The scores of 15 students in an examination were recorded as 10, 5, 8, 16, 18, 20, 8, 10, 16, 20, 18, 11 16 14 and 12. After calculating the mean, median and mode, an error is found. One of the values is wrongly written as 16 instead of 18. Which of the following measures of central tendency will change?

Options

A. Mean and median
B. Median and mode
C. Only mode
D. Mean and mode

Answer

To solve the problem, we will first calculate the mean, median, and mode of the original data set, then we will correct the error and recalculate these measures to see which ones change.

### Step 1: List the Original Scores
The original scores of the 15 students are:
10, 5, 8, 16, 18, 20, 8, 10, 16, 20, 18, 11, 16, 14, 12

### Step 2: Sort the Scores in Ascending Order
Sorting the scores gives us:
5, 8, 8, 10, 10, 11, 12, 14, 16, 16, 16, 18, 18, 20, 20

### Step 3: Calculate the Mean
The mean is calculated by summing all the scores and dividing by the number of scores.

Sum of scores = 5 + 8 + 8 + 10 + 10 + 11 + 12 + 14 + 16 + 16 + 16 + 18 + 18 + 20 + 20 = 202

Mean = Total Sum / Number of Scores = 202 / 15 ≈ 13.47

### Step 4: Calculate the Median
To find the median, we look for the middle value in the sorted list. Since there are 15 scores (an odd number), the median is the 8th score.

Median = 14 (the 8th number in the sorted list)

### Step 5: Calculate the Mode
The mode is the number that appears most frequently in the data set.

In our sorted list, the number 16 appears 3 times, which is more than any other number.

Mode = 16

### Step 6: Identify the Error
It is given that one of the values was wrongly recorded as 16 instead of 18. We will replace the incorrect score with the correct one.

### Step 7: Correct the Scores
The corrected scores are:
10, 5, 8, 18, 18, 20, 8, 10, 18, 20, 18, 11, 18, 14, 12

### Step 8: Sort the Corrected Scores
Sorting the corrected scores gives us:
5, 8, 8, 10, 10, 11, 12, 14, 18, 18, 18, 18, 20, 20

### Step 9: Recalculate the Mean
Sum of corrected scores = 5 + 8 + 8 + 10 + 10 + 11 + 12 + 14 + 18 + 18 + 18 + 18 + 20 + 20 = 202 - 16 + 18 = 204

Mean = Total Sum / Number of Scores = 204 / 15 = 13.6

### Step 10: Recalculate the Median
The median is still the 8th score in the sorted list of corrected scores.

Median = 14 (the 8th number in the sorted list remains unchanged)

### Step 11: Recalculate the Mode
Now, the mode is the number that appears most frequently in the corrected data set.

In the corrected list, the number 18 appears 4 times, which is more than any other number.

Mode = 18

### Conclusion
- Original Mean: 13.47
- Corrected Mean: 13.6 (changed)
- Original Median: 14
- Corrected Median: 14 (unchanged)
- Original Mode: 16
- Corrected Mode: 18 (changed)

### Final Answer
The measures of central tendency that will change are the Mean and Mode.