4.9 SAT Math - Mean / Median / Mode

Many of these concepts you learned way back when, in fifth grade or so, and that tends to be a problem.  They're TOO easy.  Or at least the SAT testmakers think so.  Who remembers junk you learned in grade school?  Well, you'd better.  Careless mistakes are a killer.  Some of these concepts, though easy in and of themselves, will be tested in trickier ways than you're used to as well.  What are they testing with many of these?  If you read carefully and follow directions well.  So please do.

Mean/Median/Mode

All three of these terms are types of averages, but obviously they're asking for very different things.  Mean is usually what we mean when we say 'average,' and the one that any student who has tried to frantically figure out his GPA before report cards comes out knows how to do.  We'll come back to how that will be tested in a moment.

Median

Median is the middle number when numbers are arranged least to greatest.  A median is also that cement strip in the middle of the road – median=middle.
The easiest way to find a median is to simply rearrange the numbers given from least to greatest, and then eliminate the first and last number in the list until you're left with only one number in the center.  If the original list contains an even number of numbers, you may be left with two:

Find Joe's median test score if he scored 85, 84, 85, 82, 83, 72, 96, 81, and 93 on this semester's tests.

Rearrange:  72, 81, 82, 83, 84, 85, 85, 93, 96

-SAT Math Hint: Count the number of terms in the original list and count the number of terms in you're rearranged list to make sure you got everything.
Then eliminate first and last:



And continue until you're left with one or a pair in the center:




His median test score is 84.

If you're left with two in the center, add the two numbers together and divide by two to get the median.

-SAT Math Hint: Your answer in this case will most likely NOT be one of the numbers on the original list.  That's okay.

Sara buys coffee at the local coffee store at least once a day.  On days when she's particularly busy, she visits multiple times.  She made one visit on Monday, two on Tuesday, one on Wednesday, one on Thursday, three on Friday, and two on Saturday.  What was her median number of visits for the week?

First, write down your numbers: 1, 2, 1, 1, 3, 2
Then rearrange: 1, 1, 1, 2, 2, 3
Then cross out first and last until you get to the center:
You're left with 1 and 2, so add them together and divide by 2:

She had a median number of 1.5 trips for the week.

-SAT Math Tip: A median is always found by first arranging the numbers least to greatest.  Do NOT start slashing away at pairs until you've rewritten the whole string of numbers first, including every single one.  So if there are 3 14s in your list, you better have 3 14s written down in your rearrangement.

Mode

Mode means "most."  That's it.  When they're asking you for mode, they're asking you which number in the list appears most often.

What is the mode in each of the following sets?

Set W: {4,3,3,5,2,1}
Set X: {2,7,2,9,6,7,2}
Set Y: {2,6,2,8,9,6,3}
Set Z: {1,5,7,4,8}

In Set W, every other number appears once, but 3 appears twice, so 3 is the mode.
In Set X, both 2 and 7 appear more than once, but 2 appears more often than 7 so 2 is the mode.
In Set Y, both 2 and 6 appear an equal number of times, both more often than any other number.  Set Y has two modes: 2 and 6.
In Set Z, no number appears more often, so there is no mode.

-SAT Math Tip: Remember, a mode must appear more often than any other number, not just be repeated.  There can be no mode; there can also be more than one mode if those numbers appear an equal number of times but more than any other number.

Mean

Let's talk about mean, or averages.  The formula for average is "where the total is all the numbers you were given added up and the number of things is the number of things you were given.  We can shorthand the formula to .