If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
15
Understanding the Mode in Statistics
The mode of a set of data is the value that appears most frequently in the set. A data set can have one mode, multiple modes, or no mode at all.
Analyzing the Given Scores and Mode
We are given the following set of scores: 10, 12, 13, 15, 15, 13, 12, 10, x.
We are also told that the mode of this set is 15.
Let's count the frequency of each score in the given list, excluding the value of x for now:
- Score 10 appears 2 times.
- Score 12 appears 2 times.
- Score 13 appears 2 times.
- Score 15 appears 2 times.
Currently, scores 10, 12, 13, and 15 all have the same frequency (2 times). For 15 to be the mode, it must appear more times than any other score in the set. The frequency of 15 needs to be higher than the frequency of 10, 12, and 13.
Determining the Value of x
Let's consider the possible values for x based on the options and the definition of the mode:
- If x were 10, the scores would be: 10, 12, 13, 15, 15, 13, 12, 10, 10. The frequency of 10 would be 3, and the frequency of 15 would be 2. In this case, 10 would be the mode, not 15.
- If x were 12, the scores would be: 10, 12, 13, 15, 15, 13, 12, 10, 12. The frequency of 12 would be 3, and the frequency of 15 would be 2. In this case, 12 would be the mode, not 15.
- If x were 13, the scores would be: 10, 12, 13, 15, 15, 13, 12, 10, 13. The frequency of 13 would be 3, and the frequency of 15 would be 2. In this case, 13 would be the mode, not 15.
- If x were 15, the scores would be: 10, 12, 13, 15, 15, 13, 12, 10, 15. Let's count the frequencies again:
- Score 10 appears 2 times.
- Score 12 appears 2 times.
- Score 13 appears 2 times.
- Score 15 appears 2 + 1 = 3 times.
For 15 to be the mode of the given set of scores, the value of x must increase the frequency of 15 so that it becomes the highest frequency among all scores. Setting x equal to 15 achieves this.
Final Answer Determination
Based on our analysis, the value of x must be 15 for the mode of the scores to be 15.
| Score | Initial Frequency | Frequency if x = 15 |
|---|---|---|
| 10 | 2 | 2 |
| 12 | 2 | 2 |
| 13 | 2 | 2 |
| 15 | 2 | 3 |
As the table shows, when x=15, the frequency of 15 becomes 3, while all other scores have a frequency of 2. Thus, 15 is the mode.
Revision Table: Statistics Concepts
| Concept | Definition | Example |
|---|---|---|
| Mode | The value that appears most often in a data set. | In {2, 3, 3, 5, 7}, the mode is 3. |
| Median | The middle value in a sorted data set. | In {2, 3, 7, 9, 10}, the median is 7. |
| Mean | The average of all values in a data set (sum of values divided by the count of values). | In {2, 3, 7}, the mean is \(\frac{2+3+7}{3} = 4\). |
| Frequency | The number of times a particular value appears in a data set. | In {2, 3, 3, 5, 7}, the frequency of 3 is 2. |
Additional Information: Central Tendency Measures
Mode, Median, and Mean are all measures of central tendency. They give us a single value that attempts to describe the center of a data set.
- The Mode is useful for categorical data or when identifying the most popular item.
- The Median is useful for skewed data sets as it is not affected by outliers.
- The Mean is the most commonly used measure but can be affected by outliers.
In this problem, understanding the definition of the mode and counting frequencies were key to finding the value of x.
Similar Questions
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Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
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If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
A, B, C are three sets of values of x:
A: 2, 3, 7, 1, 3, 2, 3
B: 7, 5, 9, 12, 5 3, 8
C: 4, 4, 11, 7, 2, 3, 4
Select the correct statement from among the following
