Question

In their book Introduction to Linear Regression Analysis (3rd edition, Wiley, 2001) Montgomery, Peck, and Vining present measurements on NbOCl₃ concentration from a tube-flow reactor experiment. The data, in gram-mole per liter × 10^–3, are as follows. Construct a stem-and-leaf diagram for this data. Compute the sample mean, sample standard deviation, and the sample median.

450	450	562	429	434	463	437	1258	1220
1164	1052	977	1183	1273	1303	1496	1503	1692
1752	2752	3212	3212	3378	1909	2558	2650	2808

Answer 1

Mean = Sum of observations/ Count of observations
Mean = (450 + 450 + 562 + 429 + 434 + 463 + 437 + 1258 + 1220 + 1164 + 1052 + 977 + 1183 + 1273 + 1303 + 1496 + 1503 + 1692 + 1752 + 2752 + 3212 + 3212 + 3378 + 1909 + 2558 + 2650 + 2808/ 27) = 1539.88

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Variance
Step 1: Add them up
450 + 450 + 562 + 429 + 434 + 463 + 437 + 1258 + 1220 + 1164 + 1052 + 977 + 1183 + 1273 + 1303 + 1496 + 1503 + 1692 + 1752 + 2752 + 3212 + 3212 + 3378 + 1909 + 2558 + 2650 + 2808 = 41577
Step 2: Square your answer
41577*41577 =1728646929
…and divide by the number of items. We have 27 items , 1728646929/27 = 64023960.3333
Set this number aside for a moment.
Step 3: Take your set of original numbers from Step 1, and square them individually this time
450^2 + 450^2 + 562^2 + 429^2 + 434^2 + 463^2 + 437^2 + 1258^2 + 1220^2 + 1164^2 + 1052^2 + 977^2 + 1183^2 + 1273^2 + 1303^2 + 1496^2 + 1503^2 + 1692^2 + 1752^2 + 2752^2 + 3212^2 + 3212^2 + 3378^2 + 1909^2 + 2558^2 + 2650^2 + 2808^2 = 87846177
Step 4: Subtract the amount in Step 2 from the amount in Step 3
87846177 - 64023960.3333 = 23822216.6667
Step 5: Subtract 1 from the number of items in your data set, 27 - 1 = 26
Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the variance
23822216.6667 / 26 = 916239.102
Step 7: Take the square root of your answer from Step 6. This gives you the standard deviation
957.2037

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median is the middle most observation = (n+1)/2
median = 14th observation from after ascending the data
429 434 437 450 450 463 562 977 1052 1164 1183 1220 1258 1273 1303 1496 1503 1692 1752 1909 2558
2650 2752 2808 3212 3212 3378
median = 1273