Section 6A Characterizing a Data Distribution

Section 6ASection 6ACharacterizing a Characterizing a Data DistributionData Distribution

Pages 380-388Pages 380-388

Definition -The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values.

Example: Lengths of words in the Gettysburg Address

word length Frequency

1 1

2 5

3 49

4 53

5 59

6 35

7 24

8 19

9 5

10 10

11 7

total 267

word length

Frequency

1086420

60

50

40

30

20

10

0

710

5

19

24

35

59

53

49

5

1

Histogram of word length

How do we characterize a data How do we characterize a data distribution?distribution?

Center =Average

- Mean- Mean- Median- Median- Mode- Mode

- Effect of an Outlier- Effect of an Outlier- Confusion- Confusion

Shape of a Distribution

- Number of Peaks- Number of Peaks- Symmetry or Skewness- Symmetry or Skewness- Variation- Variation

more in section 6Bmore in section 6B

AveragesAverages

The word “Average” actually has The word “Average” actually has several meanings.several meanings.

Generally – Generally –

average = average = centercenter of a of a distributiondistribution

or or typical typical representativerepresentative

6-A

The mean is what we most commonly call the average value. It is defined as follows:

The median is the middle value in the sorted data set (or halfway between the two middle values if the number of values is even).

The mode is the most common value (or group of values) in a distribution.

Measures of Center in a Measures of Center in a DistributionDistribution

6-A

sum of all valuesmeantotal number of values

Mean DistanceMean Distance6-A

PlanetPlanet Distance from sun Distance from sun (millions (millions of miles)of miles)

MercuryMercury 3636

VenusVenus 6767

EarthEarth 9393

MarsMars 142142

JupiterJupiter 484484

SaturnSaturn 887887

UranusUranus 1,7651,765

NeptuneNeptune 2,7912,791

*Pluto*Pluto 3,6543,654

MeanMean distance distance

36 + 67 + 93 + 142+ 484 + 887 + 36 + 67 + 93 + 142+ 484 + 887 + 1,765 + 2,791 + 3,654 1,765 + 2,791 + 3,654

= 9,922= 9,922

Mean distance= 9,922/ 9 Mean distance= 9,922/ 9

= = 1,102.41,102.4 million milesmillion miles

sum of all valuesmeantotal number of values

6-A

Median DistanceMedian Distance6-A


MercuryMercury 3636

VenusVenus 6767

EarthEarth 9393

MarsMars 142142


SaturnSaturn 887887




The median is the middle value in the sorted data set (or halfway between the two middle values if the number of values is even).

Median DistanceMedian Distance6-A


MercuryMercury 3636

VenusVenus 6767

EarthEarth 9393

MarsMars 142142


SaturnSaturn 887887



PlutoPluto 3,6543,654

4 below

4 above

Steps for Finding the Steps for Finding the MedianMedian

1.1. Sort the data (put it in order) !!!!!!Sort the data (put it in order) !!!!!!2.2. Count the data (Count the data (nn pieces). Decide pieces). Decide

if if nn is is oddodd or or eveneven..3.3. If If nn is is oddodd – the median will be in – the median will be in

positionposition (n+1)/2. (n+1)/2.4.4. If If nn is is eveneven – the median will be – the median will be

locatedlocated halfway between the halfway between the numbers in positions n/2 and numbers in positions n/2 and (n+1)/2.(n+1)/2.

6-A

Median Distance – ‘Real’ Planet Median Distance – ‘Real’ Planet ListList

6-A

PlanetPlanet Distance from sun Distance from sun (millions of (millions of miles)miles)

MercuryMercury 3636

VenusVenus 6767

EarthEarth 9393

MarsMars 142142


SaturnSaturn 887887



Median DistanceMedian Distance 6-A

PlanetPlanet Distance from sun (Distance from sun (millions millions of miles)of miles)

MercuryMercury 3636

VenusVenus 6767

MarsMars 9393

EarthEarth 142142


SaturnSaturn 887887



Median is halfway between 142 and 484, so

median = (142+484)/2 = 313

Comment about the Comment about the MedianMedian

1.1. The median splits the data The median splits the data into two equal-sized pieces.into two equal-sized pieces.

2.2. Half the data (50%) will be Half the data (50%) will be below the median.below the median.

3.3. Half the data (50%) will be Half the data (50%) will be above the median.above the median.

6-A

Mode DistanceMode Distance6-A


MercuryMercury 3636

VenusVenus 6767

EarthEarth 9393

MarsMars 142142


SaturnSaturn 887887




The mode is the most common value (or group of values) in a distribution.

Mode ExamplesMode Examples6-A

a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 2 6 6 8 9 10 8


a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 2 6 6 8 9 10 8

Mode is Mode is

55


a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 2 6 6 8 9 10 8

Mode is Mode is

55

BimodalBimodal


a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 2 6 6 8 9 10 8

Mode is 5Mode is 5

Bimodal Bimodal

TrimodalTrimodal

The ModeThe Mode

You may not have one!You may not have one! Could have multiple modes!Could have multiple modes! The mode is easy to spot in a The mode is easy to spot in a

graph – it occurs at the peak.graph – it occurs at the peak. The mode is the only measure The mode is the only measure

of “center” available for of “center” available for categorical datacategorical data – – e.g. gendere.g. gender

6-A

How do we characterize a data distribution?

Average

- Mean- Median- Mode

- Effect of an Outlier- Confusion


- Number of Peaks- Symmetry or Skewness- Variation

OutliersOutliers

An An outlieroutlier is an observation that is is an observation that is much higher (or much lower) than all much higher (or much lower) than all the other values in your list.the other values in your list.

i.e. – an i.e. – an extremely unusualextremely unusual observation.observation.

Note – every not every set of data has Note – every not every set of data has outliers. The minimum and maximum outliers. The minimum and maximum values are not necessarily outliers!!!values are not necessarily outliers!!!

The Effect of an OutlierDefinition: An outlier is a data value that is much higher or much lower than almost all other values.

Five graduating seniors on a college basketball team receive the following first-year contract offers to play in the National Basketball Association: $0, $0, $0, $0, $3,500,000

(0+0+0+0+3500000)mean = $700,000

5

median: 0, 0, 0, 0, $3,500,000 median: $0

mode: 0, 0, 0, 0, $3,500,000

mode: $0

Including an outlier can pull the mean significantly upward or downward.Including an outlier does not significantly affect the median.Including an outlier does not affect the mode.

A track coach wants to determine an appropriate heart rate for her athletes during their workouts. In the middle of the workout, she reads the following heart rates (beats/min) from five athletes: 130, 135, 140, 145, 325.

The Effect of an Outlier

_____________________________________________Cleary 325 is an outlier. Clearly 325 is a mistake (faulty heart monitor?)

(130+135+140+145+325)mean = 175bpm

5

median: 130, 135, 140, 145, 325

median: 140 bpm

(130+135+140+145)mean = 137.5bpm

4

Throw out the outlier?

median: 130, 135, 140, 145 median: 137.5 bpm

mode: none

mode: none


Average

- Mean- Median- Mode- Effect of an Outlier- Confusion



Mean vs. MedianMean vs. MedianA news article reports that of the 411 A news article reports that of the 411

players on the NBA roster in February, players on the NBA roster in February, 1988, only 139 “made more than the 1988, only 139 “made more than the league league average salaryaverage salary of $2.36 of $2.36 million.”million.”

Recall that the word “average” can have Recall that the word “average” can have several interpretations. In this case, is several interpretations. In this case, is $2.36 million the $2.36 million the meanmean or the or the median median salarysalary for 1988 NBA players? for 1988 NBA players? Explain.Explain.

6-A

Confusion about “Average”

A newspaper surveys wages for assembly workers and reports an average of $22 per hour. The workers at one large firm immediately request a pay raise, claiming that they work as hard as other companies but their average wage is only $19. The management rejects their request, telling them that they are overpaid because their average wage, in fact is $23 per hour. Can they both be right?

median: $19

salaries: $19, $19, $19, $19, 39

(19+19+19+19+39) $115mean = $23

5 5

Confusion about “Average”

A newspaper survey wages for assembly workers and reports an average of $22 per hour. The workers at one large firm immediately request a pay raise, claiming that they work as hard as other companies but their average wage is only $19. The management rejects their request, telling them that they are overpaid because their average wage, in fact is $23 per hour. Can they both be right?

median: $23

salaries: $6, $20, $23, $23, $23

(6+20+23+23+23) $95mean = $19

5 5

Confusion about “Average”All 100 first-year students at a small college take three courses in the Core Studies Program. The first two courses are taught in large lectures, with all 100 students in a single class. The third course is taught in ten classes of 10 students each. The students claim that the mean size of their Core Studies classes is 70. The administrators claim that the mean class size is only 25 students. Explain.

Students say my average class size is:

(100+100+ 10)70

3

Administrators say the average Core Studies class size is:

(total students enrolled in all Core Studies classes) 30025

(number of Core Studies classes) 12

mean class size per student

mean number of students per class


Average

- Mean- Median- Mode- Effect of an Outlier- Confusion



Describing a distributionDescribing a distribution6-A

Shape of a DistributionSymmetry and Skewness

Mode = Mean = Median

SYMMETRIC

A distribution is symmetric if its left half is a mirror image of its right half.

SKEWED LEFT(negatively)

Mean Mode Median


A distribution is left-skewed if its ‘tail’ is on the left.

SKEWED RIGHT(positively)

Mean Mode Median


A distribution is right-skewed if its ‘tail’ is on the right.

Symmetric and Skewed Symmetric and Skewed DistributionsDistributions

6-A

Mode = Mean = Median

SYMMETRIC

SKEWED LEFT(negatively)

Mean Mode Median

SKEWED RIGHT(positively)

Mean Mode Median

Use Mean to describe center

Use Median to describe center

Do you expect the distribution of heights of 100 women to be symmetric, left-skewed, or right-skewed? Explain.

Do you expect the distribution of speeds of cars on a road where a visible patrol car is using radar to be symmetric, left-skewed, or right skewed. Explain.


Variation = horizontal spreadVariation = horizontal spread

6-A

High variationModerate variation

Low variation

How would you expect the variation to differ between times in the Olympic marathon and times in the New York Marathon? Explain.

Describing a distributionDescribing a distribution ShapeShape

Number of peaks, symmetry/skewnessNumber of peaks, symmetry/skewness Outliers?Outliers?

CenterCenter Use Use meanmean if the data is if the data is symmetricsymmetric Use Use medianmedian is there is a is there is a strong skewstrong skew or or

are are outliersoutliers Spread

Horizontal spread – Is the data tightly clustered around the center? (low or high variation?)

6-A

HomeworkHomework

Pages 388-390Pages 388-390

# 10, 14, 18, 21, 22, 27, 28, 30, # 10, 14, 18, 21, 22, 27, 28, 30, 35, 38*35, 38*

* It is not necessary to draw the * It is not necessary to draw the sketch for this one.sketch for this one.

6-A

Section 6A Characterizing a Data Distribution

Documents

median mode

data value

mode examples mode

set of data

middle values

categorical data

number of values

data distributionpages