Question #f4b1c

1 Answer
Geoff K.
Sep 25, 2017

Using the range approximation, our estimate for the standard deviation is #s~~1.67.#

Explanation:

The range approximation says, "the standard deviation #s# for a data set is roughly proportional to the range #R# of that set." In other words:

#R/s~~["some constant"]#

Let's call this constant #c#. Since the range is the difference between the highest and lowest value, solving for #s# gives us:

#s~~(x_"max"-x_"min")/c#

The constant #c# will depend on how big the data set is (that is, how large #n# is). For small data sets #(n=5)#, we expect that constant to be #2.5#. That is, the range of our data should be about 2.5 times bigger than the standard deviation #(R~~2.5s)#. When #n=10#, the table says the #R# should be about 3 times larger than #s# #(R~~3s).#

Since our data set has 10 elements #(n=10)#, the value we'll use for our constant #c# is 3.

Start with the formula, then plug in the known values to solve for #s:#

#s~~R/c#

#color(white)s=(x_"max"-x_"min")/c#

#color(white)s=(7-2)/3#

#color(white)s=5/3#

#color(white)s~~1.67#

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