Unit 26: Statistics

The SAT won’t test on in-depth statistical methods (like calculating the standard deviation), but it will test on foundational statistical concepts and how to solve for some of them:

Example:

The average (arithmetic mean) of 6, 19, and x is 19. What is the value of x?

A) 19
B) 25
C) 31
D) 32

Strategy:

  1. Identify which statistical concept you’re being tested on.
  2. If median, order the data set (if not already ordered).
  3. Use your definitions to answer the question.

The SAT only tests a handful of statistical concepts.  The most important are:

  • Average
    • Mean
    • Median
    • Mode
  • Range
  • Standard Deviation

A quick recap:

Given the data set (lbrack;22,;38,;20,;33,;24,;40,;38;rbrack) calculate the mean, median, mode, and range.

Mean = average

To get the mean, simply add everything up and divide by the number of things.

In our example, first add up the values: (22+24+20+33+24+40+38=215)

There are seven items, so divide that sum by 7: (frac{215}7=30.71).

Median = middle

The median is simply the middle value.  

In this case, we need to reorder our data set to go from least to most:  20, 22, 24, 33, 38, 38, 40.  We want the value in the middle, with half above and half below, so here the median is 33.

What if you have an even number of items in your data set, like (lbrack3,5,9,7rbrack) How do you find the median then?  Take the two middle numbers and find the mean (average).  

So (5+7=12), then (frac{12}2=6) is our median.  

Mode = Most often

The mode is the number that appears most often in the data set.  

In our example above, 38 is the only value that appears more than once, so it is the mode.  

Note that there can be more than one mode in a set, but this is highly unlikely to be tested on the SAT.

Range = difference

The range of a set is the difference between the smallest and largest values – we ignore   everything else.  

In our example, 20 is our smallest number, and 40 is our largest, so (40-20=20) is our range.

Standard Deviation

Standard deviation can be a little tricky when you are first introduced to it, but it is simply a measure of how dispersed, or far apart from each other, the data points in a set are.  

Low or small standard deviation: Sets where the numbers are clustered near the mean of the set.  

High or large standard deviation: Sets where lots of the numbers are far from the average.

Some examples help to make this clear.

Below are charts detailing the measured weights (in kg) of groups of wombats divided by sex.

The mean for the females is around 24.  Notice how all the values are very close to the average – this represents a low standard deviation.  For the males, the mean is about 25; However, only a couple of our values are close to the mean.  This data set has a much higher standard deviation.

The SAT will not make you calculate the actual standard deviation.  This is a complex job best left for computers.  They will make the answer painfully obvious as long as you have an understanding of what standard deviation is.  

You might also be presented data in graphical form and asked to evaluate standard deviation.