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Understanding Z-Scores: Statistical Transformations and Interpretations

Learn how z-scores, linear transformations, and rescaling data enable meaningful comparisons and analyses in statistics. Explore examples showcasing shifting and rescaling techniques with practical applications.

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Understanding Z-Scores: Statistical Transformations and Interpretations

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  1. Chapter 6 Part 1Using the Mean and Standard Deviation Together z-scores 68-95-99.7 rule Changing units (shifting and rescaling data)

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

  3. z-score corresponding to y

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

  5. 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.

  6. Z-scores add to zero

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

  8. Changing Units of Measurement How shifting and rescaling data affect data summaries

  9. x* Shifts data by a a Changes scale 0 x Shifting and rescaling: linear transformations • Original data x1, x2, . . . xn • Linear transformation: x* = a + bx, (intercept a, slope b)

  10. 0 0 32 150 0 12 100 9/5 40 2.54 Linear Transformationsx* = a+ b x Examples: Changing • from feet (x) to inches (x*): x*=12x • from dollars (x) to cents (x*): x*=100x • from degrees celsius (x) to degrees fahrenheit (x*): x* = 32 + (9/5)x • from ACT (x) to SAT (x*): x*=150+40x • from inches (x) to centimeters (x*): x* = 2.54x

  11. Shifting data only: b = 1x* = a + x • Adding the same value a to each value in the data set: • changes the mean, median, Q1 and Q3 by a • The standard deviation, IQR and variance are NOT CHANGED. • Everything shifts together. • Spread of the items does not change.

  12. weights of 80 men age 19 to 24 of average height (5'8" to 5'10") x = 82.36 kg NIH recommends maximum healthy weight of 74 kg. To compare their weights to the recommended maximum, subtract 74 kg from each weight; x* = x – 74 (a=-74, b=1) x* = x – 74 = 8.36 kg Shifting data only: b = 1x* = a + x (cont.) • No change in shape • No change in spread • Shift by 74

  13. Original x data: x1, x2, x3, . . ., xn Summary statistics: mean x median m 1st quartile Q1 3rd quartile Q3 stand dev s variance s2 IQR x* data: x* = a + bx x1*, x2*, x3*, . . ., xn* Summary statistics: new mean x* = a + bx new median m* = a+bm new 1st quart Q1*= a+bQ1 new 3rd quart Q3* = a+bQ3 new stand dev s* = b  s new variance s*2 = b2 s2 new IQR* = b  IQR Shifting and Rescaling data: x* = a + bx, b > 0

  14. weights of 80 men age 19 to 24, of average height (5'8" to 5'10") x = 82.36 kg min=54.30 kg max=161.50 kg range=107.20 kg s = 18.35 kg Change from kilograms to pounds: x* = 2.2x (a = 0, b = 2.2) x* = 2.2(82.36)=181.19 pounds min* = 2.2(54.30)=119.46 pounds max* = 2.2(161.50)=355.3 pounds range*= 2.2(107.20)=235.84 pounds s* = 18.35 * 2.2 = 40.37 pounds Rescaling data: x* = a + bx, b > 0 (cont.)

  15. 4 student heights in inches (x data) 62, 64, 74, 72 x = 68 inches s = 5.89 inches Suppose we want centimeters instead: x* = 2.54x (a = 0, b = 2.54) 4 student heights in centimeters: 157.48 = 2.54(62) 162.56 = 2.54(64) 187.96 = 2.54(74) 182.88 = 2.54(72) x* = 172.72 centimeters s* = 14.9606 centimeters Note that x* = 2.54x = 2.54(68)=172.2 s* = 2.54s = 2.54(5.89)=14.9606 Example of x* = a + bx not necessary!UNC method Go directly to this. NCSU method

  16. x data: Percent returns from 4 investments during 2003: 5%, 4%, 3%, 6% x = 4.5% s = 1.29% Inflation during 2003: 2% x* data: Inflation-adjusted returns. x* = x – 2% (a=-2, b=1) x* data: 3% = 5% - 2% 2% = 4% - 2% 1% = 3% - 2% 4% = 6% - 2% x* = 10%/4 = 2.5% s* = s = 1.29% x* = x – 2% = 4.5% –2% s* = s = 1.29% (note! that s* ≠ s – 2%) !! Example of x* = a + bx not necessary! Go directly to this

  17. Example • Original data x: Jim Bob’s jumbo watermelons from his garden have the following weights (lbs): 23, 34, 38, 44, 48, 55, 55, 68, 72, 75 s = 17.12; Q1=38, Q3 =68; IQR = 68 – 38 = 30 • Melons over 50 lbs are priced differently; the amount each melon is over (or under) 50 lbs is: • x* = x  50 (x* = a + bx, a=-50, b=1) -27, -16, -12, -6, -2, 5, 5, 18, 22, 25 s* = 17.12; Q*1 = 38 - 50 =-12, Q*3 = 68 - 50 = 18 IQR* = 18 – (-12) = 30 NOTE: s* = s, IQR*= IQR

  18. Z-scores: a special linear transformation a + bx Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs. Question 1. A student’s tuition charge is $941.25. What is the z-score of this tuition? x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50

  19. Z-scores: a special linear transformation a + bx (cont.) Example. At a community college, if a student takes x credit hours the tuition is x* = $250 + $35x. The credit hours taken by students in an Intro Stats class have mean x = 15.7 hrs and standard deviation s = 2.7 hrs. Question 2. Roger is a student in the Intro Stats class who has a course load of x = 13 credit hours. The z-score is z = (13 – 15.7)/2.7 = -2.7/2.7 = -1. What is the z-score of Roger’s tuition? Roger’s tuition is x* = $250 + $35(13) = $705 Since x* = $250+$35(15.7) = $799.50; s* = $35(2.7) = $94.50 The z-score does not depend on the unit of measurement. This is why z-scores are so useful!!

  20. SUMMARY: Linear Transformations x* = a + bx • Linear transformations do not affect the shape of the distribution of the data -for example, if the original data is right-skewed, the transformed data is right-skewed

  21. SUMMARY: Shifting and Rescaling data, x* = a + bx, b > 0

  22. 68-95-99.7 rule Mean and Standard Deviation (numerical) Histogram (graphical) 68-95-99.7 rule

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

  24. 68-95-99.7 rule: 68% within 1 stan. dev. of the mean 68% 34% 34% y-s y y+s

  25. 68-95-99.7 rule: 95% within 2 stan. dev. of the mean 95% 47.5% 47.5% y-2s y y+2s

  26. 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

  27. 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

  28. 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

  29. 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

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

  31. End of Chapter 6 Part 1.Next: Part 2 Normal Models

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