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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
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
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
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
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
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
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
Shape of a DistributionSymmetry and Skewness
A distribution is left-skewed if its values are more spread out on the left (outliers?).
(note positioning of mean, median, and mode.)
SKEWED RIGHT(positively)
Mean Mode Median
Shape of a DistributionSymmetry and Skewness
A distribution is right-skewed if its values are more spread out on the right (outliers?).
(note positioning of mean, median, and mode.)
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.
Shape of a DistributionSymmetry and Skewness
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
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.
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
Homework
Pages 388-391
#14,16,18,20, 28,29,30,31