Question
The box plot represents the distribution of the amount of change, in cents, that 50 people were carrying when surveyed.
<p>Box plot from 0 to 220 by 20's. Amount of change in cents. Whisker from 0 to 13. Box from 13 to 85 with vertical line at 25. Whisker from 85 to 203.</p>
The box plot represents the distribution of the same data set, but with the maximum, 203, removed.
<p>Box plot from 0 to 220 by 20's. Amount of change in cents. Whisker from 0 to 13. Box from 13 to 85 with vertical line at 25. Whisker from 85 to 97.</p>
The median is 25 cents for both plots. After examining the data, the value 203 is removed because it was an error in recording.
Explain why the median remains the same when 203 cents was removed from the data set.
When 203 cents is removed from the data set, does the mean remain the same? Explain your reasoning.
<p>Box plot from 0 to 220 by 20's. Amount of change in cents. Whisker from 0 to 13. Box from 13 to 85 with vertical line at 25. Whisker from 85 to 203.</p>
The box plot represents the distribution of the same data set, but with the maximum, 203, removed.
<p>Box plot from 0 to 220 by 20's. Amount of change in cents. Whisker from 0 to 13. Box from 13 to 85 with vertical line at 25. Whisker from 85 to 97.</p>
The median is 25 cents for both plots. After examining the data, the value 203 is removed because it was an error in recording.
Explain why the median remains the same when 203 cents was removed from the data set.
When 203 cents is removed from the data set, does the mean remain the same? Explain your reasoning.
Answers
Answered by
GPT-4o mini
To understand why the median remains the same when the value of 203 cents is removed from the data set, we must first clarify what the median represents. The median is the middle value of a data set when it is arranged in ascending order. For a data set with an even number of observations, the median is calculated as the average of the two middle values.
In this case, the median is reported as 25 cents for both box plots. Since the value of 203 cents is much larger than the median, its removal does not affect the middle values of the other observations in the data set.
When 203 is removed, the remaining 49 values still keep the same middle values relative to the overall distribution. Therefore, the two middle values in the new data set (which is now with 49 observations) remain in a position that results in a median of 25 cents. As long as these middle values do not change, the median will remain the same.
Regarding the mean, the mean is calculated as the average of all values in the data set. Since removing 203 cents, which is a high outlier, lowers the overall total of the data but retains the count of observations (albeit reducing it by one), the mean will likely decrease. This is because the removal of the 203 cent value, which pulls the average up due to its high value, affects the total sum of the data set.
For example, if the original data set values add up to a certain total (including 203), removing 203 means the new total will be less, and when you divide by the new total number of observations (49 instead of 50), the mean decreases.
In summary, the median remains the same when 203 is removed because it does not influence the middle values of the remaining observations, whereas the mean is likely to decrease because the high outlier removed lowers the overall sum of values included in the average.
In this case, the median is reported as 25 cents for both box plots. Since the value of 203 cents is much larger than the median, its removal does not affect the middle values of the other observations in the data set.
When 203 is removed, the remaining 49 values still keep the same middle values relative to the overall distribution. Therefore, the two middle values in the new data set (which is now with 49 observations) remain in a position that results in a median of 25 cents. As long as these middle values do not change, the median will remain the same.
Regarding the mean, the mean is calculated as the average of all values in the data set. Since removing 203 cents, which is a high outlier, lowers the overall total of the data but retains the count of observations (albeit reducing it by one), the mean will likely decrease. This is because the removal of the 203 cent value, which pulls the average up due to its high value, affects the total sum of the data set.
For example, if the original data set values add up to a certain total (including 203), removing 203 means the new total will be less, and when you divide by the new total number of observations (49 instead of 50), the mean decreases.
In summary, the median remains the same when 203 is removed because it does not influence the middle values of the remaining observations, whereas the mean is likely to decrease because the high outlier removed lowers the overall sum of values included in the average.