A group of 15 students recorded the number of books they read over the summer. The table b… — SAT

A group of $15$ students recorded the number of books they read over the summer. The table below shows the distribution of the number of books read.

Number of books	Number of students
$1$	$1$
$2$	$3$
$3$	$5$
$4$	$5$
$18$	$1$

If the student who read $18$ books is removed from the dataset, which of the following statements best describes the effect on the mean and the median of the dataset?

The mean decreases by $1$ , and the median remains the same.Answer
B
The mean remains the same, and the median decreases by $1$ .
C
Both the mean and the median decrease by $1$ .
D
Both the mean and the median remain the same.

Answer

The mean decreases by

1

, and the median remains the same.

The correct answer states that the mean decreases by

1

and the median remains the same. Removing the outlier of

18

reduces the sum from

60

42

and the number of terms from

15

14

, resulting in a new mean of

3

(down from

4

). The median is determined by the middle values of the dataset; both the

8\text{th}

value of the

15

-element set and the average of the

7\text{th}

and

8\text{th}

values of the

14

-element set are

3

, meaning the median does not change.

Step-by-Step Solution

Calculate the mean of the original dataset containing

15

students.

The sum of the books read is

(1 \times 1) + (2 \times 3) + (3 \times 5) + (4 \times 5) + (18 \times 1) = 1 + 6 + 15 + 20 + 18 = 60

. The mean is

\frac{60}{15} = 4

To find the initial baseline mean before the outlier is removed.

Find the median of the original dataset.

With

15

data points, the median is the

8\text{th}

value when ordered. Counting from the smallest value, the first value is

1

, the next

3

values are

2

, and the next

5

values are

3

(covering the

5\text{th}

through

9\text{th}

positions). Thus, the

8\text{th}

value is

3

, so the median is

3

To find the initial baseline median before the outlier is removed.

Calculate the new mean after removing the student who read

18

books.

The new number of students is

14

. The new sum of books read is

60 - 18 = 42

. The new mean is

\frac{42}{14} = 3

. The mean decreased by

4 - 3 = 1

To determine the change in the mean.

Find the new median after removing the student who read

18

books.

With

14

data points, the median is the average of the

7\text{th}

and

8\text{th}

values. Both the

7\text{th}

and

8\text{th}

values are in the group of

3

s (covering the

5\text{th}

through

9\text{th}

positions). Therefore, the new median is

\frac{3 + 3}{2} = 3

, meaning the median remains the same.

To determine the change in the median.

Key Concept

Understanding how outliers affect measures of center, specifically the mean (which is sensitive to outliers) and the median (which is resistant to outliers).

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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