Mean Median Mode - Digital SAT Math: Concepts, 20+ Practice Questions & Shortcuts

Key Concepts & Worked Archetypes

Concept 1

Concept 1: The Arithmetic Mean

The mean is the sum of all values divided by the total count of values.

Sum of values: $\text{Sum} = \text{Mean} \times n$
Effect of adding a constant $k$ to every term: $\text{New Mean} = \text{Old Mean} + k$
Effect of multiplying every term by $k$: $\text{New Mean} = \text{Old Mean} \times k$

📘 Traditional Algebraic Method

Sum the data points and divide by the number of entries $n$.

⚡ SAT Speed Trick & Desmos Hack

Use the Desmos 'mean()' function by typing the list into a variable, e.g., L = [x1, x2, x3], then typing mean(L).

💡 Worked SAT Archetype Example

Problem: The mean of 5 numbers is 12. If a 6th number, 18, is added, what is the new mean?

📘 Step-by-Step Textbook Solution:

Step 1

Calculate the original sum: $5 \times 12 = 60$

Step 2

Add the new number to the sum: $60 + 18 = 78$

Step 3

Divide by the new count: $78 / 6 = 13$

Step 4

The new mean is 13.

⚡ Speed / Desmos Tactic:

Step 1

Define list L = [12, 12, 12, 12, 12, 18]

Step 2

Type mean(L) in a new line

Step 3

Read result 13

Concept 2

Concept 2: Median and Ordering

The median is the middle value of a data set when ordered from least to greatest.

For odd $n$: $\text{Median} = \text{value at position } (n+1)/2$
For even $n$: $\text{Median} = \text{average of values at } n/2 \text{ and } (n/2)+1$
The median is resistant to outliers, unlike the mean.

📘 Traditional Algebraic Method

Sort the data set in ascending order and identify the middle term.

⚡ SAT Speed Trick & Desmos Hack

Use the Desmos 'median()' function on a defined list.

💡 Worked SAT Archetype Example

Problem: Find the median of the set: {15, 3, 8, 12, 20, 5}.

📘 Step-by-Step Textbook Solution:

Step 1

Sort the set: {3, 5, 8, 12, 15, 20}

Step 2

Identify the two middle terms: 8 and 12

Step 3

Calculate the average: $(8 + 12) / 2 = 10$

Step 4

The median is 10.

⚡ Speed / Desmos Tactic:

Step 1

Define list L = [15, 3, 8, 12, 20, 5]

Step 2

Type median(L) in a new line

Step 3

Read result 10

Concept 3

Concept 3: Mode and Frequency

The mode is the value that appears most frequently in a data set.

A set can have one mode, multiple modes (multimodal), or no mode.
If all values appear with the same frequency, there is no mode.
The mode is the peak of a frequency histogram.

📘 Traditional Algebraic Method

Create a frequency table to count occurrences of each value.

⚡ SAT Speed Trick & Desmos Hack

Use the 'dotplot' or 'histogram' visualization in Desmos if data is provided in a table.

💡 Worked SAT Archetype Example

Problem: Find the mode of: {4, 7, 4, 9, 7, 4, 2}.

📘 Step-by-Step Textbook Solution:

Step 1

Count occurrences of 4: 3 times

Step 2

Count occurrences of 7: 2 times

Step 3

Count occurrences of 9: 1 time

Step 4

The value with the highest count is 4.

⚡ Speed / Desmos Tactic:

Step 1

Define list L = [4, 7, 4, 9, 7, 4, 2]

Step 2

Use the sort function to group values: sort(L)

Step 3

Observe the most frequent value is 4

Concept 4

Concept 4: Mean vs Median Sensitivity

The mean is sensitive to extreme values (outliers), while the median is robust.

If the distribution is skewed right, $\text{Mean} > \text{Median}$
If the distribution is skewed left, $\text{Mean} < \text{Median}$
If the distribution is symmetric, $\text{Mean} \approx \text{Median}$

📘 Traditional Algebraic Method

Compare the impact of adding a large value to the mean versus the median.

⚡ SAT Speed Trick & Desmos Hack

Test with a small set: {1, 2, 3} vs {1, 2, 100}.

💡 Worked SAT Archetype Example

Problem: Which measure changes more when an outlier is added to a set?

📘 Step-by-Step Textbook Solution:

Step 1

Original set {1, 2, 3}: Mean = 2, Median = 2

Step 2

Add outlier 100: {1, 2, 3, 100}

Step 3

New Mean: $106 / 4 = 26.5$

Step 4

New Median: $(2 + 3) / 2 = 2.5$

Step 5

The mean changed by 24.5, while the median changed by 0.5.

⚡ Speed / Desmos Tactic:

Step 1

Define L1 = [1, 2, 3]

Step 2

Define L2 = [1, 2, 3, 100]

Step 3

Compare mean(L1), mean(L2) and median(L1), median(L2)

Practice Questions (20)

Question 1 Basic Calculation

Easy

The set of numbers $\{4, 8, 12, 16, 20\}$ represents the number of items sold by a store over 5 days. What is the mean of this set?

📘 Step-by-Step Algebraic Solution

Step 1

Sum the values: $4 + 8 + 12 + 16 + 20 = 60$

Step 2

Divide by the count of numbers: $60 / 5$

Step 3

Result: $12$

⚡ Desmos Shortcut / Speed Hack

Step 1

Observe the set is an arithmetic sequence.

Step 2

For an arithmetic sequence, the mean equals the median.

Step 3

The middle number is 12.

Question 2 Basic Calculation

Easy

What is the median of the data set $\{15, 22, 15, 30, 18\}$?

📘 Step-by-Step Algebraic Solution

Step 1

Sort the list: $15, 15, 18, 22, 30$

Step 2

Identify the middle value in the sorted list.

Step 3

The middle value is $18$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Type 'median([15, 22, 15, 30, 18])' into Desmos.

Step 2

Read the output.

Question 3 Basic Calculation

Medium

A data set consists of 6 numbers. If the sum of the numbers is 150 and the mean is 25, what is the value of the 6th number if the sum of the first 5 numbers is 120?

📘 Step-by-Step Algebraic Solution

Step 1

Total sum = $150$

Step 2

Sum of first 5 = $120$

Step 3

6th number = $150 - 120 = 30$

⚡ Desmos Shortcut / Speed Hack

Step 1

Recognize that the mean of 25 for 6 numbers implies a total of $6 \times 25 = 150$.

Step 2

Subtract the known sum from the total: $150 - 120 = 30$.

Question 4 Basic Calculation

Medium

The mode of a set of 7 integers is 10. If the set is $\{8, 10, 12, x, 10, 15, 9\}$, what must be true about $x$?

📘 Step-by-Step Algebraic Solution

Step 1

Count current frequencies: 10 appears twice, others appear once.

Step 2

To keep 10 as the mode, $x$ cannot create a new mode.

Step 3

If $x = 10$, the mode is still 10. If $x \neq 10$, the mode is still 10.

Step 4

Actually, $x$ can be anything, but the question implies $x$ is not 10 to maintain the current set structure.

⚡ Desmos Shortcut / Speed Hack

Step 1

Check the options.

Step 2

If $x=10$, 10 appears 3 times. If $x=5$, 10 appears 2 times. Both keep 10 as the mode.

Step 3

The most restrictive condition is $x \neq 10$ is not required, but $x$ cannot be a number that appears twice.

Question 5 Basic Calculation

Hard

A set of 5 positive integers has a mean of 10, a median of 11, and a mode of 12. What is the smallest possible value for the smallest integer in the set?

📘 Step-by-Step Algebraic Solution

Step 1

Set is $a, b, 11, 12, 12$.

Step 2

Sum: $a + b + 11 + 12 + 12 = 50$.

Step 3

$a + b = 15$.

Step 4

To minimize $a$, maximize $b$. Since median is 11, $b \leq 11$.

Step 5

If $b = 11$, $a = 4$. If $b = 12$, median would change. So $b=11, a=4$. Wait, check constraints.

Step 6

Smallest $a$ is 3 if $b=12$ (but mode is 12, so $b$ can be 12).

⚡ Desmos Shortcut / Speed Hack

Step 1

Sum is 50.

Step 2

$a + b + 11 + 12 + 12 = 50 \implies a + b = 15$.

Step 3

Since $b \leq 11$, $a \geq 4$. If $b=12$, $a=3$.

Question 6 Missing Value Algebra

Easy

The mean of $\{5, 10, x, 20\}$ is 12. What is the value of $x$?

📘 Step-by-Step Algebraic Solution

Step 1

$(5 + 10 + x + 20) / 4 = 12$

Step 2

$35 + x = 48$

Step 3

$x = 13$

⚡ Desmos Shortcut / Speed Hack

Step 1

Total sum must be $4 \times 12 = 48$.

Step 2

$48 - (5 + 10 + 20) = 13$.

Question 7 Missing Value Algebra

Easy

If the median of $\{2, 5, x, 10, 12\}$ is 7, what is $x$?

📘 Step-by-Step Algebraic Solution

Step 1

Sorted list: $2, 5, x, 10, 12$ (assuming $5 < x < 10$).

Step 2

The middle value is $x$.

Step 3

Therefore $x = 7$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Look at the sorted order.

Step 2

The middle position is the 3rd number.

Step 3

$x$ must be 7.

Question 8 Missing Value Algebra

Medium

A set of 4 numbers has a mean of 10 and a median of 10. If the smallest number is 6, what is the largest number?

📘 Step-by-Step Algebraic Solution

Step 1

$6 + x + y + z = 40$.

Step 2

$(x + y) / 2 = 10 \implies x + y = 20$.

Step 3

$6 + 20 + z = 40 \implies z = 14$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Total sum is 40.

Step 2

Sum of middle two is 20.

Step 3

$40 - 6 - 20 = 14$.

Question 9 Missing Value Algebra

Medium

The mean of 5 numbers is 20. If one number is removed, the mean of the remaining 4 numbers is 18. What was the removed number?

📘 Step-by-Step Algebraic Solution

Step 1

Sum of 5 = $5 \times 20 = 100$.

Step 2

Sum of 4 = $4 \times 18 = 72$.

Step 3

Removed number = $100 - 72 = 28$.

⚡ Desmos Shortcut / Speed Hack

Step 1

$5 \times 20 - 4 \times 18 = 100 - 72 = 28$.

Question 10 Missing Value Algebra

Hard

A set of 5 distinct positive integers has a mean of 12 and a median of 10. What is the maximum possible value of the largest integer?

📘 Step-by-Step Algebraic Solution

Step 1

Set is $a, b, 10, d, e$. Sum = 60.

Step 2

To maximize $e$, minimize $a, b, d$.

Step 3

$a=1, b=2$ (distinct positive). $d=11$ (must be $> 10$).

Step 4

$1 + 2 + 10 + 11 + e = 60$.

Step 5

$24 + e = 60 \implies e = 36$. Wait, check constraints.

Step 6

$d$ must be $> 10$. Smallest $d$ is 11. $e = 60 - 24 = 36$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Sum = 60.

Step 2

$1 + 2 + 10 + 11 + e = 60$.

Step 3

$e = 36$.

Question 11 Data Interpretation

Easy

The table shows test scores: 70, 80, 80, 90, 100. What is the mode?

📘 Step-by-Step Algebraic Solution

Step 1

80 appears twice.

Step 2

All other scores appear once.

Step 3

Mode is 80.

⚡ Desmos Shortcut / Speed Hack

Step 1

Scan for the number that repeats most.

Question 12 Data Interpretation

Easy

A dot plot shows values 1, 2, 2, 3, 4. What is the median?

📘 Step-by-Step Algebraic Solution

Step 1

List: 1, 2, 2, 3, 4.

Step 2

Middle is the 3rd value.

Step 3

Median is 2.

⚡ Desmos Shortcut / Speed Hack

Step 1

Count dots from both ends to the middle.

Question 13 Data Interpretation

Medium

In a frequency table, the value 5 appears 3 times, 10 appears 2 times, and 15 appears 1 time. What is the mean?

📘 Step-by-Step Algebraic Solution

Step 1

Sum = $(5 \times 3) + (10 \times 2) + (15 \times 1) = 15 + 20 + 15 = 50$.

Step 2

Total count = $3 + 2 + 1 = 6$.

Step 3

Mean = $50 / 6 = 8.33$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Use Desmos list: [5, 5, 5, 10, 10, 15].

Step 2

mean([5, 5, 5, 10, 10, 15]).

Question 14 Data Interpretation

Medium

A histogram shows 5 students scored 60-70, 10 scored 70-80, and 5 scored 80-90. What is the median range?

📘 Step-by-Step Algebraic Solution

Step 1

Total = 20. Median is between 10th and 11th.

Step 2

1st-5th are 60-70.

Step 3

6th-15th are 70-80.

Step 4

10th and 11th are in 70-80.

⚡ Desmos Shortcut / Speed Hack

Step 1

Cumulative frequency: 5, 15, 20.

Step 2

10th and 11th fall in the second bin.

Question 15 Data Interpretation

Hard

A set of data has a mean of 50 and a standard deviation of 0. What is the median?

📘 Step-by-Step Algebraic Solution

Step 1

SD = 0 implies all values are identical.

Step 2

If all values are $x$, mean = $x$.

Step 3

Since mean = 50, all values are 50.

Step 4

Median of a set of 50s is 50.

⚡ Desmos Shortcut / Speed Hack

Step 1

SD=0 means constant data.

Step 2

Mean = Median = Mode = Constant.

Question 16 Conceptual/Properties

Easy

If every number in a set is increased by 5, what happens to the mean?

📘 Step-by-Step Algebraic Solution

Step 1

New sum = $\sum (x_i + 5) = \sum x_i + 5n$.

Step 2

New mean = $(\sum x_i + 5n) / n = \text{mean} + 5$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Test with {1, 2, 3}. Mean = 2.

Step 2

New set {6, 7, 8}. Mean = 7.

Step 3

$7 - 2 = 5$.

Question 17 Conceptual/Properties

Easy

Which measure of central tendency is most affected by an outlier?

📘 Step-by-Step Algebraic Solution

Step 1

Mean is sensitive to extreme values because it sums all data.

Step 2

Median and mode are positional or frequency-based.

⚡ Desmos Shortcut / Speed Hack

Step 1

Recall: Mean is 'pulled' by outliers.

Question 18 Conceptual/Properties

Medium

If a set of data is perfectly symmetric, which of the following is true?

📘 Step-by-Step Algebraic Solution

Step 1

In a symmetric distribution, the left and right sides balance.

Step 2

This balance point is both the mean and the median.

⚡ Desmos Shortcut / Speed Hack

Step 1

Think of a normal distribution curve.

Question 19 Conceptual/Properties

Medium

If every number in a set is multiplied by 2, what happens to the median?

📘 Step-by-Step Algebraic Solution

Step 1

If $x_1 < x_2 < x_3$, then $2x_1 < 2x_2 < 2x_3$.

Step 2

The middle value $x_2$ becomes $2x_2$.

⚡ Desmos Shortcut / Speed Hack

Step 1

Test with {1, 2, 3}. Median = 2.

Step 2

New set {2, 4, 6}. Median = 4.

Question 20 Conceptual/Properties

Hard

A set of 10 numbers has a mean of 20. If we add a new number 30 to the set, what is the new mean?

📘 Step-by-Step Algebraic Solution

Step 1

Old sum = $10 \times 20 = 200$.

Step 2

New sum = $200 + 30 = 230$.

Step 3

New mean = $230 / 11 \approx 20.9$.

⚡ Desmos Shortcut / Speed Hack

Step 1

New value is 10 above the old mean.

Step 2

Spread that 10 over 11 items: $10/11 \approx 0.9$.

Step 3

$20 + 0.9 = 20.9$.

Official SAT Exam Blueprint & Weightage

Metric	Exam Pattern & Weightage
Question Frequency	1-3 questions per test module
Module 1 Appearance	Medium Frequency (Foundational tests)
Module 2 Appearance	High Frequency (Hard module score differentiator)
Pattern Analysis	College Board frequently tests the relationship between mean and sum, and the effect of outliers on measures of center.

🏛️

Official SAT PYQ Drill Bank (2023–2026)

We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Mean Median Mode.

Updated for 2026 Testing Season

Revision, Traps & Desmos Syntax

Mean

$\bar{x} = \frac{\sum x_i}{n}$

Used to find the average of a data set.

Sum Property

$\sum x_i = \bar{x} \cdot n$

Crucial for 'missing value' problems.

🚨 Top SAT Traps & Misconceptions

⚠️ SAT Trap: The 'Average of Averages' Trap

Never average two means unless the sample sizes are identical. Always calculate the total sum first.

⚠️ SAT Trap: The 'Missing Value' Trap

Forgetting to account for the new total count $n$ when adding a value to a set.

⚡ Essential Desmos Cheatsheet

🎯 List Processing

L = [x1, x2, ...]

Always define your data as a list to use mean(), median(), and total() functions.

3-Level Mock Test (30 Questions)

🟢 Level 1: Foundation

10 Qs · Sub-600 Score

🟡 Level 2: Target 700+

10 Qs · 600–740 Score

🔴 Level 3: 800-Mastery

10 Qs · 750–800 Score

Question 1 Level 1: Foundation

The set of numbers is $\{2, 4, 4, 6, 9\}$. What is the mode of this set?

Question 2 Level 1: Foundation

What is the mean of the set $\{10, 20, 30, 40\}$?

Question 3 Level 1: Foundation

Find the median of the set $\{3, 1, 7, 5, 9\}$.

Question 4 Level 1: Foundation

If the mean of $\{x, 5, 8\}$ is 7, what is the value of $x$?

Question 5 Level 1: Foundation

What is the median of $\{12, 15, 12, 18, 20, 15\}$?

Question 6 Level 1: Foundation

A student scored 80, 90, and 70 on three tests. What is the mean score?

Question 7 Level 1: Foundation

Which measure of central tendency is most affected by an outlier?

Question 8 Level 1: Foundation

Find the mode of the data: $2, 3, 3, 4, 5, 5, 5, 6$.

Question 9 Level 1: Foundation

The mean of 5 numbers is 10. What is their sum?

Question 10 Level 1: Foundation

What is the median of $\{10, 2, 38, 4, 5\}$?

Question 11 Level 2: Target 700+

The mean of a set of 6 numbers is 12. If one number, 17, is removed, what is the mean of the remaining 5 numbers?

Question 12 Level 2: Target 700+

A data set has a mean of 20 and a median of 18. If every value in the set is increased by 5, what are the new mean and median?

Question 13 Level 2: Target 700+

The mean of 4 numbers is $x$. If a 5th number, $y$, is added, what is the new mean?

Question 14 Level 2: Target 700+

In a set of 5 distinct positive integers, the median is 10 and the mean is 12. What is the maximum possible value for the largest integer?

Question 15 Level 2: Target 700+

The mode of $\{4, 7, 4, 9, x, 9, 9\}$ is 9. What is the smallest possible integer value for $x$?

Question 16 Level 2: Target 700+

A set of 10 numbers has a mean of 15. A set of 20 numbers has a mean of 30. What is the mean of the combined 30 numbers?

Question 17 Level 2: Target 700+

If the median of $\{2, 5, 8, x, 15\}$ is 7, what is $x$?

Question 18 Level 2: Target 700+

The average of 5 consecutive integers is 10. What is the largest integer?

Question 19 Level 2: Target 700+

If the mean of $x, x+2, x+4$ is 10, what is the median?

Question 20 Level 2: Target 700+

A set has 4 numbers. The mean is 10 and the range is 8. If the smallest number is 6, what is the largest?

Question 21 Level 3: 800 Mastery

A set of 10 integers has a mean of 10.5. If the sum of the 5 smallest integers is 20, what is the maximum possible value of one of the 5 largest integers?

Question 22 Level 3: 800 Mastery

The mean of a set of $n$ numbers is 20. If each number $x_i$ is replaced by $2x_i + 3$, what is the new mean?

Question 23 Level 3: 800 Mastery

In a set of 7 numbers, the mean is 15 and the median is 15. If the 3 smallest numbers are 10, 11, 12, what is the maximum possible value of the largest number?

Question 24 Level 3: 800 Mastery

The mean of 5 numbers is 10. If the median is 12 and the mode is 15, what is the smallest possible value for the smallest number?

Question 25 Level 3: 800 Mastery

A set of 4 numbers has a mean of 10 and a median of 10. If the range is 10, what is the set?

Question 26 Level 3: 800 Mastery

If the mean of $x_1, x_2, x_3, x_4$ is $M$, what is the mean of $x_1-k, x_2-k, x_3-k, x_4-k$?

Question 27 Level 3: 800 Mastery

The mean of 10 numbers is 20. If one number is replaced by 40, the new mean is 22. What was the original number?

Question 28 Level 3: 800 Mastery

A set of 5 numbers has mean 10, median 10, and mode 10. If the numbers are integers and the range is 4, what is the set?

Question 29 Level 3: 800 Mastery

If the mean of $a, b, c$ is 10 and the mean of $c, d, e$ is 20, what is the mean of $a, b, c, d, e$?

Question 30 Level 3: 800 Mastery

The mean of 5 numbers is 10. If we add a 6th number, the mean becomes 12. What is the 6th number?