area % Favored Number surveyed

1 40 1000

2 30 3000

3 50 800

A recent survey of new diet cola reported the following percentages of people who liked the taste.

Find the weighted mean of the percentages.

Example (cont.)

x1 = .40

x2 = .30

x3 = .50

w1 = 1000

w2 = 3000

w3 = 800

Use formula:

{.40(1000) + .30(3000) + .50(800)} / {1000+3000+800}

= 1700/4800

= 0.354 = 35.4%

Spread

• Range is a very basic measure of spread (Max – Min).– 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• Range without outlier: 94 – 25 = 69

Spread

• Interquartile Range (IQR)– First Quartile (Q1)

• 25th Percentile

– Third Quartile (Q3)• 75th Percentile

• IQR = Q3 – Q1– Center (Middle) 50% of the values

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

Top 15 Populations US Cities 2004New York, N.Y. 810

Los Angeles, Calif. 385

Chicago, Ill. 286

Houston, Tex. 201

Philadelphia, Pa. 147

Phoenix, Ariz. 142

San Diego, Calif. 126

San Antonio, Tex. 124

Dallas, Tex. 121

San Jose, Calif. 90

Detroit, Mich. 90

Indianapolis, Ind. 78

Jacksonville, Fla. 78

San Francisco, Calif. 74

* Populations were all divided by 10,000.

Example – Top City Populations

• Order the values (14 values) 74 78 78 90 90 121 124 126 142 147

201 286 385 810

Lower Half = 74 78 78 90 90 121 124 Q1 = Median of lower half = 90

• Upper Half = 126 142 147 201 286 385 810– Q3 = Median of upper half = 201

• IQR = Q3 – Q1 = 201 - 90 = 111

August High Temps (8/1–8/13)

• Order the values (13 values)

71 76 81 81 85 86 90 90 91 93 93 96 96

• Lower Half = 71 76 81 81 85 86– Q1 = Median of lower half = 81

• Upper Half = 90 90 91 93 93 96 96– Q3 = Median of upper half = 93

• IQR = Q3 – Q1 = 93 - 81 = 12

August High Temps (8/14–8/25)

• Order the values (12 values) 76 77 77 79 81 83 84 85 86 88 91 93

• Lower Half = 76 77 77 79 81 83– Q1 = Median of lower half = 78

• Upper Half = 84 85 86 88 91 93– Q3 = Median of upper half = 87

• IQR = Q3 – Q1 = 87-78 = 9

Five Number Summary

• Minimum

• Q1

• Median

• Q3

• Maximum

Examples

• Vikings (as of 1/9)– Min = 13 – Q1 = 20– Median = 27 – Q3 = 31 – Max = 38

• Colts (as of 1/9)– Min = 14– Q1 = 24– Median = 34– Q3 = 41– Max = 51

Graph of Five Number Summary

• Boxplot– Box between Q1 and Q3– Line in the box marks the median– Lines extend out to minimum and

maximum

• Best used for comparisons

• Use this simpler method

Example – Vikings & Colts

• Boxplot of Vikings scores– Box from 20 to 31– Line in box 27– Lines extend out from box from 14 and 38

• Boxplot of Colts scores– Box from 24 to 41– Line in box at 34– Lines extend out from box to 14 and 51

Side by Side Boxplots of Vikings Scores and Colts Scores

Spread

• Standard deviation– “Average” spread from mean – Most common measure of spread– Denoted by letter s– Make a table when calculating by hand

Standard Deviation

1

)()()(

1

1

1

222

21

2

2

n

xxxxxx

n

xx

xxn

s

n

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

x )( xx

97.31111

71.26425.29869.10

s

2)( xx

Example - Vikings

• Find the standard deviation of the scores of Vikings games given the following statistic:

2436.732)( 2 xx

77.6117

2436.732

1

)( 2

n

xxs

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• s without outlier: 21.70

Which summaries should you use with different distributions?

• The appropriate measures of center and spread when your distribution is symmetric are:

– Mean– Standard deviation

• The appropriate measures of center and spread when your distribution is skewed are:

– Median– IQR

Comparing Variance

• When comparing the variance for two sets of numbers find the coefficient of variation:

• Formula = Cvar = =

– Then compare the percentages.

x

s

Standardizing (first look)

• I got a 85 on my English test and you got a 36 on your Spanish test. Who did better?

• How can we compare things that come from different scales?

• Standardizing– Use z formula (called z-score)

s

xxz

Standardizing

Z=standardized score

X = raw score

X-bar = mean of raw scores

S = sample standard deviation

• So what does this mean for our test scores?

s

xxz

Standardizing

• I got a 85 on my English test and you got a 35 on your Spanish test. Who did better?

• Now I need to give you more information.

• The English class’s tests had a mean of 83 and a standard deviation of 3.

• The Spanish tests had a mean of 30 and a standard deviation of 2.

Standardizing

5.22/52

3035

667.03/23

8385

z

z

Comparing Standardized Scores

• I scored .667 standard deviations above the mean on my English test where you scored 2.5 standard deviations above the mean on your Spanish test.

• Comparatively you scored better on your exam.