Type mean(L) or median(L) for a list. With a frequency table, plain mean() gives the wrong answer. Here’s the fix, with SAT examples.
mean(...) and median(...).Why this matters on the SAT
The SAT asks about the center of a data set in a few ways: the mean or median of a list of numbers, the mean of a frequency table, or how one unusual value shifts the center. Desmos saves you the adding. Type raw numbers once as a list, or type values and their frequencies as two paired columns, and every calculation works from what you typed.
Two words to have ready. The mean is the total of the values divided by how many values there are. The median is the middle value once the data are in order. With an even number of values there are two middle ones, and the median is their mean.
SAT example
The number of minutes nine students spent completing a task is shown below.
What is the mean number of minutes?
Give the list a name:
A=[6,10,13,2,15,22,10,4,4]
Then type mean(A). Desmos shows , and the fraction toggle turns that into the exact value:
If a follow-up question asked for the median, median(A) would use the same list, with no retyping.
Here you’ll work with measures of center for one-variable data. Spread, standard deviation, and data displays come in the next lesson.
Look at how the question hands you the data. Raw numbers go in a list. Values with counts go in a value-frequency table. And sometimes hand reasoning gets you there first.
The question gives the raw numbers, like the nine task times above.
The numbers are out of order, like , , , , so sorting or adding them in your head invites a slip.
You need more than one statistic, like both the mean and the median, from the same numbers.
The question shows each value next to how many times it occurs.
Writing out every repeat would take forever, like typing nine times and twelve times.
Two data sets share the same values but have different frequencies, like two classes taking the same test.
Use hand reasoning when the answer is already in view. A sorted list shows you its middle, the biggest frequency points straight to the mode, and cumulative counting (adding up frequencies) can find the median faster than typing a longer expression. Leave regression out of this one: a frequency table describes one variable, so there are no input and output pairs to fit a model to.
A sorted list has values, and the question asks only for the median. A different question gives six distinct values whose frequencies add up to , and it asks for the mean. Which method would you pick for each?
Here's a new set of raw data:
Name it:
B=[14,9,18,12,16,11,20,12]
Then type mean(B) and median(B). Desmos returns
and
You don't have to sort the numbers first. median(B) puts them in order for you and then finds the middle. In order, they're . With values there's no single middle one, so the median is the mean of the th and th values:
Change the in list to , without retyping the rest of the list. What are the new mean and median? Which one changed?
Typing the numbers again inside each statistic, once for the mean and once for the median. Every retype is another chance to drop or mistype a value. Define the list once, check it against the numbers in the question, and reuse its name.
A frequency tells you how many times a value occurs. Take this table:
| Value | Frequency |
|---|---|
It describes one , three s, four s and two s. Put the values in x_1 and their frequencies in y_1. The mean of all those numbers is a weighted mean:
In Desmos, type
total(x_1*y_1)/total(y_1)
The top multiplies each value by its frequency and adds the results. That's the same as adding all ten numbers one by one:
The bottom counts every observation:
So the mean is
Here's the part that trips people up. The table has rows, but it holds observations. Divide by and you're treating each value as if it showed up only once. Count observations, not rows.
A student divides the weighted total by the four table rows and gets . What went wrong, and what should the denominator be?
Typing total(x_1)/total(y_1). That adds each value only once and ignores how often it occurs. Here it gives , which can't be right when of the numbers are or more. Multiply the paired columns first, then total.
Stay with the same table, and add a column showing which positions each value fills once all numbers are in order:
| Value | Frequency | Positions in order |
|---|---|---|
| through | ||
| through | ||
| through |
With observations, the median sits halfway between positions and . Both of those positions hold a , so
The greatest frequency is , and it belongs to the value , so
Here's how to find a frequency-table median every time:
For the mode, find the greatest frequency and report the value that goes with it. If several values tie for the greatest frequency, they're all modes. If every value occurs equally often, there's no unique mode.
A frequency table has values , , , and with frequencies , , , and . What are the median and the mode?
Picking the middle row of the table as the median. In the table above, the middle rows are and , which would give , but the real median is . Rows are values, and some values fill more positions than others, so let the frequencies tell you where the middle is.
Reporting the greatest frequency as the mode. In the table above, is how often appears, so the mode is , not . The mode is always one of the data values.
Worked example
Data set P and data set Q each consist of test scores. The table shows how many times each score occurs.
| Score | Data set P frequency | Data set Q frequency |
|---|---|---|
Which statement correctly compares the means and medians of the two data sets?
The mean of data set P is greater, and the medians are equal.
The means are equal, and the median of data set P is less.
The mean of data set P is less, and the medians are equal.
The mean and median of data set P are both less than the mean and median of data set Q.
Step 1
Put the scores in x_1, P's frequencies in y_1, and Q's frequencies in y_2. Each frequency belongs to the score in its row, so keep the rows lined up.
Step 2
Type one weighted-mean expression for each data set:
total(x_1*y_1)/total(y_1)
and
total(x_1*y_2)/total(y_2)
Desmos gives
and
So P's mean is lower than Q's.
Step 3
Each data set has observations, so each median sits between positions and .
For P, the s fill positions through and the s fill through . Both middle positions hold :
For Q, the running count reaches by the end of the s, and the s fill positions through . Both middle positions hold :
P's median is lower too.
Step 4
Both comparisons point the same way:
and
P is lower on both, so the answer is D.
Deciding from one row, like the row, where Q has and P has . Every row feeds the mean, and the median depends on where the middle positions land, so use the whole table.
Notice the split. The calculator did the heavy lifting on the means, and cumulative counting was quicker for the medians. One SAT question can reward both. Matching centers wouldn't make two data sets alike, though. They could still differ in spread and shape, which comes up in the next lesson.
The table is already in the calculator. Add the weighted-mean expression, then submit the mean.
Finish the solution
The table summarizes a data set.
| Value | Frequency |
|---|---|
What is the mean of the data set?
Your turn. Before you type anything, look at how the data are given and which center the question asks for.
Practice problem
The recorded travel times, in minutes, for eight trips are
What is the median travel time, in minutes?
Practice problem
The table summarizes the number of points scored by a player in each of games.
| Points | Frequency |
|---|---|
What is the mean number of points scored per game?
Practice problem
The table summarizes delivery times over days.
| Delivery time (minutes) | Frequency |
|---|---|
One additional delivery takes minutes and is added to the data set. Which statement correctly compares the mean, median, and mode of the new data set with those of the original data set?
Finish the lesson
Finish the remaining questions correctly to complete this lesson.
mean(...) and median(...).total(x_1*y_1)/total(y_1).Next lesson
Compare variability and interpret one-variable data displays without confusing center and spread.
Start next lessonPractice
456 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.
Start practice