Mean, median, and frequency tables

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Difficulty: Intermediate
Estimated time: 50 minutes
Domains: Problem Solving and Data Analysis
Techniques: StatisticsListsMeanMedianModeFrequency-tablesWeighted-mean

What you’ll learn

Enter raw data once as a Desmos list and reuse it with mean and median.
Calculate a weighted mean from value and frequency columns.
Explain why total frequency is the denominator of a frequency-table mean.
Locate a median from cumulative frequency, including an even number of observations.
Identify the mode as the value with the greatest frequency.
Compare the centers of two data sets without using regression.
Choose hand reasoning when it gives the result faster than calculator setup.

Prerequisites

Tables, lists, and calculated columns0%
Create lists, preserve paired table rows, and calculate with table columns.

Why this matters on the SAT

Reuse one data entry for every center calculation

SAT questions can ask for the mean or median of a raw data set, the weighted mean of a frequency table, or how one unusual value changes a measure of center. Desmos reduces repeated arithmetic: enter raw values once as a list, or enter values and frequencies as paired columns, then calculate from those saved data.

The mean is the total of the values divided by the number of observations. The median is the middle value, or the mean of the two middle values, after the data are ordered.

SAT example

The number of minutes nine students spent completing a task is shown below.

6,\ 10,\ 13,\ 2,\ 15,\ 22,\ 10,\ 4,\ 4

What is the mean number of minutes?

Fast Desmos solution

Define A=[6,10,13,2,15,22,10,4,4] and enter mean(A). Desmos displays $9.555\ldots$ . Use the fraction toggle to see the exact value

\boxed{\frac{86}{9}}.

This lesson focuses on measures of center for one-variable data. The next lesson covers spread, standard deviation, and data displays. Regression is not part of either calculation here because there is no relationship between an input variable and an output variable to model.

Calculator loads as you approach

The saved list lets you calculate the requested mean without retyping the data.

When should you use Desmos?

Choose the shortest method that preserves the information the question gives.

Use a list when…

the problem gives raw data values;
the values are unsorted or easy to mistype; or
you need several statistics from the same data.

Use a value-frequency table when…

each distinct value appears with a frequency;
expanding the data would require many repeated values; or
two data sets share values but have different frequencies.

Use hand reasoning when a sorted raw list makes the median immediate, the mode is visible from the greatest frequency, or cumulative frequency locates the median faster than a calculator expression. Do not use regression. A frequency table summarizes one variable; it does not give paired explanatory and response data.

Check your understanding:

A sorted list contains $11$ values, and the question asks only for the median. A different question gives six distinct values with frequencies totaling $80$ and asks for the mean. Which method is stronger for each problem?

1. Calculate from one reusable list

Suppose the raw data are

14,\ 9,\ 18,\ 12,\ 16,\ 11,\ 20,\ 12.

Define B=[14,9,18,12,16,11,20,12]. Then enter mean(B) and median(B). Desmos returns

\operatorname{mean}(B)=14

and

\operatorname{median}(B)=13.

The median calculation automatically orders the values. Because there are $8$ values, it averages the fourth and fifth ordered values:

\frac{12+14}{2}=13.

Check your understanding:

Change the value $20$ in list $B$ to $28$ . Without retyping the list, determine the new mean and median. Which statistic changes?

Common mistake:

Retyping the data inside every statistic and changing or omitting one value. Define one named list, compare it with the prompt, and reuse that same list.

Calculator loads as you approach

Define the list once, then reuse its name for each statistic.

2. Calculate a weighted mean from frequencies

A frequency tells you how many times a value occurs.

Value	Frequency
$2$	$1$
$4$	$3$
$6$	$4$
$8$	$2$

Put the values in x_1 and their matching frequencies in y_1. The weighted mean is

\frac{\text{sum of the value-frequency products}}{\text{total frequency}}.

In Desmos, enter total(x_1*y_1)/total(y_1). The numerator multiplies every value by its frequency and adds the products:

2(1)+4(3)+6(4)+8(2)=54.

The denominator counts all observations:

1+3+4+2=10.

Therefore, the mean is

\frac{54}{10}=\boxed{5.4}.

The number of rows is $4$ , but the data set contains $10$ observations. Dividing by $4$ would treat each distinct value as if it appeared once.

Check your understanding:

A student divides the weighted total by the four table rows and gets $13.5$ . Explain the error and give the correct denominator.

Common mistake:

Entering total(x_1)/total(y_1). This adds each distinct value only once and ignores how often it occurs. Multiply the paired columns before taking the numerator total.

Calculator loads as you approach

The value and frequency columns stay paired in every product.

3. Read median and mode from a frequency table

Value	Frequency	Observation positions
$2$	$1$	$1$
$4$	$3$	$2$ through $4$
$6$	$4$	$5$ through $8$
$8$	$2$	$9$ through $10$

There are $10$ observations, so the median lies halfway between positions $5$ and $6$ . Both positions contain $6$ , so $\operatorname{median}=6$ . The greatest frequency is $4$ , attached to $6$ , so $\operatorname{mode}=6$ .

For a frequency-table median:

Add frequencies to find the total observations.
Identify the middle position, or two middle positions for an even total.
Add frequencies from the smallest value upward until you reach those positions.
Read the value or values there.

For the mode, find the greatest frequency and report its corresponding value. If several values share the greatest frequency, all are modes. If every value occurs equally often, there is no unique mode.

Check your understanding:

A frequency table has values $1$ , $3$ , $5$ , and $7$ with frequencies $2$ , $5$ , $4$ , and $3$ . What are the median and mode?

Common mistake:

Choosing the middle table row as the median. Rows represent distinct values, not equal groups of observations. Frequencies determine where the middle observations fall.

Common mistake:

Reporting the greatest frequency as the mode. The mode is the data value associated with that frequency.

4. Example: Compare means and medians

Worked example

Data set P and data set Q each consist of $24$ test scores.

Score	Data set P frequency	Data set Q frequency
$60$	$6$	$2$
$70$	$7$	$4$
$80$	$5$	$5$
$90$	$4$	$7$
$100$	$2$	$6$

Which statement correctly compares the means and medians of the two data sets?

A
The mean of data set P is greater, and the medians are equal.
B
The means are equal, and the median of data set P is less.
C
The mean of data set P is less, and the medians are equal.
D
The mean and median of data set P are both less than the mean and median of data set Q.

Step 1

Keep the shared values aligned

Enter scores in x_1, P frequencies in y_1, and Q frequencies in y_2. Each frequency must remain in the row of its score.

Step 2

Calculate both means

Use total(x_1*y_1)/total(y_1) and total(x_1*y_2)/total(y_2). Desmos gives

\operatorname{mean}(P)=\frac{905}{12}\approx75.42

and

\operatorname{mean}(Q)=\frac{1015}{12}\approx84.58.

The mean of P is less than the mean of Q.

Calculator loads as you approach

Read the expression results; the plotted table points are not the statistics being compared.

Step 3

Locate both medians

Positions $12$ and $13$ determine each median. For P, both contain $70$ , so $\operatorname{median}(P)=70$ . For Q, both contain $90$ , so $\operatorname{median}(Q)=90$ .

Step 4

Match the complete comparison

Both comparisons point in the same direction:

\operatorname{mean}(P)<\operatorname{mean}(Q)

and

\operatorname{median}(P)<\operatorname{median}(Q).

Therefore, the answer is D.

Common mistake:

Comparing only the frequencies in the final row. Every value and frequency contributes to the mean, and the middle observation positions determine the median.

The calculator is strongest for the weighted means. Cumulative counting is clearer for the medians. One SAT problem can reward both methods. Equal centers still would not prove that two data sets have the same spread or shape.

Finish the solution

The value-frequency table is already entered. Add the weighted-mean expression and submit the mean.

Finish the weighted mean

Finish the solution

The table summarizes a data set.

Value	Frequency
$12$	$2$
$16$	$5$
$20$	$4$
$24$	$1$

What is the mean of the data set?

First steps

Identify the value and frequency columns.
Enter one expression that divides the weighted total by the total frequency.
Submit the mean.

Finish it

Your answer

Calculator loads as you approach

Use the entered table to calculate the mean.

Practice problems

SAT practice problems

Choose the shortest reliable method. Use lists for raw data, value-frequency tables for weighted means, cumulative counting for frequency-table medians, and the greatest frequency for the mode.

Find a median from raw data

Practice problem

The recorded travel times, in minutes, for eight trips are

18,\ 12,\ 15,\ 23,\ 19,\ 12,\ 21,\ 17.

What is the median travel time, in minutes?

Your answer

Calculator loads as you approach

Use the calculator to determine the requested median.

Find a weighted mean

Practice problem

The table summarizes the points scored by a player in each of $10$ games.

Points	Frequency
$3$	$2$
$6$	$4$
$9$	$3$
$12$	$1$

What is the mean number of points scored per game?

Your answer

Calculator loads as you approach

Use the calculator to determine the requested mean.

Compare center after adding a value

Practice problem

The table summarizes delivery times over $21$ days.

Delivery time (minutes)	Frequency
$12$	$3$
$13$	$4$
$14$	$6$
$15$	$5$
$16$	$3$

One additional delivery takes $30$ minutes and is added to the data set. Which statement correctly compares the new mean, median, and mode with the originals?

Calculator loads as you approach

Use the calculator only if it shortens your reasoning.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Define a raw data list once, verify every entry, and reuse its name with mean(...) and median(...).
For a frequency table, use total(x_1*y_1)/total(y_1) for the mean.
The denominator is total frequency because it counts all observations.
Locate a median by cumulative positions, not the middle table row.
The mode is the value with the greatest frequency.
Use hand reasoning when ordered values or cumulative counts reveal the answer faster.
Measures of center do not describe spread or shape.

Next lesson

Spread, standard deviation, and data displays

Compare variability and interpret one-variable data displays without confusing center and spread.

Start next lesson

Free Practice

Free question bank practice problems

One-variable data: distributions and measures of center and spread