SAT Math Lesson Plan 12: Measures of Central Tendency

Lesson Summary

This 45-minute lesson focuses on understanding and applying measures of central tendency—mean, median, mode, and range—as well as more advanced concepts such as weighted averages and quartiles. These concepts account for approximately 5–10% of the questions on the SAT Math section. Students will learn how to compute these values, interpret them in the context of real-world data, and use them to answer SAT-style questions. The lesson includes a Warm-Up, six detailed examples, a Review with worked-out problems, and a 10-question quiz with solutions. This is Lesson 12 in a 35-lesson series designed to build strong SAT Math foundations through guided instruction, interactive tools, and strategic problem solving.

Lesson Objectives

• Define and calculate the mean, median, mode, and range of a data set.
• Understand and compute weighted averages.
• Interpret and determine quartiles and the interquartile range.
• Apply all these concepts to solve real-world and SAT-style problems.

Common Core Standards

• CCSS.MATH.CONTENT.HSS.ID.A.2 – Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets.
• CCSS.MATH.CONTENT.HSS.ID.A.3 – Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Prerequisite Skills

• Basic arithmetic operations: addition, subtraction, multiplication, division
• Comfort working with fractions, decimals, and percentages
• Ability to interpret data from tables and charts

Key Vocabulary

• Mean: The sum of all values divided by the number of values.
• Median: The middle value in a sorted data set.
• Mode: The most frequent value(s) in a data set.
• Range: The difference between the highest and lowest values.
• Weighted Average: An average that assigns importance to certain values based on their “weight.”
• Quartiles: Values that divide a data set into four equal parts: Q1 (lower quartile), Q2 (median), and Q3 (upper quartile).
• Categorical Data: Data that represents groups or categories (e.g., colors, brands, or preferences). For this type of data, the mode is the only applicable measure of central tendency.

Warm Up

Start with a basic review of how to compute the mean, median, mode, and range from a small data set. These fundamentals will support your understanding of more complex examples later in the lesson.

Problem: The math scores for five students are: 72, 85, 88, 91, 92

Step 1: Find the mean

$$\text{Mean} = \frac{72 + 85 + 88 + 91 + 92}{5} = \frac{428}{5} = 85.6$$

Step 2: Find the median

The data is already ordered. The middle number is the third number: 88

Step 3: Find the mode

Each value occurs only once. There is no mode.

Step 4: Find the range

$$\text{Range} = 92 - 72 = 20$$

Summary: Mean = 85.6, Median = 88, Mode = none, Range = 20

Teach

On the SAT, measures of central tendency—particularly mean and median—appear regularly in the context of interpreting data sets, charts, and word problems. You’ll need to understand when to use each measure, how to calculate them efficiently, and how to interpret their meaning in real-world or abstract scenarios. Some questions may involve estimating values from a graph, comparing two data sets, or evaluating the effect of an outlier.

In the following examples, you will learn how to:

• Quickly calculate mean, median, mode, and range
• Identify when a data set is skewed and how that affects the mean and median
• Use weighted averages in common SAT contexts like grades and test scores
• Work with quartiles to understand data spread
• Apply mode to categorical data when numerical measures are not applicable

Each example will include step-by-step solutions, visual representations when appropriate, and insights into how the SAT may present similar problems.

Example 1: Calculating the Mean

This problem demonstrates how to find the mean from a simple data set.

Problem: The ages of six students are: 16, 17, 17, 18, 18, 19. What is the mean age?

Step 1:

Add all the values: $$16 + 17 + 17 + 18 + 18 + 19 = 105$$

Step 2:

Divide the total by the number of values: $$\frac{105}{6} = 17.5$$

Answer: The mean age is 17.5 years.

Example 2: Finding the Median and Mode

This problem helps identify the middle and most frequent values in a data set.

Problem: The following values represent the number of goals scored by a soccer team over seven games: 2, 1, 3, 2, 4, 2, 5

Step 1: Order the data

1, 2, 2, 2, 3, 4, 5

Step 2: Median

The middle value is the fourth number: 2

Step 3: Mode

The number 2 occurs three times, more than any other value.

Answer: Median = 2, Mode = 2

Example 3: Determining the Range and Impact of an Outlier

This example demonstrates how an outlier affects the range and mean.

Problem: The quiz scores of five students are: 70, 75, 80, 85, 100

Step 1: Range

$$\text{Range} = 100 - 70 = 30$$

Step 2: Mean

$$\text{Mean} = \frac{70 + 75 + 80 + 85 + 100}{5} = \frac{410}{5} = 82$$

Step 3: Remove the outlier (100) and recalculate

$$\frac{70 + 75 + 80 + 85}{4} = \frac{310}{4} = 77.5$$

Observation: The outlier (100) increases the mean from 77.5 to 82 and also increases the range significantly.

Example 4: Calculating a Weighted Average

Weighted averages appear often in SAT contexts like grades, prices, and statistics.

Problem: A student’s grade is based on two quizzes and one final exam. The two quizzes together have a weight of 2, and the final exam has a weight of 3. The student scores 85 on both quizzes and 90 on the final exam. What is the weighted average score?

General Formula:

$$\text{Weighted Average} = \frac{(w_1 \times x_1) + (w_2 \times x_2)}{w_1 + w_2}$$

Where $w$ is the weight assigned to each score $x$.

Step 1:

Assign the values:
- Score of 85 (quizzes), weight = 2
- Score of 90 (final exam), weight = 3

Step 2:

$$\text{Weighted Average} = \frac{(2 \times 85) + (3 \times 90)}{2 + 3} = \frac{170 + 270}{5} = \frac{440}{5} = 88$$

Answer: The weighted average score is 88.

Example 5: Understanding Quartiles

This example demonstrates how to calculate quartiles and the interquartile range.

Problem: The following data set represents test scores: 60, 62, 67, 70, 72, 75, 77, 80, 85

Step 1: Find the median (Q2)

The middle value = 72 → This is Q2.

Step 2: Find Q1 (lower quartile)

Lower half: 60, 62, 67, 70 → Median of this half = $\frac{62 + 67}{2} = 64.5$

Step 3: Find Q3 (upper quartile)

Upper half: 75, 77, 80, 85 → Median = $\frac{77 + 80}{2} = 78.5$

Step 4: Interquartile Range (IQR)

$$\text{IQR} = Q3 - Q1 = 78.5 - 64.5 = 14$$

Answer: Q1 = 64.5, Q2 = 72, Q3 = 78.5, IQR = 14

Example 6: Real-World Application

This example reflects the type of data interpretation problem you might see on the SAT.

Problem: A company tracks the number of hours employees worked each week: 38, 40, 42, 44, 40, 39, 41

Step 1: Mean

$$\text{Total} = 38 + 40 + 42 + 44 + 40 + 39 + 41 = 284 \Rightarrow \text{Mean} = \frac{284}{7} \approx 40.57$$

Step 2: Median

Ordered: 38, 39, 40, 40, 41, 42, 44 → Median = 40

Step 3: Mode

40 appears twice → Mode = 40

Step 4: Range

44 - 38 = 6

Answer: Mean ≈ 40.57, Median = 40, Mode = 40, Range = 6

Example 7: Mode and Categorical Data

This example shows how mode is used with categorical (non-numeric) data.

Problem: Twenty students were asked to name their favorite type of fruit. The results are summarized below:

Question: What is the most common fruit choice among the students?

Step 1:

Identify the fruit with the highest frequency.

Step 2:

Strawberry was chosen by 6 students, more than any other fruit.

Answer: The mode is Strawberry.

Note: Mean, median, and range do not apply to this kind of data. For categorical data, the mode is the only valid measure of central tendency.

Review

This lesson explored how to compute and interpret the four main measures of central tendency: mean, median, mode, and range. You also learned how to apply weighted averages in real-world contexts and how to determine quartiles and the interquartile range (IQR). These concepts help describe the center and spread of data and are frequently tested on the SAT in the form of word problems, data tables, and graphs.

Here’s a summary of what was covered:

• How to compute the mean, median, mode, and range of a data set
• When to use each measure based on the context and data distribution
• How to calculate and interpret weighted averages
• How to find quartiles and the interquartile range (IQR)
• How outliers can affect the mean and range
• How to identify the mode in categorical (non-numeric) data

Example 1: Basic Central Tendency

Problem: Find the mean, median, and mode for the data set: 12, 14, 14, 15, 17

Step 1: Mean

$$\frac{12 + 14 + 14 + 15 + 17}{5} = \frac{72}{5} = 14.4$$

Step 2: Median

Ordered data → middle value is 14

Step 3: Mode

14 appears twice → Mode = 14

Answer: Mean = 14.4, Median = 14, Mode = 14

Example 2: Weighted Average

Problem: A student’s course grade is based on: Homework (25%) = 92, Midterm (35%) = 85, Final (40%) = 88

Step 1:

$$\text{Weighted Average} = (92 \times 0.25) + (85 \times 0.35) + (88 \times 0.4)$$ $$= 23 + 29.75 + 35.2 = 87.95$$

Answer: Final grade = 87.95

Example 3: Quartiles and IQR

Problem: Given the data set: 55, 60, 65, 70, 75, 80, 85, 90

Step 1: Q2 (median)

Even number of data points: $$\frac{70 + 75}{2} = 72.5$$

Step 2: Q1

Lower half: 55, 60, 65, 70 → Q1 = $\frac{60 + 65}{2} = 62.5$

Step 3: Q3

Upper half: 75, 80, 85, 90 → Q3 = $\frac{80 + 85}{2} = 82.5$

Step 4: IQR

$$82.5 - 62.5 = 20$$

Answer: Q1 = 62.5, Q2 = 72.5, Q3 = 82.5, IQR = 20

Example 4: Mode with Categorical Data

Problem: A group of 15 people were asked their favorite soft drink brand. Here are the results:

Question: Which soft drink is the most popular?

Step 1:

Identify the brand with the highest frequency.

Step 2:

Root Beer appears most often, with 5 preferences.

Answer: The mode is Root Beer.

Note: Because the data is categorical, you cannot calculate a mean, median, or range. Mode is the only meaningful measure in this case.

Multimedia Resources

For additional practice and support on measures of central tendency—including mean, median, mode, range, weighted averages, and quartiles—explore this curated collection of videos, tutorials, and worksheets from Media4Math:

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson12

Quiz

Directions: Choose the best answer for each question. Show your work on a separate sheet if needed.

1. The numbers in a data set are: 10, 15, 20, 25, 30. What is the mean?
   a) 15
   b) 20
   c) 22
   d) 25
2. What is the median of this data set? 8, 12, 14, 18, 22
   a) 12
   b) 13
   c) 14
   d) 15
3. Twenty students were surveyed about their favorite sport. The results are: Soccer (6), Basketball (5), Baseball (3), Tennis (4), Volleyball (2). What is the mode?
   a) Soccer
   b) Basketball
   c) Baseball
   d) Tennis
4. Find the range of this data set: 22, 18, 24, 20, 25
   a) 5
   b) 6
   c) 7
   d) 8
5. If the mean of five numbers is 12, what is their total sum?
   a) 60
   b) 50
   c) 62
   d) 56
6. A student scored 80 on a test worth 40% and 90 on a test worth 60%. What is their weighted average?
   a) 84
   b) 85
   c) 86
   d) 88
7. Which measure is most affected by an outlier?
   a) Median
   b) Mode
   c) Mean
   d) Range
8. Given the data set: 50, 55, 60, 65, 70, 75, 80. What is Q1?
   a) 55
   b) 57.5
   c) 60
   d) 62.5
9. The data set 70, 72, 75, 78, 100 has a large outlier. What is the median?
   a) 75
   b) 78
   c) 100
   d) 80
10. If a number is added to a data set and the mean increases, what can you conclude about the number added?
    a) It was lower than the previous mean
    b) It was equal to the mean
    c) It was higher than the previous mean
    d) It was the mode

Answer Key

1. Answer: b) 20
   $$\text{Mean} = \frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20$$
2. Answer: c) 14
   Ordered data → 8, 12, 14, 18, 22 → Middle value is 14
3. Answer: a) Soccer
   Soccer has the highest frequency (6). Since this is categorical data, the mode is the appropriate measure.
4. Answer: b) 6
   Range = 25 - 19 = 6
5. Answer: a) 60
   Mean × Number of values = 12 × 5 = 60
6. Answer: c) 86
   Weighted average = (80 × 0.4) + (90 × 0.6) = 32 + 54 = 86
7. Answer: c) Mean
   The mean is sensitive to extreme values (outliers).
8. Answer: b) 57.5
   Lower half = 50, 55, 60 → Q1 = $\frac{55 + 60}{2} = 57.5$
9. Answer: a) 75
   Ordered list = 70, 72, 75, 78, 100 → Median = 75
10. Answer: c) It was higher than the previous mean
    Adding a number above the current mean increases the mean.