Penggambaran Data Secara Numerik

Announcement ,[object Object],[object Object],[object Object]

Introduction to Probability and Statistics Eleventh Edition Robert J. Beaver • Barbara M. Beaver • William Mendenhall Presentation designed and written by: Barbara M. Beaver with minor change by Joon Jin Song

Introduction to Probability and Statistics Eleventh Edition Chapter 2 Describing Data with Numerical Measures Some graphic screen captures from Seeing Statistics ® Some images © 2001-(current year) www.arttoday.com

Describing Data with Numerical Measures ,[object Object],[object Object],[object Object],[object Object]

Measures of Center ,[object Object]

Arithmetic Mean or Average ,[object Object],where n = number of measurements

Example ,[object Object],If we were able to enumerate the whole population, the population mean would be called   (the Greek letter “ mu ”).

Median ,[object Object],[object Object],[object Object],.5( n + 1)

Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Median = 4 th largest measurement Median = (5 + 6)/2 = 5.5 — average of the 3 rd and 4 th measurements

Mode ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],[object Object],[object Object],The number of quarts of milk purchased by 25 households: 0 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5

[object Object],Extreme Values Applet ,[object Object]

Extreme Values Skewed left: Mean < Median Skewed right: Mean > Median Symmetric: Mean = Median

Measures of Variability ,[object Object]

The Range ,[object Object],[object Object],[object Object],[object Object],R = 14 – 5 = 9. ,[object Object]

The Variance ,[object Object],[object Object],4 6 8 10 12 14

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

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

Two Ways to Calculate the Sample Variance Use the Definition Formula: Sum 25 5 14 60 0 45 1 -1 8 9 -3 6 9 3 12 16 -4 5

Two Ways to Calculate the Sample Variance Use the Calculational Formula: Sum 196 14 465 45 64 8 36 6 144 12 25 5

[object Object],[object Object],[object Object],[object Object],Some Notes Applet

Using Measures of Center and Spread: Tchebysheff’s Theorem Given a number k greater than or equal to 1 and a set of n measurements, at least 1-(1/ k 2 ) of the measurement will lie within k standard deviations of the mean. ,[object Object],[object Object],[object Object],[object Object]

Using Measures of Center and Spread: The Empirical Rule ,[object Object],[object Object],[object Object],[object Object],[object Object]

Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Shape? Skewed right

[object Object],[object Object],[object Object],44.9  32.19 44.9  21.46 44.9  10.73  ks  .997 At least .89 50/50 (1.00) 12.71 to 77.09 3  .95 At least .75 49/50 (.98) 23.44 to 66.36 2  .68 At least 0 31/50 (.62) 34.17 to 55.63 1 Empirical Rule Tchebysheff Proportion in Interval Interval k ,[object Object],[object Object],[object Object]

Example The length of time for a worker to complete a specified operation averages 12.8 minutes with a standard deviation of 1.7 minutes. If the distribution of times is approximately mound-shaped, what proportion of workers will take longer than 16.2 minutes to complete the task? .025 .475 .475 95% between 9.4 and 16.2 47.5% between 12.8 and 16.2 (50-47.5)% = 2.5% above 16.2

[object Object],[object Object],[object Object],Approximating s

Approximating s R = 70 – 26 = 44 Actual s = 10.73 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Measures of Relative Standing ,[object Object],[object Object],4 x = 9 lies z = 2 std dev from the mean . s s Suppose s = 2. s

z -Scores ,[object Object],[object Object],[object Object],[object Object],Not unusual Outlier Outlier z -3 -2 -1 0 1 2 3 Somewhat unusual

Measures of Relative Standing ,[object Object],(100- p ) % p % p -th percentile x

Examples ,[object Object],BUREAU OF LABOR STATISTICS 2002  Median  Lower Quartile (Q 1)  Upper Quartile (Q 3 ) $319 is the 10 th percentile. $319 90% 10% 50 th Percentile 25 th Percentile 75 th Percentile

Quartiles and the IQR ,[object Object],[object Object],[object Object],[object Object]

Calculating Sample Quartiles ,[object Object],[object Object],[object Object],once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation. .75( n + 1) .25( n + 1)

Example ,[object Object],[object Object],[object Object],Position of Q 1 = .25(18 + 1) = 4.75 Position of Q 3 = .75(18 + 1) = 14.25 ,[object Object],[object Object]

Example ,[object Object],[object Object],[object Object],Position of Q 1 = .25(18 + 1) = 4.75 Position of Q 3 = .75(18 + 1) = 14.25 ,[object Object],[object Object],[object Object],[object Object]

Using Measures of Center and Spread: The Box Plot The Five-Number Summary: Min Q 1 Median Q 3 Max ,[object Object],[object Object],[object Object]

Constructing a Box Plot ,[object Object],[object Object],[object Object],Q 1 m Q 3

Constructing a Box Plot ,[object Object],[object Object],[object Object],[object Object],* Q 1 m Q 3

Constructing a Box Plot ,[object Object],Q 1 m Q 3 *

Example Amt of sodium in 8 brands of cheese: 260 290 300 320 330 340 340 520 Applet m = 325 Q 3 = 340 m Q 1 = 292.5 Q 3 Q 1

Example IQR = 340-292.5 = 47.5 Lower fence = 292.5-1.5(47.5) = 221.25 Upper fence = 340 + 1.5(47.5) = 411.25 Applet Outlier: x = 520 m Q 3 Q 1 *

Interpreting Box Plots ,[object Object],[object Object],[object Object]

Key Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Key Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Key Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object]

Penggambaran Data Secara Numerik

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