PISA math scores 2009 Below are some summary statistics available at www.pisa.oecd.org showing mean, standard deviation, 5th, 10th, 25th, 75th, 90th, and 95th percentiles for 2009 PISA math scores of 15-year-old students in various Organisation for Economic Co-operation and Development (OECD) countries, and also China.

a) Since you do not have sufficient information to construct standard boxplots, construct modified boxplots instead, showing whiskers extending to 5th and 95th percentiles on each side of the box, with the box constructed as usual, but marking off the mean rather than the median inside the box. Use such plots to compare the countries and then summarize the noticeable differences among countries.

b) Do the distributions appear to be close to symmetric or very skewed? Explain. Are the mean and standard deviation useful summary numbers for these data?

c) If the distributions are mound-shaped and not very skewed, then the empirical rule (see Chapter 3) should hold. Calculate the mean + two standard deviations and the mean – two standard deviations for Canada and the U.S. Can you check if more than 90% of scores lie between these two numbers for each country? Explain. Can you confirm that between 50% and 80% of the data fall within the mean ± one standard deviation for Canada and for the U.S.? Explain.

Country Mean StDev 5th | 10th 25th 75th 90th 95th Australia 514 Canada 527 Finland 541 France 497 101 Germany 513 Japan 529 Korea 546 Mexico 419 8 94 357 392 451 580 634 665 88 379 413 468 588 638 665 82 399 431 487 599 644 669 321 361 429 570 622 652 98 347 380 443 585 638 666 94 370 407 468 595 595 648 677 89 397 430 486 609 659 689 79 289 318 366 472 520 547 UK 492 87 348 380 434 552 606 635 USA 487 91 337 368 425 551 607 637 China 674 726 757 600 103 421 462 531 462 531 674 Source: Based on data from OECD PISA 2009 online database https://www. acer.org/ozpisa/publications-and-data # 2009, accessed on Feb 22, 2013.