Statistics: the 68-95-99 rule

Tutoring statistics, rules of thumb can be key. The tutor mentions a real-life application of the 68-95-99 rule.

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.”

This morning I was out observing the sprinkler’s reach, not wanting it to touch the sidewalk, if possible. Of course, it occasionally did.

Making some quick observations, I surmised the following:

Sprinkler’s mean reach: 8ft.

68% of the time it reaches about 6 inches more or less than the mean.

95% of the time if reaches about 10 inches either way from the mean.

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.

I wondered: Can I model the standard deviation from those observations?

The 68-95-99 rule states that, for a normal population with mean μ and standard deviation σ,

68% of the measurements fall within μ±σ

95% within μ±1.96σ

99.7% within μ±2.75σ

Imagining a standard deviation of 6 inches, the observations don’t fit well. With μ=8, σ=5, however, they fall out

68% within 8ft±5inches (est. 6 inches)

95% within 8ft±9.8inches (est. 10 inches)

99.7% within 8ft±13.75inches (est 14 inches)

In this context, σ is difficult to estimate at a glance, since it lies decidedly in the wet patch. My estimate of 6 inches for σ could easily be much better imagined as 5 inches.



Harnett, Donald and James L. Murphy. Statistical Analysis. Don Mills: Addison-Wesley, 1986.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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