A group of 25 students took a history quiz, and their scores are summarized in the frequency table below. | Score | Frequency | | :---: | :---: | | 6 | 2 | | 7 | 4 | | 8 | 10 | | 9 | 6 | | 10 | 3 | The teacher decides to remove the scores of the 2 students who scored 6 because they were absent on th

A group of 25 students took a history quiz, and their scores are summarized in the frequen… — SAT

A group of $25$ students took a history quiz, and their scores are summarized in the frequency table below.

Score	Frequency
6	2
7	4
8	10
9	6
10	3

The teacher decides to remove the scores of the $2$ students who scored $6$ because they were absent on the day of the quiz and took it later under different conditions. Which of the following statements describes the effect of removing these two scores on the mean, median, and standard deviation of the quiz scores?

The mean increases, the median remains unchanged, and the standard deviation decreases.Answer
B
The mean increases, the median increases, and the standard deviation decreases.
C
The mean remains unchanged, the median remains unchanged, and the standard deviation increases.
D
The mean increases, the median remains unchanged, and the standard deviation increases.

Answer

The mean increases, the median remains unchanged, and the standard deviation decreases.

The correct answer is the statement indicating that the mean increases, the median remains unchanged, and the standard deviation decreases. Removing the two lowest scores (

6

) eliminates the values that pull the average down, resulting in an increased mean. The median remains at

8

because the middle position of the sorted scores still lies within the class of

8

s. The standard deviation decreases because the overall spread of the data is reduced when the lowest and most distant values are removed.

Step-by-Step Solution

Calculate the original mean and locate the original median.

Original Mean =

8.16

; Original Median =

8

The original sum of scores is

(2 \times 6) + (4 \times 7) + (10 \times 8) + (6 \times 9) + (3 \times 10) = 204

. With

25

students, the mean is

204 / 25 = 8.16

. The median is the

13\text{th}

value in the sorted list of

25

scores. Looking at the cumulative frequencies, the first

2

scores are

6

, the next

4

are

7

(cumulative

6

), and the next

10

are

8

(cumulative

16

). Thus, the

13\text{th}

value is

8

Calculate the new mean and locate the new median after removing the two scores of

6

New Mean

\approx 8.35

; New Median =

8

Removing the two scores of

6

reduces the sum by

12

(new sum =

192

) and the count of students to

23

. The new mean is

192 / 23 \approx 8.35

, which is an increase. The new median is the

12\text{th}

value of the remaining

23

sorted scores. The first

4

scores are

7

, and the next

10

are

8

(cumulative

14

). The

12\text{th}

value is still

8

, so the median remains unchanged.

Analyze the change in the standard deviation.

The standard deviation decreases.

Standard deviation measures the spread of the data. The scores of

6

are the minimum scores and are the furthest from the mean of the distribution. Removing these extreme values reduces the variability and concentrates the remaining scores closer to the mean, which decreases the standard deviation.

Key Concept

Analyzing the effects of removing extreme data values on the mean, median, and standard deviation of a dataset.

Question

Answer

Step-by-Step Solution

Key Concept

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