Introduction / Context:
This question tests knowledge of three key measures of central tendency: mean, median and mode. We are given a small data set and asked to compute each measure, then sum them up. Careful ordering of data and correct definitions are crucial to arrive at the correct combined value.
Given Data / Assumptions:
- Data set: 6, 3, 8, 4, 3, 11, 7, 4, 5, 4.
- Number of observations = 10.
- We must find:
- Mode – the most frequent value.
- Median – the middle value when data is ordered (or average of two middle values).
- Mean – the arithmetic average.
- Finally, we add mode + median + mean.
Concept / Approach:
We follow these steps:
- Sort the data in ascending order for median and mode.
- Identify the mode by highest frequency.
- For median with 10 observations, take the average of the 5th and 6th values.
- Compute mean as total sum divided by 10.
- Add the three results.
Step-by-Step Solution:
Step 1: Sort the data.
Original: 6, 3, 8, 4, 3, 11, 7, 4, 5, 4.
Sorted: 3, 3, 4, 4, 4, 5, 6, 7, 8, 11.
Step 2: Find the mode (most frequent value).
3 appears twice, 4 appears three times, 5, 6, 7, 8, 11 appear once each.
So mode = 4.
Step 3: Find the median.
There are 10 observations (even number), so median = average of 5th and 6th values.
5th value = 4, 6th value = 5.
Median = (4 + 5) / 2 = 9 / 2 = 4.5.
Step 4: Compute the mean.
Sum all values: 6 + 3 + 8 + 4 + 3 + 11 + 7 + 4 + 5 + 4.
6 + 3 = 9, +8 = 17, +4 = 21, +3 = 24, +11 = 35, +7 = 42, +4 = 46, +5 = 51, +4 = 55.
Mean = 55 / 10 = 5.5.
Step 5: Sum mode, median and mean.
Mode + Median + Mean = 4 + 4.5 + 5.5 = 14.
Verification / Alternative check:
We can confirm the sorted list and double-check the counts for the mode.
The median is correctly taken as the average of the 5th and 6th values for an even-sized data set.
The sum 4 + 4.5 + 5.5 clearly equals 14 with no rounding issues.
Why Other Options Are Wrong:
13.5, 13 and 14.5 would require at least one of mode, median or mean to be miscalculated.
For example, if someone mistakenly took median as 5 (instead of 4.5), the sum would be 4 + 5 + 5.5 = 14.5, not 14.
Common Pitfalls:
Not sorting the data correctly or misidentifying the 5th and 6th terms can lead to an incorrect median.
Sometimes, students confuse mode with median or forget to divide by the number of observations when computing the mean.
Final Answer:
The sum of the mode, the median and the mean of the given data is 14.
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