Numerical measures stat ppt @ bec doms

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Chapter Goals

Chapter Topics ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Summary Measures Center and Location Mean Median Mode Other Measures of Location Weighted Mean Describing Data Numerically Variation Variance Standard Deviation Coefficient of Variation Range Percentiles Interquartile Range Quartiles

Measures of Center and Location Center and Location Mean Median Mode Weighted Mean Overview

Mean (Arithmetic Average) ,[object Object],[object Object],[object Object],n = Sample Size N = Population Size

Mean (Arithmetic Average) ,[object Object],[object Object],[object Object],(continued) 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 0 1 2 3 4 5 6 7 8 9 10 Mean = 4

Median ,[object Object],[object Object],[object Object],[object Object],0 1 2 3 4 5 6 7 8 9 10 Median = 3 0 1 2 3 4 5 6 7 8 9 10 Median = 3

Mode ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 5 0 1 2 3 4 5 6 No Mode

Weighted Mean ,[object Object],Example : Sample of 26 Repair Projects Weighted Mean Days to Complete: Days to Complete Frequency 5 4 6 12 7 8 8 2

[object Object],Review Example House Prices: $2,000,000 500,000 300,000 100,000 100,000

Summary Statistics ,[object Object],[object Object],[object Object],[object Object],House Prices: $2,000,000 500,000 300,000 100,000 100,000 Sum 3,000,000

[object Object],[object Object],[object Object],Which measure of location is the “best”?

Shape of a Distribution ,[object Object],[object Object],Mean = Median = Mode Mean < Median < Mode Mode < Median < Mean Right-Skewed Left-Skewed Symmetric (Longer tail extends to left) (Longer tail extends to right)

Other Location Measures ,[object Object],[object Object],[object Object],[object Object],Other Measures of Location Percentiles Quartiles 1 st quartile = 25 th percentile 2 nd quartile = 50 th percentile = median 3 rd quartile = 75 th percentile

Percentiles ,[object Object],Example: The 60 th percentile in an ordered array of 19 values is the value in 12 th position:

Quartiles ,[object Object],25% 25% 25% 25% Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Example: Find the first quartile (n = 9) Q1 = 25 th percentile, so find the (9+1) = 2.5 position so use the value half way between the 2 nd and 3 rd values, so Q1 = 12.5 25 100 Q1 Q2 Q3

Box and Whisker Plot ,[object Object],Minimum -- Q1 -- Median -- Q3 -- Maximum Example : 25% 25% 25% 25%

Shape of Box and Whisker Plots ,[object Object],[object Object]

Distribution Shape and Box and Whisker Plot Right-Skewed Left-Skewed Symmetric Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

Box-and-Whisker Plot Example ,[object Object],[object Object],0 2 3 5 27 Min Q1 Q2 Q3 Max

Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range

[object Object],Variation Same center, different variation

Range ,[object Object],[object Object],Range = x maximum – x minimum 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Example:

[object Object],[object Object],Disadvantages of the Range 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5 1 ,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4, 5 1 ,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4, 120 Range = 5 - 1 = 4 Range = 120 - 1 = 119

Interquartile Range ,[object Object],[object Object],[object Object]

Interquartile Range Median (Q2) X maximum X minimum Q1 Q3 Example: 25% 25% 25% 25% 12 30 45 57 70 Interquartile range = 57 – 30 = 27

[object Object],[object Object],[object Object],Variance

Standard Deviation ,[object Object],[object Object],[object Object],[object Object],[object Object]

Calculation Example: Sample Standard Deviation Sample Data (X i ) : 10 12 14 15 17 18 18 24 n = 8 Mean = x = 16

Comparing 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 ,[object Object],[object Object],[object Object],[object Object],Population Sample

Comparing Coefficient of Variation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Both stocks have the same standard deviation, but stock B is less variable relative to its price

[object Object],[object Object],The Empirical Rule X 68%

[object Object],[object Object],The Empirical Rule 99.7% 95%

[object Object],[object Object],[object Object],[object Object],[object Object],Tchebysheff’s Theorem within At least

[object Object],[object Object],Standardized Data Values

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Standardized Population Values

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Standardized Sample Values

Using Microsoft Excel ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],Using Excel (continued)

Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000,000 500,000 300,000 100,000 100,000

Numerical measures stat ppt @ bec doms

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Numerical measures stat ppt @ bec doms