AMC 8 · 2010 · #4

Easy mode Grade 6

Problem

Look at this list of numbers: $2, 3, 0, 3, 1, 4, 0, 3$ .

Find the mean (the average). Find the median (the middle value once they are in order). Find the mode (the number that appears most often).

What do you get when you add the mean, the median, and the mode together?

Pick an answer.

(A)

6.5

(B)

(C)

7.5

(D)

8.5

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Take the data set $2, 3, 0, 3, 1, 4, 0, 3$. Compute its mean, median, and mode, then add the three results together.

Givens: Data set: $2, 3, 0, 3, 1, 4, 0, 3$ ($8$ values); Three summary statistics to compute: mean, median, mode; Answer choices: (A) $6.5$, (B) $7$, (C) $7.5$, (D) $8.5$, (E) $9$

Unknowns: The single value $\text{mean} + \text{median} + \text{mode}$

Understand

Restated: Take the data set $2, 3, 0, 3, 1, 4, 0, 3$. Compute its mean, median, and mode, then add the three results together.

Givens: Data set: $2, 3, 0, 3, 1, 4, 0, 3$ ($8$ values); Three summary statistics to compute: mean, median, mode; Answer choices: (A) $6.5$, (B) $7$, (C) $7.5$, (D) $8.5$, (E) $9$

Plan

Primary tool: #2 Make a Systematic List

Secondary: #7 Identify Subproblems

The data is given in a jumbled order, so the first move is Tool #2 (Make a Systematic List): sort the eight numbers from smallest to largest. That single ordered list makes the median and mode obvious by inspection and keeps us from miscounting duplicates. Tool #7 (Identify Subproblems) is the natural frame for the whole problem because the question is really three small problems — find mean, find median, find mode — followed by one easy addition.

Execute — Answer: C

#2 Make a Systematic List 5.MD.B.2 Step 1

Sort the data from smallest to largest so the middle values and any repeats are easy to spot.

$$2, 3, 0, 3, 1, 4, 0, 3 \;\Rightarrow\; 0, 0, 1, 2, 3, 3, 3, 4$$

💡 Lining up data on a number line — exactly what a Grade 5 line plot does — makes the order easy to read.

#7 Identify Subproblems 6.SP.B.5 Step 2

Find the mode: scan the sorted list for the value that appears most often.
$0$ appears twice, $3$ appears three times, every other value appears once.
So the mode is $3$.

$$\text{mode} = 3$$

💡 Picking the most frequent value is the simplest Grade 6 measure-of-center reading.

#7 Identify Subproblems 6.SP.B.5 Step 3

Find the median.
With $8$ values (an even count) the median is the average of the $4$th and $5$th terms in the sorted list.
Those are $2$ and $3$.

$$\text{median} = \dfrac{2 + 3}{2} = 2.5$$

💡 Averaging the two middle values when the count is even is part of the Grade 6 "summarize a data set" standard.

#7 Identify Subproblems 6.SP.B.5 Step 4

Find the mean: add every value and divide by the count $8$.

$$\text{mean} = \dfrac{0+0+1+2+3+3+3+4}{8} = \dfrac{16}{8} = 2$$

💡 Sum-over-count is the textbook Grade 6 definition of the mean.

#7 Identify Subproblems 5.NBT.B.7 Step 5

Add the three results to answer the question.

$$\text{mean} + \text{median} + \text{mode} = 2 + 2.5 + 3 = 7.5 \;\Rightarrow\; \textbf{(C)}$$

💡 Adding whole numbers and a decimal to the tenths is a Grade 5 decimal-arithmetic skill.

[1] #2 5.MD.B.2 Sort the data from smallest to largest so the middle values and any repeats are

[2] #7 6.SP.B.5 Find the mode: scan the sorted list for the value that appears most often. $0$ a

[3] #7 6.SP.B.5 Find the median. With $8$ values (an even count) the median is the average of th

[4] #7 6.SP.B.5 Find the mean: add every value and divide by the count $8$.

[5] #7 5.NBT.B.7 Add the three results to answer the question.

Review

Reasonableness: The eight numbers all sit between $0$ and $4$, so each measure of center has to land in that range. We got mean $= 2$, median $= 2.5$, mode $= 3$ — all in $[0, 4]$ and clustered where the data piles up (three $3$s pull the mode and median upward, the two $0$s drag the mean down). Their sum $7.5$ lies between $0 + 0 + 0 = 0$ and $4 + 4 + 4 = 12$, and matches choice (C).

Alternative: Tool #1 (Draw a Diagram) via a dot plot: place dots above $0, 0, 1, 2, 3, 3, 3, 4$ on a number line. The tallest stack ($3$) is the mode by eye. The center of mass — counted by sliding from each end toward the middle — sits between the $4$th and $5$th dots, giving median $2.5$. The mean ($2$) is the balance point you can verify by checking that the total "distance" of dots left of $2$ equals the total distance of dots right of $2$. Same three numbers, same sum $7.5$.

CCSS standards used (min grade 6)

5.MD.B.2 Make a line plot to display a data set and solve problems using the data (Sorting the eight values from smallest to largest, the same ordering step a Grade 5 line plot requires.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Adding $2 + 2.5 + 3 = 7.5$ at the final step — whole numbers plus a tenths decimal.)
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures (Computing the three measures of center — mean, median, and mode — of the eight-value data set.)

⭐ This AMC 8 problem only needs Grade 6 measures of center — mean, median, and mode — you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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