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Unformatted text preview: Chapter 3
Descriptive Statistics: Numerical Methods Slide 1 Learning Objectives
1.
2.
3.
4. 5. Understand the purpose of measures of location. Be able to compute the mean, median, mode, quartiles, and various percentiles. Understand the purpose of measures of variability. Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of variation. Understand how z scores are computed and how they are used as a measure of relative location of a data value. Slide 2 (Continued)
6. 7.
8. 9. Know how Chebyshev's theorem and the empirical rule can be used to determine the percentage of the data within a specified number of standard deviations from the mean. Learn how to construct a 5number summary and a box plot. Be able to compute and interpret covariance and correlation as measures of association between two variables. Be able to compute a weighted mean. Slide 3 Major Contents
s
s
s
s
s
s Measures of Location
Measures of Variability
Measures of Relative Location and Detecting Outliers
Exploratory Data Analysis
Measures of Association Between Two Variables
The Weighted Mean and Working with Grouped Data σ
µ % x Slide 4 Measures of Location
s
s
s
s
s Mean （（（（
Median （（（（（
Mode （（（（
Percentiles （（（（（（
Quartiles （（（（（（ Slide 5 Example: Apartment Rents
Given below is a sample of monthly rent values ($)
for onebedroom apartments. The data is a sample of 70
apartments in a particular city. The data are presented
in ascending order. 425
440
450
465
480
510
575 430
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515
575 430
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580 435
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590 435
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600 435
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600 435
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600 435
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600 440
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615 Slide 6 Mean
The mean of a data set is the average of all the data values.
s If the data are from a sample, the mean is denoted by .
s x s ∑ xi
x=
n
If the data are from a population, the mean is denoted by µ (mu). ∑ xi
µ=
N Slide 7 Example: Apartment Rents
s Mean ∑ xi 34 , 356
x=
=
= 490.80
n
70 425
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510
575 430
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515
575 430
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580 435
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590 435
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525
600 435
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535
600 435
445
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549
600 435
445
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600 440
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615 440
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615 Slide 8 Median
s s The median is the measure of location most often reported for annual income and property value data.
A few extremely large incomes or property values can inflate the mean. Slide 9 Median
s s s The median of a data set is the value in the middle when the data items are arranged in ascending order.
For an odd number of observations, the median is the middle value.
For an even number of observations, the median is the average of the two middle values. Slide 10 Example: Apartment Rents
s Median Median = 50th percentile
i = (p/100)n = (50/100)70 = 35.5 Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 11 Mode
s s s
s The mode of a data set is the value that occurs with greatest frequency.
The greatest frequency can occur at two or more different values.
If the data have exactly two modes, the data are bimodal.
If the data have more than two modes, the data are multimodal. Slide 12 Example: Apartment Rents
Mode 450 occurred most frequently (7 times) Mode = 450
s 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 13 Percentiles
s s A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.
Admission test scores for colleges and universities are frequently reported in terms of percentiles. Slide 14 Percentiles
s The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 p) percent of the items take on this value or more.
• Arrange the data in ascending order.
• Compute index i, the position of the pth percentile. i = (p/100)n • If i is not an integer, round up. The p th percentile is the value in the i th position.
If i is an integer, the p th percentile is the average of the values in positions i and i +1. • Slide 15 Example: Apartment Rents
s 90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585
425
440
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480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
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525
600 435
445
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500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
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510
570
615 Slide 16 Quartiles
s
s
s
s Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile Slide 17 Example: Apartment Rents
s Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525
425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
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490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
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475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Slide 18 Measures of Variability
s s It is often desirable to consider measures of variability (dispersion), as well as measures of location.
For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each. Suppl i er B Suppl i er A
6 3. 5 5 3
2. 5
Fr e qu e nc y Fr e que nc y 4
3 2
1. 5 2 1 1 0. 5
0 0
6 7 8 9 10
D
ays 11 12 13 7 8 9 10
Days 11 12 13 Slide 19 Measures of Variability
s
s
s
s
s Range( （（ )
Interquartile Range （（（（（（（
Variance （（（（
Standard Deviation （（（（（
Coefficient of Variation （（（（（（ Slide 20 Range
s s
s The range of a data set is the difference between the largest and smallest data values.
It is the simplest measure of variability.
It is very sensitive to the smallest and largest data values. Slide 21 Example: Apartment Rents
s Range Range = largest value smallest value Range = 615 425 = 190 425
440
450
465
480
510
575 430
440
450
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485
515
575 430
440
450
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525
580 435
445
450
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490
525
590 435
445
450
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490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 435
445
460
480
500
550
600 440
450
465
480
500
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615 440
450
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510
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615 Slide 22 Interquartile Range
s s
s The interquartile range of a data set is the difference between the third quartile and the first quartile.
It is the range for the middle 50% of the data.
It overcomes the sensitivity to extreme data values. Slide 23 Example: Apartment Rents
Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 Q1 = 525 445 = 80
s 425
440
450
465
480
510
575 430
440
450
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485
515
575 430
440
450
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525
580 435
445
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490
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445
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600 435
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600 435
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600 435
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615 440
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615 Slide 24 Variance
s s The variance is a measure of variability that utilizes all the data.
It is based on the difference between the value of each observation (xi) and the mean (x for a sample, µ for a population). Slide 25 Variance
s s The variance is the average of the squared differences between each data value and the mean.
If the data set is a sample, the variance is denoted by s2. 2 s 2 ∑ ( xi − x )
s=
n −1
If the data set is a population, the variance is denoted by σ 2. ∑ ( xi − µ )
σ2 =
N 2 Slide 26 Standard Deviation
s s s s The standard deviation of a data set is the positive square root of the variance.
It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean.
If the data set is a sample, the standard deviation is denoted s. s = s2
If the data set is a population, the standard deviation is denoted σ (sigma). σ = σ2 Slide 27 Coefficient of Variation
s s The coefficient of variation indicates how large the standard deviation is in relation to the mean.
If the data set is a sample, the coefficient of variation is computed as follows: s
(100)
x
s If the data set is a population, the coefficient of variation is computed as follows: σ
(100)
µ Slide 28 Example: Apartment Rents
s Variance s
s 2 ∑
= ( xi − x ) 2
n −1 = 2 , 996.16 Standard Deviation s = s2 = 2996. 47 = 54. 74
s Coefficient of Variation s
54. 74
× 100 =
× 100 = 11.15
x
490.80 Slide 29 Measures of Relative Location
and Detecting Outliers
s
s
s
s zScores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers Slide 30 zScores
s
s The zscore is often called the standardized value.
It denotes the number of standard deviations a data value xi is from the mean. xi − x
zi =
s s s s A data value less than the sample mean will have a zscore less than zero.
A data value greater than the sample mean will have a z
score greater than zero.
A data value equal to the sample mean will have a zscore of zero. Slide 31 Example: Apartment Rents
s zScore of Smallest Value (425)
xi − x 425 − 490.80
z=
=
= −1. 20
s
54. 74 Standardized Values for Apartment Rents 1.20
0.93
0.75
0.47
0.20
0.35
1.54 1.11
0.93
0.75
0.38
0.11
0.44
1.54 1.11
0.93
0.75
0.38
0.01
0.62
1.63 1.02
0.84
0.75
0.34
0.01
0.62
1.81 1.02
0.84
0.75
0.29
0.01
0.62
1.99 1.02
0.84
0.56
0.29
0.17
0.81
1.99 1.02
0.84
0.56
0.29
0.17
1.06
1.99 1.02
0.84
0.56
0.20
0.17
1.08
1.99 0.93
0.75
0.47
0.20
0.17
1.45
2.27 0.93
0.75
0.47
0.20
0.35
1.45
2.27 Slide 32 Chebyshev’s Theorem At least (1 1/z2) of the items in any data set will be
within z standard deviations of the mean, where z is
any value greater than 1.
• At least 75% of the items must be within
z = 2 standard deviations of the mean.
• At least 89% of the items must be within
z = 3 standard deviations of the mean.
• At least 94% of the items must be within
z = 4 standard deviations of the mean. Slide 33 Example: Apartment Rents
s Chebyshev’s Theorem x Let z = 1.5 with = 490.80 and s = 54.74
At least (1 1/(1.5)2) = 1 0.44 = 0.56 or 56% of the rent values must be between
x z(s) = 490.80 1.5(54.74) = 409 and
x + z(s) = 490.80 + 1.5(54.74) = 573 Slide 34 Example: Apartment Rents
Chebyshev’s Theorem (continued) Actually, 86% of the rent values are between 409 and 573. s 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
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490
525
580 435
445
450
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525
590 435
445
450
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490
525
600 435
445
460
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535
600 435
445
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600 435
445
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600 440
450
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615 440
450
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615 Slide 35 Empirical Rule For data having a bellshaped distribution: • Approximately 68% of the data values will be within one standard deviation of the mean. Slide 36 Empirical Rule
For data having a bellshaped distribution: • Approximately 95% of the data values will be within two standard deviations of the mean. Slide 37 Empirical Rule
For data having a bellshaped distribution: • Almost all (99.7%) of the items will be within three standard deviations of the mean. Slide 38 Example: Apartment Rents
s Empirical Rule Interval Within +/ 1s
436.06 to 545.54
Within +/ 2s
381.32 to 600.28
Within +/ 3s
326.58 to 655.02 425
440
450
465
480
510
575 430
440
450
470
485
515
575 430
440
450
470
490
525
580 435
445
450
472
490
525
590 435
445
450
475
490
525
600 435
445
460
475
500
535
600 435
445
460
475
500
549
600 % in Interval
48/70 = 69%
68/70 = 97%
70/70 = 100%
435
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600 440
450
465
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615 440
450
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615 Slide 39 Detecting Outliers
s s s An outlier is an unusually small or unusually large value in a data set.
A data value with a zscore less than 3 or greater than +3 might be considered an outlier. It might be:
• an incorrectly recorded data value
• a data value that was incorrectly included in the data set
• a correctly recorded data value that belongs in the data set Slide 40 Example: Apartment Rents
s Detecting Outliers
The most extreme zscores are 1.20 and 2.27.
Using z > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents 1.20
0.93
0.75
0.47
0.20
0.35
1.54 1.11
0.93
0.75
0.38
0.11
0.44
1.54 1.11
0.93
0.75
0.38
0.01
0.62
1.63 1.02
0.84
0.75
0.34
0.01
0.62
1.81 1.02
0.84
0.75
0.29
0.01
0.62
1.99 1.02
0.84
0.56
0.29
0.17
0.81
1.99 1.02
0.84
0.56
0.29
0.17
1.06
1.99 1.02
0.84
0.56
0.20
0.17
1.08
1.99 0.93
0.75
0.47
0.20
0.17
1.45
2.27 0.93
0.75
0.47
0.20
0.35
1.45
2.27 Slide 41 Exploratory Data Analysis
s
s FiveNumber Summary
Box Plot Slide 42 FiveNumber Summary
s
s
s
s
s Smallest Value
First Quartile
Median
Third Quartile
Largest Value Slide 43 Example: Apartment Rents
s FiveNumber Summary
Lowest Value = 425 First Quartile = 450 Median = 475
Third Quartile = 525 Largest Value = 615 425
440
450
465
480
510
575 430
440
450
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485
515
575 430
440
450
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490
525
580 435
445
450
472
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525
590 435
445
450
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490
525
600 435
445
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500
535
600 435
445
460
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549
600 435
445
460
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500
550
600 440
450
465
480
500
570
615 440
450
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510
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615 Slide 44 Box Plot
s s s A box is drawn with its ends located at the first and third quartiles.
A vertical line is drawn in the box at the location of the median.
Limits are located (not drawn) using the interquartile range (IQR).
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• Data outside these limits are considered outliers.
… continued Slide 45 Box Plot (Continued)
s Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. s The locations of each outlier is shown with the symbol * . Slide 46 Example: Apartment Rents
s Box Plot Lower Limit: Q1 1.5(IQR) = 450 1.5(75) = 337.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
There are no outliers. 37
5 40
0 42
5 45
0 47
5 50
0 52
5 550 575 600 625 Slide 47 Measures of Association Between Two Variables
s
s Covariance
Correlation Coefficient Slide 48 Covariance
s s
s The covariance is a measure of the linear association between two variables.
Positive values indicate a positive relationship.
Negative values indicate a negative relationship. Slide 49 Covariance
s If the data sets are samples, the covariance is denoted by sxy.
∑ ( xi − x ) ( yi − y )
sxy =
n −1 s If the data sets are populations, the covariance is denoted σ xy
by . σ xy ∑ ( xi − µ x ) ( yi − µ y )
=
N Slide 50 Correlation Coefficient
s
s
s
s s The coefficient can take on values between 1 and +1.
Values near 1 indicate a strong negative linear relationship.
Values near +1 indicate a strong positive linear relationship.
If the data sets are samples, the coefficient is rxy. r = sxy xy
If the data sets are populations, the coefficient is .
ss
xy ρ xy = σ xy ρ xy σ xσ y Slide 51 The Weighted Mean and
Working with Grouped Data
s
s
s
s Weighted Mean
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data Slide 52 Weighted Mean
s s s When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.
In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade.
When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. Slide 53 Weighted Mean x = Σ wi xi Σ wi
where: xi = value of observation i wi = weight for observation i Slide 54 Grouped Data
s s s s The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data.
To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class.
We compute a weighted mean of the class midpoints using the class frequencies as weights.
Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. Slide 55 Mean for Grouped Data
s Sample Data ∑fM
x=
∑f
i i i s Population Data ∑fM
µ=
i i N where: fi = frequency of class i Mi = midpoint of class i Slide 56 Example: Apartment Rents
Given below is the previous sample of monthly rents
for onebedroom apartments presented here as grouped
data in the form of a frequency distribution. Rent ($) Frequency
420439
440459
460479
480499
500519
520539
540559
560579
580599
600619 8
17
12
8
7
4
2
4
2
6 Slide 57 Example: Apartment Rents
s Mean for Grouped Data
Rent ($)
420439
440459
460479
480499 500519
520539
540559
560579
580599
600619
Total fi
8
17
12
8
7
4
2
4
2
6
70 Mi
429.5
449.5
469.5
489.5
509.5
529.5
549.5
569.5
589.5
609.5 f iM i
34 , 525
3436.0
x=
= 493. 21
7641.5
70
5634.0 This approximation
3916.0 differs by $2.41 from
3566.5
2118.0 the actual sample
1099.0 mean of $490.80.
2278.0
1179.0
3657.0
34525.0 Slide 58 Variance for Grouped Data
s Sample Data ∑ f i ( Mi − x ) 2
s2 =
n −1
s Population Data ∑ f i ( Mi − µ ) 2
σ2 =
N Slide 59 Example: Apartment Rents
s Variance for Grouped Data s 2 = 3, 017.89
s Standard Deviation for Grouped Data s = 3, 017.89 = 54. 94
This approximation differs by only $.20 from the actual standard deviation of $54.74. Slide 60 Exercises
1. The average value of a data set is called the a. mean b. median c. mode d. range e. percentile Slide 61 s Correct Answer:
a. mean Slide 62 2. The most frequently occurring data value in a data set is the a. median b. arithmetic mean c. population parameter d. range e. mode Slide 63 s Correct Answer:
e. mode Slide 64 3. A measure of central location which splits the data set into two equal groups is called the a. mean b. mode c. median d. standard deviation e. none of the above Slide 65 s Correct Answer:
c. median Slide 66 4. The coefficient of variation is a. the same as the variance b. a measure of central tendency c. a measure of absolute variability d. a measure of relative variability e. none of the above Slide 67 s Correct Answer:
d. a measure of relative variability Slide 68 5. The sum of the deviation of the individual data elements from their mean is always a. equal to zero b. equal to one c. negative d. positive e. none of the above Slide 69 s Correct Answer:
a. equal to zero Slide 70 6. The ratio of the standard deviation to the mean is a. the variance
b. the range c. the coefficient of variation d. always greater than 1 e. none of the above Slide 71 s Correct Answer:
c. the coefficient of variation Slide 72 s 7. If a covariance is negative then, in general, as a. x increases, y increases. b. y increases, x increases. c. x increases, y decreases. d. None of the above Slide 73 s Correct Answer:
c. x increases, y decreases. Slide 74 s 8. A sample of five price/earnings ratios for companies in the Services sector follows. 37 11 14 17 12
Find the sample standard deviation. a. 9.62 b. 10.76 c. 11.5 d. 115.7 e. None of the above Slide 75 s Correct Answer:
b. 10.76 Slide 76 9. A sample of five price/earnings ratios for companies in the Services sector follows. 37 11 14 17 12
Find the sample median. a. 10.76 b. 14 c. 18 d. 18.2 e. None of the above Slide 77 s Correct Answer:
b. 14 Slide 78 10. The wages of social workers at a large nonprofit agency average $25,000 with a standard deviation of $2000. At least what percentage of social workers earns between $21,000 and $29,000. a. 68% b. 75% c. 89% d. 95% e. The answer cannot be determined with the given information. Slide 79 s Correct Answer:
b. 75% Slide 80 11. The wages of social workers at a large nonprofit agency average $25,000 with a standard deviation of $2000. Assuming that the distribution of wages is mound shaped, approximately what percentage of social workers earns between $21,000 and $29,000. a. 68% b. 75% c. 89% d. 95% e. The answer cannot be determined with the given information. Slide 81 s Correct Answer:
d. 95% Slide 82 12. At a new baseball stadium, seats are priced as follows: Section Number of Seats
Price Lower Box 15,000
$25.00 Upper Box 20,000
$15.00 Bleachers 5,000 $10.00 40,000 Find the weighted mean price of a seat at the stadium. a. $15.00 b. $16.67 c. $17.42 d. $18.13 Slide 83 s Correct Answer:
d. $18.13 Slide 84 ...
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