04 Describing Data Graphically and Numerically Part 3

# Example the mean number of days required to fill

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Example: The mean number of days required to fill orders is 10.3 days for both of the suppliers. Which supplier would you prefer?

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Range 26 Simplest measure of variation Difference between the largest and the smallest observations: Range = Largest value – Smallest value Example: For our previous example data set: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 The range is 3925 – 3310 = 615
Disadvantages of Range 27 Ignores the way in which data are distributed: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 Range = 3925 – 3310 = 615 3310 3500 3500 3500 3500 3500 3500 3500 3500 3500 3500 3925 Range = 3925 – 3310 = 615 Sensitive to outliers: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 Range = 3925 – 3310 = 615 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 10000

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Interquartile Range 28 A measure of variability that overcomes the dependency on extreme values is the interquartile range (IQR ) IQR is the difference between the third quartile, Q 3 , and the first quartile, Q 1 IQR = Q 3 Q 1 IQR is is the range for the middle 50% of the data. Example: For our previous example data set: 3310 3355 3450 3480 3480 3490 3520 3540 3550 3650 3730 3925 The IQR is 3600 – 3465 = 135 Q 1 =3465 Q 3 =3600
Variance 29 The variance is a measure of variability that utilizes all the data. The variance is based on the difference between the value of each observation (x i ) and the mean (this difference is also called deviation about the mean ) Population variance: Sample variance: N μ) (x σ N 1 i 2 i 2 = - = 1 - n ) x (x s n 1 i 2 i 2 = - =

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Standard Deviation 30 The standard deviation is defined to be the positive square root of the variance. Most commonly used measure of variation Has the same units as the original data Population standard deviation: Sample standard deviation: N μ) (x σ N 1 i 2 i = - = 1 - n ) x (x s n 1 i 2 i = - =
Example 31 Sample data: 10 12 14 15 17 18 18 24 n = 8 Mean = x = 16 4.3095 7 130 1 8 16) (24 16) (14 16) (12 16) (10 1 n ) x (24 ) x (14 ) x (12 ) x (10 s 2 2 2 2 2 2 2 2 = = - - + + - + - + - = - - + + - + - + - =

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Comparing Standard Deviations 32 Same mean, but different standard deviations: Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B Data A Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 Data C
Coefficient of Variation 33 Measures relative variation Shows variation relative to mean Usually in percentage (%) Is useful to compare the variability of two or more sets of data measured in different units two or more sets of data that have different standard deviations and different means. C.V. for population: C.V. for sample: 100% x s CV = 100% μ σ CV =

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Comparing Coefficients of Variation 34 Stock A: Average price last year = \$50 Standard deviation = \$5 Stock B: Average price last year = \$100 Standard deviation = \$5 5% 100% * \$100 \$5 100% * x s CV B = = = 10% 100% * \$50 \$5 100% * x s CV A = = = Both stocks have the same standard deviation, but stock B is less variable relative to its price
The Empirical Rule 35 If the data distribution is bell-shaped, then the interval contains about 68% of the values in the population or the sample.

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