Data set consists of 12 values and has mean . Every value in data set is between and , inclusive.
Data set consists of all 12 values in data set together with the two additional values and .
Which choice gives all the statements that must be true?
I. The mean of data set is .
II. The standard deviation of data set is greater than the standard deviation of data set .
III. The median of data set is equal to the median of data set .
For questions that compare related data sets, track each measure separately. Use totals to analyze the mean, compare distances from an unchanged mean to analyze standard deviation, and write the positions of the middle values after sorting to analyze the median. Do not assume that adding values changes every measure; determine exactly how the new values affect each one.
Hints
Use the definition of mean
Translate the mean of into the sum of its 12 values. Then find the sum of the two new values.
Locate the added values
Compare the distances of and from with the greatest possible distance of an original value from .
Track the middle positions
Consider where the two new values appear when all values are arranged from least to greatest. Then compare the middle positions for data sets with 12 and 14 values.
Desmos Guide
Use reasoning first
Algebra and position counting are faster than Desmos here because the data values are not specified. Desmos can be used to verify the pattern with an example.
Enter an example list
Enter L=[46,47,48,49,50,50,51,51,52,52,53,54]. Its mean is , and every value is between and .
Add the two outer values
Enter M=[42,46,47,48,49,50,50,51,51,52,52,53,54,58]. Then enter mean(L), mean(M), median(L), median(M), stdev(L), and stdev(M) to verify that the means and medians match while the standard deviation increases.
Step-by-step Explanation
Check the mean
Because data set has 12 values with mean , the sum of its values is . The two added values have sum
Therefore, the sum of the 14 values in data set is , so its mean is .
Compare the spread
Every original value is at most 4 units from . Each new value, and , is 8 units from . Since the mean stays at and the new values are farther from the mean than every original value, data set has a greater standard deviation.
Compare the medians
In sorted order, is below every value in data set , and is above every value in data set . For the 12 values in , the median is the average of the 6th and 7th values. For the 14 values in , the median is the average of the 7th and 8th values, which are those same original 6th and 7th values. Thus, the medians are equal.