3.4: Order of Operations in Expressions and Formulas
Learning Outcomes
- Use Order of Operations in Statistics Formulas.
We have already encountered the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In this section, we will give some additional examples where the order of operations must be used properly to evaluate statistics.
Example 3.4.1
The sample standard deviation asks us to add up the squared deviations, take the square root and divide by one less than the sample size. For example, suppose that there are three data values: 3, 5, 10. The mean of these values is 6. Then the standard deviation is:
![\[s=\sqrt{\frac{\left(3-6\right)^2+\left(5-6\right)^2+\left(10-6\right)^2}{3-1}}\nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-ffd9bc5624edf0f1da2345024ac86ca9_l3.png "Rendered by QuickLaTeX.com")
Evaluate this number rounded to the nearest hundredth.
Solution
The first thing in the order of operations is to do what is in the parentheses. We must subtract:
![\[3-6=-3,\:\:\:5-6\:=\:-1,\:\:\:10-6=4 \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-148225fb728abcd2217b14c8e8fcf12c_l3.png "Rendered by QuickLaTeX.com")
We can substitute the numbers in to get:
![\[=\sqrt{\frac{\left(-3\right)^2+\left(-1\right)^2+\left(4\right)^2}{3-1}}\nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-3c3481b0b8dc7dee99ee0e5bb2521094_l3.png "Rendered by QuickLaTeX.com")
Next, we exponentiate:
![\[\left(-3\right)^2=9,\:\:\:\left(-1\right)^2=1,\:\:\:4^2=16 \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-a0c67a74eed445458a028fdd6305fa4e_l3.png "Rendered by QuickLaTeX.com")
Substitute these in to get:
![\[\sqrt{\frac{9+1+16}{3-1}} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-55a9507cf8b70ef819ab667388ab0e02_l3.png "Rendered by QuickLaTeX.com")
We can now perform the addition inside the square root to get:
![\[\sqrt{\frac{26}{3-1}} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-a5831bcc9c897c4ab19e76297879aa96_l3.png "Rendered by QuickLaTeX.com")
Next, perform the subtraction of the denominator to get:
![\[\sqrt{\frac{26}{2}} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-80e68814c29de7146b2e9e439f9d2bc7_l3.png "Rendered by QuickLaTeX.com")
We can divide to get:
![\[\sqrt{13} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-e45c61bb2b537c62005cd6f5700c99fc_l3.png "Rendered by QuickLaTeX.com")
We don’t want to do this by hand, so in a calculator or computer type in:
![\[13^{0.5} = 3.61 \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-ce0e08d7904f9a34f37b5e89897c5c12_l3.png "Rendered by QuickLaTeX.com")
Example 3.4.2
When calculating the probability that a value will be less than 4.6 if the value is taken randomly from a uniform distribution between 3 and 7, we have to calculate:
![\[\left(4.6-3\right)\times\frac{1}{7-3} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-19b0f4b16088e481613113f1fd235188_l3.png "Rendered by QuickLaTeX.com")
Find this probability.
Solution
We can use a calculator or computer, but we must be very careful about the order of operations. Notice that there are implied parentheses due to the fraction bar. The answer is:
![\[\dfrac{(4.6 - 3) \times 1}{7-3} \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-2382f4a6b8f0104e297a09e4adfbacc8_l3.png "Rendered by QuickLaTeX.com")
Using technology, we get:
![\[\left(4.6-3\right)\times\frac{1}{7-3}\:=\:0.4 \nonumber\]](https://pressbooks.montgomerycollege.edu/app/uploads/quicklatex/quicklatex.com-84df76fd302b6966dea6b377a91d975f_l3.png "Rendered by QuickLaTeX.com")
Exercise
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