Section 6A Characterizing a Data Distribution pages 380 - 391

Section 6ASection 6ACharacterizing a Data DistributionCharacterizing a Data Distribution

pages 380 - 391pages 380 - 391

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.

ex1/381 Eight grocery stores sell the PR energy bar for the following prices:

$1.09, $1.29, $1.35, $1.79, $1.49, $1.59, $1.39, $1.29

Price Frequency1.09 11.29 21.35 11.39 11.49 11.59 11.79 1

Price Distribution

0

0.5

1

1.5

2

2.5

1.09 1.29 1.35 1.39 1.49 1.59 1.79

Price of PR Bar

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

AverageAverage

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

Shape of a DistributionShape of a Distribution

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

more in section 6Bmore in section 6B

The mean is what we most commonly call the average value.

What do we mean by AVERAGE?

sum of all valuesmean =

total number of values

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

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

ex1/381 Eight grocery stores sell the PR energy bar for the following prices:

$1.09, $1.29, $1.35, $1.79, $1.49, $1.59, $1.39, $1.29

median: $1.09, $1.29, $1.29, $1.35, $1.39, $1.49, $1.59, $1.79

(1.09+1.29+1.35+1.79+1.49+1.59+1.39+1.29)mean = $1.41

8

(1.35+1.39)$1.37

2 median: $1.37

mode: $1.09, $1.29, $1.29, $1.35, $1.39, $1.49, $1.59, $1.79

mode: $1.29

17/389 High temperatures (oF) during a 15 day period in Alaska in March: 15, 11, 10, 9, 0, 2, 4, 5, 5, 7, 10, 12, 15, 18, 19

o(15+11+10+9+0+2+4+5+5+7+10+12+15+18+19)mean = 9.5( F)

15

median: 0, 2, 4, 5, 5, 7, 9, 10, 10, 11, 12, 15, 15, 18, 19

median: 10 (oF)

mode: 0, 2, 4, 5, 5, 7, 9, 10, 10, 11, 12, 15, 15, 18, 19

modes: 5, 10, 15

trimodal

Temperature Distribution

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Temperature (F)

Nu

mb

er

of

Da

y

17/389 High temperatures (oF) during a 15 day period in Alaska in March: 15, 11, 10, 9, 0, 2, 4, 5, 5, 7, 10, 12, 15, 18, 19

Mean – balancing pointMean – balancing pointMedian – middle pointMedian – middle pointMode – high point(s)Mode – high point(s)

How do we characterize a data distribution?

Average

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

Shape of a Distribution

- Number of Peaks- Symmetry or Skewness- Variation

more in section 6B

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

ex/382 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, 3500000

median: $0

mode: 0, 0, 0, 0, 3500000mode: $0

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

ex2/383 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, 325median: 140

bpm

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

4

Throw out the outlier?

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

mode: none

mode: none


Average




more in section 6B

Confusion about “Average”

ex3/383 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 mean: $23

salaries: $19, $19, $19, $19, outlier

(19+19+19+19+x)23 = mean =

5

23×5= 76 + x

115 - 76 = x

39 = xsalaries: $19, $19, $19, $19, $39

Confusion about “Average”

ex3/383 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 mean: $19

salaries: outlier, $20, $23, $23, $23

(x+20+23+23+23)19 = mean =

5

19×5=89 + x

95 - 89 = x

$6 = xsalaries: $6, $20, $23, $23, $23

Confusion about “Average”ex4/383 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




more in section 6B

Shape of a DistributionUse a smooth curve

Temperature Distribution

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Temperature (F)

Nu

mb

er

of

Da

yShape of a Distribution

Number of Peaks

0

1

2

3

4

5

6

7

A B C D F

Letter Grades

Fre

quency

letter frequencyA 2B 4C 6D 0F 5


Average




more in section 6B

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.

(note positioning of mean, median, and mode.)

SKEWED LEFT(negatively)

Mean Mode Median


A distribution is left-skewed if its values are more spread out on the left (outliers?).


SKEWED RIGHT(positively)

Mean Mode Median


A distribution is right-skewed if its values are more spread out on the right (outliers?).


ex6/387 Do you expect the distribution of heights of 100(20) women to be symmetric, left-skewed, or right-skewed? Explain.

ex6/387 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.



Average




more in section 6B

Shape of a DistributionVariation

Low variation Moderate variation High variation

Variation describes how widely data values are spread out about the center of distribution.

ex7/388 How would you expect the variation to differ between times in the Olympic marathon and times in the New York Marathon? Explain.

Shape of a DistributionNumber of Peaks, Symmetry/Skewness, Variation

27/389 The exam scores on a 100-point exam where 50 students got an A, 20 students got a B, and 5 students got a C.

ex5/385 The heights of all students at Virginia Tech.

ex5/385 The numbers of people with a particular last digit (0 through 9) in their Social Security Number.

a) number of peaks

b) symmetric, left-skewed, or right-skewed

c) small or large variation.


Average




more in section 6B

Homework

Pages 388-391

#14,16,18,20, 28,29,30,31

Section 6A Characterizing a Data Distribution pages 380 - 391

Documents

median mode

outlier confusionshape

data value

average mean

middle values

bthe mean

average value

distribution number