110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct?
Analyzing Statistical Measures After Data Modification
This question requires us to analyze how modifying a single data point affects the median, mean, and mode of a given data set. We need to calculate these measures for the original data and the modified data and then evaluate the given statements.
Original Data Set Analysis
The original data set is:
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
First, let's sort the data in ascending order:
| 36 | 41 | 43 | 61 | 71 | 92 | 93 | 95 | 99 | 110 | 110 | 127 |
Calculating the Old Median
There are 12 data points (n=12). Since n is even, the median is the average of the 6th and 7th terms in the sorted list.
Old Median = \(\frac{92 + 93}{2} = \frac{185}{2} = 92.5\).
Calculating the Old Mean
Sum of the original data = 110 + 41 + 43 + 95 + 127 + 99 + 61 + 92 + 71 + 93 + 110 + 36 = 978.
Old Mean = \(\frac{\text{Sum of data}}{\text{Number of data points}} = \frac{978}{12} = 81.5\).
Calculating the Old Mode
The mode is the value that appears most frequently. In the original data set, the number 110 appears twice, while all other numbers appear only once.
Old Mode = 110.
Modified Data Set Analysis
The data point 93 is replaced by 94. The new data set is:
110, 41, 43, 95, 127, 99, 61, 92, 71, 94, 110, 36.
Let's sort the modified data:
| 36 | 41 | 43 | 61 | 71 | 92 | 94 | 95 | 99 | 110 | 110 | 127 |
Calculating the New Median
The number of data points is still 12 (n=12). The new median is the average of the 6th and 7th terms.
New Median = \(\frac{92 + 94}{2} = \frac{186}{2} = 93\).
Calculating the New Mean
The sum changes by replacing 93 with 94. The change in sum is \(94 - 93 = 1\).
New Sum = Original Sum + 1 = 978 + 1 = 979.
New Mean = \(\frac{979}{12} \approx 81.5833\).
The difference between the new mean and the old mean is \(\frac{979}{12} - \frac{978}{12} = \frac{1}{12}\).
Calculating the New Mode
In the modified data set, the number 110 still appears twice, and all other numbers appear once.
New Mode = 110.
Evaluating the Statements
Statement 1: The difference between new median and old median is 1.
Difference = New Median - Old Median = \(93 - 92.5 = 0.5\).
Since \(0.5 \neq 1\), statement 1 is incorrect.
Statement 2: The difference between new mean and old mean is less than 0·1.
Difference = New Mean - Old Mean = \(\frac{1}{12}\).
As a decimal, \(\frac{1}{12} \approx 0.0833\).
Since \(0.0833 < 0.1\), statement 2 is correct.
Statement 3: The difference between new mode and old mode is zero.
Difference = New Mode - Old Mode = \(110 - 110 = 0\).
Statement 3 is correct.
Conclusion
Based on the analysis, statements 2 and 3 are correct.
Therefore, the correct option is the one that includes statements 2 and 3 only.
Similar Questions
What is the median of the distribution ?
- What is the mode of the distribution?
- Which measure of central tendency is least affected by the presence of extreme observations in the data?
- To find the average ratio like price/unit, work done/hour, kilometre/hour under certain conditions, the suitable measure of central tendency applicable is
- The frequency distribution of marks of 100 candidates in a particular examination is as follows:What are the average marks of the candidates?
Marks Number of Candidates More than 10 100 More than 20 75 More than 30 60 More than 40 40 - The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?
- A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?
- Which one of the following is a positional average?
- What is the value of \(u\)?
- What is the value of \(v\)?
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
