With hot, dry weather comes also the job of irrigation, for those who choose to do so. My wife wants a nice yard this year. Being a house-husband, my answer is “yes.”<\/p>\n
This morning I was out observing the sprinkler’s reach, not wanting it to touch the sidewalk, if possible. Of course, it occasionally did.<\/p>\n
Making some quick observations, I surmised the following:<\/p>\n
Sprinkler’s mean reach: 8ft.<\/p>\n
68% of the time it reaches about 6 inches more or less than the mean.<\/p>\n
95% of the time if reaches about 10 inches either way from the mean.<\/p>\n
99.7% of the time it reaches about 14 inches beyond mean, which never happened this morning, suggested the sidewalk. I estimated the sprinkler had done about 300 oscillations. So just under 1 in 300 times, the reach might be that.<\/p>\n
I wondered: Can I model the standard deviation from those observations?<\/p>\n
The 68-95-99 rule states that, for a normal population with mean \u03bc and standard deviation \u03c3,<\/p>\n
68% of the measurements fall within \u03bc\u00b1\u03c3<\/p>\n
95% within \u03bc\u00b11.96\u03c3<\/p>\n
99.7% within \u03bc\u00b12.75\u03c3<\/p>\n
Imagining a standard deviation of 6 inches, the observations don’t fit well. With \u03bc=8, \u03c3=5, however, they fall out<\/p>\n
68% within 8ft\u00b15inches (est. 6 inches)<\/p>\n
95% within 8ft\u00b19.8inches (est. 10 inches)<\/p>\n
99.7% within 8ft\u00b113.75inches (est 14 inches)<\/p>\n
In this context, \u03c3 is difficult to estimate at a glance, since it lies decidedly in the wet patch. My estimate of 6 inches for \u03c3 could easily be much better imagined as 5 inches.<\/p>\n
Cheers:)<\/p>\n
Source:<\/p>\n
Harnett, Donald and James L. Murphy. Statistical Analysis<\/i>. Don Mills: Addison-Wesley, 1986.<\/p>\n