Data set consists of 12 values and has mean and standard deviation . Two of the values in data set are 4 and 22. Data set is created by replacing 4 and 22 with 10 and 16. Which choice best compares the mean and standard deviation of data set with those of data set ?

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Super-Hard SAT Math Question 7 | Free SAT Prep | Aniko

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Question 7·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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Data set $A$ consists of 12 values and has mean $m$ and standard deviation $s$ . Two of the values in data set $A$ are 4 and 22. Data set $B$ is created by replacing 4 and 22 with 10 and 16.

Which choice best compares the mean and standard deviation of data set $B$ with those of data set $A$ ?

When a data set is changed by replacing a few values, first check whether the total sum changes; if it doesn't, the mean stays the same. For standard deviation, remember it measures how far values are from the mean. If you replace values that are far from the mean with values closer to the mean, the standard deviation decreases—and vice versa.

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Start with the mean

The mean equals the total sum divided by the number of values. Compare what's being removed ( $4 + 22$ ) with what's being added ( $10 + 16$ ).

Think about what standard deviation measures

Standard deviation measures how spread out the data is from the mean. Values far from the mean increase spread; values close to the mean decrease spread.

Compare distances from the mean

The original values 4 and 22 are farther apart (spread over 18 units). The new values 10 and 16 are closer together (spread over only 6 units). How does this affect the overall spread?

Desmos Guide

Create a sample data set A

Enter a list that includes 4 and 22:

$A = [4, 8, 10, 12, 12, 13, 13, 14, 15, 17, 20, 22]$

(This is just an example—the actual values don't matter as long as 4 and 22 are included.)

Calculate mean and standard deviation for A

Enter:

$\operatorname{mean}(A)$

$\operatorname{stdev}(A)$

Note the values (mean ≈ 13.33, stdev ≈ 4.71 for this example).

Create data set B and compare

Replace 4 with 10 and 22 with 16:

$B = [10, 8, 10, 12, 12, 13, 13, 14, 15, 17, 20, 16]$

Calculate:

$\operatorname{mean}(B)$

$\operatorname{stdev}(B)$

Confirm that the mean is the same (≈ 13.33) and the standard deviation is smaller (≈ 3.20).

Step-by-step Explanation

Determine what happens to the mean

The mean is calculated as: total sum ÷ number of values.

Only two values change:

Removed: $4 + 22 = 26$
Added: $10 + 16 = 26$

Since the sum stays the same and there are still 12 values, the mean remains $m$ .

Understand what affects standard deviation

Standard deviation measures spread—how far values are from the mean.

Values far from the mean → increase standard deviation
Values close to the mean → decrease standard deviation

The 10 unchanged values contribute the same amount to the spread. So we only need to compare the old pair (4 and 22) with the new pair (10 and 16).

Compare how the old and new values affect spread

The original values 4 and 22 are at the extremes—they're 18 units apart from each other.

The new values 10 and 16 are much closer together—only 6 units apart.

Since the mean stays the same, replacing extreme values (far from each other) with moderate values (close together) reduces the overall spread of the data.

Conclude the comparison

Mean: Stays equal to $m$ (same total sum, same count)
Standard deviation: Decreases (extreme values replaced with values closer to the center)

The answer is: The mean of data set $B$ is equal to $m$ , and the standard deviation of data set $B$ is less than $s$ .

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Step-by-step Explanation

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