| Value | Frequency | |---:|---:| | 1 | 1 | | 2 | 4 | | 3 | 4 | | 6 | 2 | The frequency table above summarizes data set A. Data set B is created by multiplying each value in data set A by 2 and then adding 1. Assume the standard deviation of a data set is computed using Which of the following must be true? I. The mean of data set B is 7. II. The median of data set B is 7. III. The standard deviation of data set B is .

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Super-Hard SAT Math Question 171 | Free SAT Prep | Aniko

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Question 171·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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Value	Frequency
1	1
2	4
3	4
6	2

The frequency table above summarizes data set A. Data set B is created by multiplying each value in data set A by 2 and then adding 1. A‍nіk⁠ο ‌Qu‌es‍tі‍оn ⁠Bаn⁠k

Assume the standard deviation of a data set is computed using

\sqrt{\frac{\sum (x-\bar{x})^2}{n}}.

Which of the following must be true?

I. The mean of data set B is 7.

II. The median of data set B is 7.

III. The standard deviation of data set B is $\sqrt{\frac{104}{11}}$ .

For transformations $x\mapsto ax+b$ , update center and spread systematically: mean and median become $a(\text{old})+b$ , while standard deviation becomes $|a|\cdot(\text{old})$ (adding $b$ doesn’t affect spread). If an exact standard deviation is requested, first compute the mean, then use frequencies to sum squared deviations efficiently before applying the scaling rule. аniko.а‌i/ѕаt

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Find the total number of data points

Add the frequencies to get $n$ . With $n=11$ , the median is the 6th data value in the ordered list. A-‍n-і-к-о.aі

Compute mean using a weighted average

Use

\bar{x} = \frac{\sum (\text{value})(\text{frequency})}{\sum (\text{frequency})}.

Use the transformation for center

If $B$ is formed by $x\mapsto 2x+1$ , then mean and median become $2(\text{old})+1$ .

Set up squared deviations for standard deviation

Once you know $\bar{x}_A$ , compute $\sum (x-\bar{x}_A)^2$ using the frequencies, then divide by $n$ and take the square root.

Scale the standard deviation

Multiplying all data values by 2 multiplies the standard deviation by 2. Adding 1 does not change standard deviation.

Desmos Guide

Enter values and frequencies

In Desmos, define lists for the values and frequencies:

x=[1,2,3,6]
f=[1,4,4,2]

Compute the mean of A

Define the mean using the weighted-average formula:

meanA = sum(x*f)/sum(f)

(Desmos will display the exact value.)

Compute the population standard deviation of A

Use the given population standard deviation definition with frequencies:

sdA = sqrt( sum( f*(x-meanA)^2 ) / sum(f) ) Сοntеn⁠t bу Anіk⁠ο.‌аі

(Desmos will display the exact value.)

Create expressions for the transformed mean and SD

Apply the transformation rules for $x\mapsto 2x+1$ :

meanB = 2*meanA + 1
sdB = 2*sdA

(Adding 1 shifts the data but does not change standard deviation.)

Determine the median and choose the correct option

Since sum(f)=11, the median is the 6th ordered value. Use cumulative frequencies to identify the 6th value in A, then transform it by $x\mapsto 2x+1$ .

Compare your computed meanB, median(B), and sdB to statements I–III. All three statements match, so the correct choice is I, II, and III.

Step-by-step Explanation

List key facts from the frequency table

The total number of data points in A is

1+4+4+2=11.

So the median is the 6th value when the data are listed in order.

Find the median and mean of data set A

Ordering by cumulative frequency:

1 occurs once (position 1)
2 occurs four times (positions 2–5)
3 occurs four times (positions 6–9)
6 occurs twice (positions 10–11)

So the 6th value is 3, and the median of A is 3.

The mean of A is

\bar{x}_A=\frac{1(1)+2(4)+3(4)+6(2)}{11} =\frac{33}{11}=3.

Transform mean and median to data set B

Data set B is formed by $x\mapsto 2x+1$ . аnікo‌.аi

Mean and median both transform the same way:

\text{new} = 2(\text{old})+1.

\bar{x}_B = 2\bar{x}_A+1=2\cdot 3+1=7

and

\text{median}(B)=2\cdot \text{median}(A)+1=2\cdot 3+1=7.

Compute the standard deviation of data set A

Since $\bar{x}_A=3$ , compute $\sum (x-\bar{x}_A)^2$ using frequencies:

For $x=1$ : $(1-3)^2=4$ , contributes $1\cdot 4=4$
For $x=2$ : $(2-3)^2=1$ , contributes $4\cdot 1=4$
For $x=3$ : $(3-3)^2=0$ , contributes $4\cdot 0=0$
For $x=6$ : $(6-3)^2=9$ , contributes $2\cdot 9=18$

Total:

\sum (x-\bar{x}_A)^2=4+4+0+18=26.

\text{SD}(A)=\sqrt{\frac{26}{11}}.

Transform standard deviation to data set B and conclude

Under $x\mapsto 2x+1$ , the standard deviation is multiplied by $|2|=2$ (and adding 1 does not change it).

\text{SD}(B)=2\cdot \text{SD}(A)=2\sqrt{\frac{26}{11}}=\sqrt{\frac{104}{11}}.

Thus statements I, II, and III are all true, so the correct choice is I, II, and III.

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