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Section 3.4 Measures of Position Standard Score • A z score or standard score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. – The symbol for a standard score is z. – The z score represents the number of standard deviations that a data value falls above or below the mean. X−X z= s z= X −µ σ SAT Scores The following data represent a sample of the SAT scores. 798 789 675 669 658 647 642 639 637 635 631 624 624 616 608 606 605 601 599 597 597 595 593 589 589 577 574 570 568 555 555 554 554 552 552 550 549 547 544 539 534 532 530 530 527 527 521 519 518 513 512 511 509 508 499 498 498 497 488 488 482 480 479 477 476 473 470 467 464 461 452 452 451 448 447 445 443 442 418 417 414 414 413 413 409 406 404 403 401 395 394 389 382 381 364 360 351 348 334 319 The mean of the data is 513.0 and the standard deviation is 94.7. Example What is more unusual: 1.A student who scored 75 on a statistics test that had a mean of 65 and a standard deviation of 6 2.The person who scored 351 on the SAT z Facts • When all data for a variable are transformed into z scores, the resulting distribution will have a mean of 0 and a standard deviation of 1. Percentiles • Percentiles divide the data set into 100 equal groups. Finding the Value • The formula for finding a value in a data set that corresponds to a given percentile is k i= (n + 1) 100 – where n is the total number of values and k is the percentile. Example • For the data set of the city miles per gallon rating for the Saturn Ion, find the value corresponding to the 25th percentile and the 60th percentile. • The data in ascending value: 17 18 18 19 20 22 23 23 24 24 25 26 Percentile Formula The percentile corresponding to a given value x is computed by using the following formula: (number of values below x) Percentile = ×100 total number of values Round this number to the nearest integer Example • For the data set of the city miles per gallon rating for the Saturn Ion, find the percentile rank of a car that had 24 MPG. • The data in ascending value: 17 18 18 19 20 22 23 23 24 24 25 26 Quartile • Quartiles divide the data set into four equal groups. • The four groups are separated by Q1, Q2, Q 3. • Note that Q1 is the same as the 25th percentile, Q2 is the same as the 50th percentile, and Q3 is the same as the 75th percentile. Finding the Quartiles Step 1: Arrange the data in order from lowest to highest. Step 2: Find the median of the data values. This is Q2. Step 3: Find the median of the data values that fall below Q2. This is Q1. Step 4: Find the median of the data values that fall above Q2. This is Q3. Example • Find the quartiles for the items on the Taco Bell menu: • The data in ascending order: 180 290 290 340 430 450 470 540 600 Interquartile Range • The interquartile range (IQR) is defined as the difference between Q1 and Q3. • It is the range of the middle 50% of the data. – The interquartile range is used to determine outliers. – It is used as a measure of variability in exploratory data analysis. Deciles • Deciles divide the data set into 10 equal groups. • Deciles are denoted D1, D2, D3 … D9. • The deciles correspond to the percentiles: P10, P20, P30 … P90. Outliers • An outlier is an extremely high or extremely low data value when compared to the rest of the data values. Identifying Outliers Step 1: Arrange the data in order and find Q1 and Q3. Step 2: Find the interquartile range. Step 3: Multiply the IQR by 1.5. Step 4: Subtract the value obtained in step 3 from Q1 and add the value to Q3. Step 5: Check the data set for any data value that is smaller than Q1 − 1.5(IQR) and larger than Q3 + 1.5(IQR). Example • Check the following data set for outliers. 2 23 29 31 37 41 43 71