Unformatted text preview: Chapter 3
Descriptive Statistics: Numerical
Measures
Part A
• Measures of Location
• Measures of Variability Measures of Location
•
•
•
•
• Mean
Median
Mode
Percentiles
Quartiles If the measures are computed for data from a sample,
they are called sample statistics.
If the measures are computed for data from a population,
they are called population parameters. A sample statistic is referred to
as the point estimator of the
corresponding population parameter. Mean
• The mean of a data set is the average of all the data values.
x
• The sample mean, , is the point estimator of the population mean, µ . Sample Mean x x= ∑x Sum of the values
of the n observations
i n
Number of
observations
in the sample Population Mean µ µ= ∑x Sum of the values
of the N observations
i N
Number of
observations in
the population Sample Mean Example: Apartment Rents Seventy efficiency apartments
were randomly sampled in
a small college town. The
monthly rent prices for
these apartments are listed
on the next slide. Sample Mean Apartment Rent Sample Data
445
440
465
450
600
570
510 615
440
450
470
485
515
575 430
440
525
490
580
450
490 590
525
450
472
470
445
435 435
425
450
475
490
525
600 600
445
460
475
500
535
435 460
575
435
500
549
475
445 600
445
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500
550
435 440
450
465
570
500
480
430 615
450
480
465
480
510
440 Sample Mean
∑x
x= 34, 356
=
= 490.80
n
70 445
440
465
450
600
570
510 615
440
450
470
485
515
575 430
440
525
490
580
450
490 590
525
450
472
470
445
435 i 435
425
450
475
490
525
600 600
445
460
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500
535
435 460
575
435
500
549
475
445 600
445
460
480
500
550
435 440
450
465
570
500
480
430 615
450
480
465
480
510
440 Median The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. 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. Median For an odd number of observations: 26 18 27 12 14 27 19
12 14 18 19 26 27 27
the median is the middle value.
Median = 19 7 observations
in ascending order Median For an even number of observations: 26 18 27 12 14 27 30 19
12 14 18 19 26 27 27 30 8 observations
in ascending order the median is the average of the middle two values.
Median = (19 + 26)/2 = 22.5 Median
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Mode 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. Mode
450 occurred most frequently (7 times)
Mode = 450
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Percentiles 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. Percentiles
• 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. Percentiles 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. 80th Percentile
i = (p/100)n = (80/100)70 = 56
Averaging the 56th and 57th data values:
80th Percentile = (535 + 549)/2 = 542
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 80 Percentile
th “At least 80% of the items take on a value of 542 or less.”
56/70 = .8 or 80%
425
440
450
465
480
510
575 “At least 20% of the items take on a value of 542 or more.”
14/70 = .2 or 20% 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 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile 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
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Measures of Variability 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. Measures of Variability
•
•
•
•
• Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation Range 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. Range
Range = largest value smallest value
Range = 615 425 = 190
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Interquartile Range 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. Interquartile Range
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 Q1 = 525 445 = 80
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 Note: Data is in ascending order. 440
450
465
480
500
570
615 440
450
465
480
510
570
615 Variance 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 ( for a sample,
x µ for a population). Variance The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows:
∑ ( xi − x ) 2
s2 =
n −1 ∑ ( xi − µ )
σ=
N
2 for a
sample for a
population 2 Standard Deviation 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 interpreted than the variance. Standard Deviation The standard deviation is computed as follows: s = s2 σ = σ2 for a
sample for a
population Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: s ×100 %
x σ ×100 %
µ for a
sample for a
population Variance, Standard Deviation,
And Coefficient of Variation • Variance
s2 ∑( x
= − x )2
i
= 2, 996.16
n −1 • Standard Deviation
s = s 2 = 2996.16 = 54.74 • Coefficient of Variation the standard
deviation is
about 11% of the mean s 54.74 × 100 % = × 100 % = 11.15% x 490.80 Chapter 3
Descriptive Statistics: Numerical
Measures Part B • Measures of Distribution Shape, Relative Location, and Detecting Outliers
• Exploratory Data Analysis
• Measures of Association Between Two Variables
• The Weighted Mean and Working with Grouped Data •
•
•
•
• Measures of Distribution Shape,
Relative Location, and Detecting
Outliers
Distribution Shape
zScores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers Distribution Shape: Skewness
• An important measure of the shape of a distribution is called skewness.
• The formula for computing skewness for a data set is somewhat complex.
• Skewness can be easily computed using statistical software.
• Excel’s SKEW function can be used to compute the skewness of a data set. Distribution Shape: Skewness
s Symmetric (not skewed)
• Skewness is zero.
• Mean and median are equal. Relative Frequency .35 Skewness = 0 .30
.25
.20
.15
.10
.05
0 Distribution Shape: Skewness
• Moderately Skewed Left – Skewness is negative.
– Mean will usually be less than the median.
Relative Frequency .35 Skewness = − .31 .31 .30
.25
.20
.15
.10
.05
0 Distribution Shape: Skewness
• Moderately Skewed Right – Skewness is positive.
– Mean will usually be more than the median. Relative Frequency .35 Skewness = .31 .30
.25
.20
.15
.10
.05
0 Distribution Shape: Skewness
s Highly Skewed Right
• Skewness is positive (often above 1.0).
• Mean will usually be more than the median. Relative Frequency .35 Skewness = 1.25 .30
.25
.20
.15
.10
.05
0 Distribution Shape: Skewness
• Example: Apartment Rents Seventy efficiency apartments
were randomly sampled in
a small college town. The
monthly rent prices for
these apartments are listed
in ascending order on the next slide. Distribution Shape: Skewness
• Example: Apartment 425
430
Rents 430 435 435 435
440
450
465
480
510
575 440
450
470
485
515
575 440
450
470
490
525
580 445
450
472
490
525
590 445
450
475
490
525
600 445
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 Distribution Shape: Skewness Relative Frequency .35 Skewness = .92 .30
.25
.20
.15
.10
.05
0 zScores 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 zScores An observation’s zscore is a measure of the relative location of the observation in a data set. 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 zscore greater than zero. A data value equal to the sample mean will have a zscore of zero. zScores
• zScore of Smallest Value (425)
x i − 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 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. Chebyshev’s Theorem
75% At least of the data values must be z = 2 standard deviations within of the mean.
89% At least of the data values must be z = 3 standard deviations within of the mean.
94% At least of the data values must be z = 4 standard deviations within of the mean. Chebyshev’s Theorem
For example: 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
(Actually, 86% of the rent values are between 409 and 573.) Empirical Rule For data having a bellshaped distribution:
68.26% of the values of a normal random variable +/ 1 standard deviation are within of its mean.
95.44% of the values of a normal random variable
+/ 2 standard deviations are within of its mean.
99.72% of the values of a normal random variable
+/ 3 standard deviations are within of its mean. Empirical Rule
99.72%
95.44%
68.26% µ – 3σ
µ – 1σ
µ – 2σ µ µ + 3σ
µ + 1σ
µ + 2σ x Detecting Outliers 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 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 Measures of Association
Between Two Variables
• Covariance
• Correlation Coefficient Covariance The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Covariance The covariance is computed as follows:
sxy = ∑ ( xi − x ) ( yi − y )
n −1 for
samples σ xy ∑ ( xi − µ x ) ( yi − µ y )
=
N for
populations Correlation Coefficient 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. Correlation Coefficient The correlation coefficient is computed as follows: rxy = sxy ρ xy = sx s y for
samples σ xy
σ xσ y for
populations Correlation Coefficient Correlation is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it does not mean that one variable is the cause of the other. Covariance and Correlation
Coefficient A golfer is interested in investigating
the relationship, if any, between driving
distance and 18hole score.
Average Driving
Distance (yds.)
277.6
259.5
269.1
267.0
255.6
272.9 Average
18Hole Score
69
71
70
70
71
69 Covariance and Correlation
Coefficient
x y 277.6
259.5
269.1
267.0
255.6
272.9 69
71
70
70
71
69 ( x i − x ) ( y i − y ) ( x i − x )( y i − y ) 10.65 7.45 2.15 0.05
11.35 5.95 Average
267.0 70.0
Std. Dev. 8.2192 .8944 1.0 1.0 0 0 1.0
1.0 10.65 7.45 0 0
11.35 5.95
Total 35.40 Covariance and Correlation
Coefficient
• Sample Covariance
∑ (x i − x )( y i − y ) = −35.40 = − 7.08
sxy =
n−1
6−1
• Sample Correlation Coefficient
sxy
−7.08
rxy =
=
= .9631
sx sy (8.2192)(.8944) ...
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 Spring '09
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