After seeing countless commercials claiming one can get cheaper car insurance from an online company, a local insurance agent was concerned that he might lose some customers. To investigate, he randomly selected profiles (type of car, coverage, driving record, etc.) for 10 of his clients and checked online price quotes for their policies. The comparisons are shown in the table.

His statistical software produced the following summaries (where PriceDiff = Local - Online):

At first, the insurance agent wondered whether there was some kind of mistake in this output. He thought the Pythagorean Theorem of Statistics should work for finding the standard deviation of the price differences€”in other words, that SD(Local €“ Online) = ˆšSD2(Local) + SD2(Online).
But when he checked, he found that ˆš(229.2812) + (256.2672)2 = 343.864,not 175.663 as given by the software. Tell him where his mistake is.

Local Online PriceDiff 391 872 602 488 903 677 1021 1270 703 789 568 270 37 326 72 -249 80 451 1229 605 783 844 907 10810 712 702 10 Variable Local Online PriceDiff Count Mean 10 10 10 799.200 753.300 45.900 StdDev 229.281 256.267 175.663