Question 30·Hard·One-Variable Data Distributions; Measures of Center and Spread
A set of 11 test scores has a mean of 60 and a median of 55.
After an additional score is added, the mean of the 12 scores becomes 62 and the median becomes 57.
What is the value of the additional score?
For questions where a new data point is added or removed and the means are given, immediately convert each mean into a total by multiplying mean × number of data points. Set up a simple equation: (old total) + (added value) = (new total), then solve for the unknown. Use other statistics like the median only as a quick reasonableness check (for example, if the mean increases, the added value should be larger than the original mean) rather than trying to reconstruct all individual data values.
Hints
Connect mean to the total sum
Remember that mean (average) is . If you know the mean and how many values there are, you can multiply them to get the total sum.
Compute the original and new totals
Use the means to find the total of the original 11 scores, and then the total of the 12 scores after the new one is added. Multiply mean number of scores in each case.
Relate the two totals
Once you know the total before and after adding the new score, think about how those totals are connected. The new total should equal the old total plus the additional score; solve that simple equation for the unknown score.
Desmos Guide
Compute the additional score with one expression
In a Desmos expression line, type 1262 - 1160 and press Enter. The value that Desmos outputs is the additional test score that was added.
Step-by-step Explanation
Find the total of the original 11 scores
The mean (average) equals the sum of the values divided by the number of values.
For the 11 original scores with mean 60:
Find the total of the 12 scores after adding one more
After the additional score is added, there are 12 scores with mean 62.
So the new total is:
Use the change in total to get the added score
Let be the additional score. The new total equals the old total plus this new score:
Subtract 660 from both sides:
So, the value of the additional score is 84.