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2.4 (cont.) Using the Mean and Standard Deviation Together

Learn how to analyze data distribution and spread using the mean, standard deviation, z-scores, and the 68-95-99.7 rule. Explore numerical and graphical representations like histograms. Practice calculating z-scores and making comparisons based on standard deviations.

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2.4 (cont.) Using the Mean and Standard Deviation Together

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  1. 2.4 (cont.)Using the Mean and Standard Deviation Together 68-95-99.7 rule z-scores

  2. 68-95-99.7 rule Mean and Standard Deviation (numerical) Histogram (graphical) 68-95-99.7 rule

  3. The 68-95-99.7 rule; applies only to mound-shaped data

  4. 68-95-99.7 rule: 68% within 1 stan. dev. of the mean 68% 34% 34% y-s y y+s

  5. 68-95-99.7 rule: 95% within 2 stan. dev. of the mean 95% 47.5% 47.5% y-2s y y+2s

  6. Example: textbook costs 286 291 307 308 315 316 327 328 340 342 346 347 348 348 349 354 355 355 360 361 364 367 369 371 373 377 380 381 382 385 385 387 390 390 397 398 409 409 410 418 422 424 425 426 428 433 434 437 440 480

  7. Example: textbook costs (cont.) 286 291 307 308 315 316 327 328 340 342 346 347 348 348 349 354 355 355 360 361 364 367 369 371 373 377 380 381 382 385 385 387 390 390 397 398 409 409 410 418 422 424 425 426 428 433 434 437 440 480

  8. Example: textbook costs (cont.) 286 291 307 308 315 316 327 328 340 342 346 347 348 348 349 354 355 355 360 361 364 367 369 371 373 377 380 381 382 385 385 387 390 390 397 398 409 409 410 418 422 424 425 426 428 433 434 437 440 480

  9. Example: textbook costs (cont.) 286 291 307 308 315 316 327 328 340 342 346 347 348 348 349 354 355 355 360 361 364 367 369 371 373 377 380 381 382 385 385 387 390 390 397 398 409 409 410 418 422 424 425 426 428 433 434 437 440 480

  10. The best estimate of the standard deviation of the men’s weights displayed in this dotplot is • 10 • 15 • 20 • 40 7

  11. Z-scores: Standardized Data Values Measures the distance of a number from the mean in units of the standard deviation

  12. z-score corresponding to y

  13. Recent NC State and Clemson Home Football Attendance

  14. Warmup Warmup vs. During a recent season the average attendance at NC State’s football games was 54,000 with a standard deviation of 2,900. During the same season average attendance at Clemson’s football games was 81,000 with a standard deviation of 4,400. For which team is it more unusual to have a game attendance of 65,000?

  15. Warmup-cont. Warmup-cont. vs. For which team is it more unusual to have attendance of 65,000? 65,000 is 3.79 standard deviations above NC State’s mean of 54,000 65,000 is 3.64 standard deviations below Clemson’s mean of 81,000 NC State’s z-score is farther from 0 than Clemson’s z-score. Therefore it is more unusual for NC State to have attendance of 65,000.

  16. Exam 1: y1 = 88, s1 = 6; exam 1 score: 91 Exam 2: y2 = 88, s2 = 10; exam 2 score: 92 Which score is better?

  17. Comparing SAT and ACT Scores • SAT Math: Eleanor’s score 680 SAT mean =500 sd=100 • ACT Math: Gerald’s score 27 ACT mean=18 sd=6 • Eleanor’s z-score: z=(680-500)/100=1.8 • Gerald’s z-score: z=(27-18)/6=1.5 • Eleanor’s score is better.

  18. Z-scores add to zero

  19. In 2014-15 the mean tuition at 4-yr public colleges/universities in the U.S. was $6185 with a standard deviation of $1804. In NC the mean tuition was $4320. What is NC’s z-score? • 1.03 • -1.03 • 2.39 • 1865 • -1865 1

  20. End of Section 2.4

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