Spread, standard deviation, and box plots in Desmos

Two data sets can have the same mean and still tell very different stories. One can huddle close to its mean while the other sprawls out. SAT questions check whether you can keep two ideas apart: center, where the data sit, and spread, how far apart the values are.

You'll measure spread two ways. The range is the largest value minus the smallest. Standard deviation tells you how far the values typically sit from the mean. Small means the values cluster tightly around the mean. Large means they're spread out.

Fast Desmos solution

Both lists balance around $10$, so both means are $10$. Now line the lists up in order. Every value in Q that isn't $10$ sits twice as far from $10$ as its partner in P: $2$ is $8$ away, while $6$ is only $4$ away. Q is more spread out, so it has the greater standard deviation. The answer is C.

The graph shows P as the lower row of dots and Q as the upper row. You can read the answer from the picture before you look at any stdevp(...) value. The numbers only confirm it.

Good news: SAT questions usually ask you to read and compare standard deviation, not to calculate it by hand. Desmos has two versions, stdev(...) for a sample and stdevp(...) for a whole population. When you compare two data sets of the same size, or ask how a shift or a multiplier changes the spread, both versions point to the same answer. So we'll stick with stdevp(...), and use it only to check what you've already reasoned out.

Turn a raw list into a picture

Say you have

Type the values as a list, then use dotplot(R) to see every value. The smallest is $4$ and the largest is $11$, so

Once you've read the dot plot, replace dotplot(R) with boxplot(R) to see the same data squeezed into a box plot. Show only one picture at a time, or they'll sit on top of each other.

The dot plot shows all three $8$s. The box plot makes the median and the overall spread quick to read, but the counts are gone.

A full comparison answers two separate questions:

1. Center: Which data set sits higher or lower?
2. Spread: Which data set is more spread out?

Try it on these two:

and

Both have mean $5$. Both have the same range, too:

So are they equally spread out? No. A keeps three values right at the mean, with only $0$ and $10$ out at the ends. B puts four values out at the ends, far from the mean. So B has the greater standard deviation.

Here's why that can happen. Range looks at only the two end values. Standard deviation looks at where all the values sit around the mean. So two data sets can share a range and still have different standard deviations.

When you describe a comparison, talk about center and spread one at a time. If the means match but the standard deviations don't, the data sets share a center but differ in spread. If the means differ but the standard deviations match, the centers sit in different places but the spread is equal. And if both differ, say so for each.

You can often answer these questions without typing a single value. The whole idea fits in four words: adding slides, multiplying stretches.

Add the same number to every value

Say every student gets $5$ bonus points. Every score goes up $5$, and so does the mean. The scores keep the same distances from each other and from the mean, so the spread doesn't change.

In general, adding $c$ to every value raises the mean, the median, the minimum and the maximum by $c$. The range and the standard deviation stay the same.

Multiply every value by the same number

Now double every score. The mean doubles, and so does every gap: two scores that were $3$ apart are now $6$ apart.

In general, multiplying every value by $k$ multiplies the mean and the median by $k$. It stretches or squeezes every distance by $|k|$, so the range and the standard deviation get multiplied by $|k|$.

What if $k$ is negative? The data also flip, like a mirror image, so their order reverses. But a distance can't be negative, and the spread can't be either. That's why spread uses $|k|$.

See both in one list

Start with

Its mean is $8$ and its range is $12-4=8$. Then define F=E+7 and G=-3E. F is a slide: it runs from $11$ to $19$, with mean $15$ and the same spread as E. G flips and stretches: its values run from $-12$ down to $-36$, its mean is $-24$, and its range, $24$, is three times E's. Its standard deviation is three times E's, too.