A set of nine data points is shown below. Which statement is true if a tenth data point of 45 is added to the data set?
8 11 12 10 9 7 5 3 9
A. The mean and median will both increase.
B. The mean will increase and the median will decrease.**
C. The mean will increase and the median will remain the same.
D. The mean and the median will both decrease.
C. The mean will increase and the median will remain the same.
before 45 is added: mean is 8.22 median is 9
after 45 is added: mean is11.9 median is 9
To determine the effect of adding a tenth data point of 45 to a given dataset, we need to compare the mean and median before and after the addition.
Step 1: Calculate the mean of the initial dataset
The sum of the initial data points is 8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9 = 74.
Divide the sum by the number of data points (9) to find the mean: 74 / 9 = 8.22 (rounded to two decimal places).
Step 2: Find the median of the initial dataset
To find the median, we need to arrange the data points in ascending order: 3, 5, 7, 8, 9, 9, 10, 11, 12.
Since the number of data points is odd, the median is the middle value, which is 9.
Step 3: Calculate the mean and median after adding the tenth data point (45)
The new sum is 74 + 45 = 119.
The new count of data points is 10.
Step 4: Calculate the new mean
Divide the new sum (119) by the new count of data points (10): 119 / 10 = 11.9.
Step 5: Find the new median
Since the dataset now has an even number of data points (10), the median is the average of the two middle values. The dataset in ascending order is 3, 5, 7, 8, 9, 9, 10, 11, 12, 45. The two middle values are 9 and 10, so their average is (9 + 10) / 2 = 9.5.
Therefore, the correct statement is B. The mean will increase and the median will decrease if a tenth data point of 45 is added to the dataset.
To find the answer to this question, we need to understand how the mean and median are calculated and how a new data point affects these measures.
The mean is calculated by adding up all the values in the data set and dividing the sum by the number of data points. The median is the middle value of a sorted data set, or the average of the two middle values if there are an even number of data points.
In this case, we have a set of nine data points: 8, 11, 12, 10, 9, 7, 5, 3, 9. To find the mean, we add up all these values and divide by 9:
(8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9) / 9 = 84 / 9 = 9.33 (rounded to two decimal places)
The median can be found by first arranging the data set in ascending order: 3, 5, 7, 8, 9, 9, 10, 11, 12. Since there are nine values, the median is the middle value, which is 9.
Now, if we add a tenth data point of 45 to the data set, the new data set would be: 8, 11, 12, 10, 9, 7, 5, 3, 9, 45.
To find the new mean, we add up all these values and divide by 10:
(8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9 + 45) / 10 = 119 / 10 = 11.9 (rounded to one decimal place)
To find the new median, we arrange the data set in ascending order: 3, 5, 7, 8, 9, 9, 10, 11, 12, 45. Since there are ten values, the median is the average of the two middle values, which are 9 and 10. The new median is (9 + 10) / 2 = 9.5.
Comparing the new mean and the new median to the original values, we can see that the mean has increased from 9.33 to 11.9, while the median has decreased from 9 to 9.5.
Therefore, the correct statement is B. The mean will increase and the median will decrease.
the median cannot decrease if a new high value is added.
It will be the average of the two new middle values, which cannot be less than the current middle value.