The arithmetic mean of 11 observations is 11. The arithmetic mean of the first 6 observations is 10.5 and the arithmetic mean of the last 6 observations is 11.5. What is the sixth observation?
11.0
Finding the Sixth Observation Using Arithmetic Mean Properties
This problem involves using the concept of the arithmetic mean to find a specific observation within a dataset. We are given the overall mean of 11 observations and the means of two overlapping subsets (the first 6 and the last 6 observations). The key is that the sixth observation is included in both of the subsets.
Understanding the Arithmetic Mean
The arithmetic mean (or average) of a set of observations is calculated by summing up all the observations and dividing by the number of observations. Mathematically, the formula is:
\(\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}\)
From this formula, we can also find the sum of observations if we know the mean and the number of observations:
\(\text{Sum of observations} = \text{Mean} \times \text{Number of observations}\)
Calculating the Sums of Observations
Let the 11 observations be \(x_1, x_2, \dots, x_{11}\). We are given the following information:
- Arithmetic mean of all 11 observations is 11.
- Arithmetic mean of the first 6 observations (\(x_1, \dots, x_6\)) is 10.5.
- Arithmetic mean of the last 6 observations (\(x_6, \dots, x_{11}\)) is 11.5.
Using the formula for the sum of observations, we can calculate the total sum for each group:
| Group of Observations | Number of Observations | Arithmetic Mean | Sum of Observations |
|---|---|---|---|
| All 11 observations (\(x_1, \dots, x_{11}\)) | 11 | 11 | \(11 \times 11 = 121\) |
| First 6 observations (\(x_1, \dots, x_6\)) | 6 | 10.5 | \(6 \times 10.5 = 63\) |
| Last 6 observations (\(x_6, \dots, x_{11}\)) | 6 | 11.5 | \(6 \times 11.5 = 69\) |
So, we have:
- Sum of all 11 observations = 121
- Sum of first 6 observations = 63
- Sum of last 6 observations = 69
Finding the Sixth Observation
Let's look at the sums. The sum of the first 6 observations includes \(x_1 + x_2 + x_3 + x_4 + x_5 + x_6\). The sum of the last 6 observations includes \(x_6 + x_7 + x_8 + x_9 + x_{10} + x_{11}\).
If we add the sum of the first 6 and the sum of the last 6, we get:
\((\text{Sum of first 6}) + (\text{Sum of last 6}) = (x_1 + \dots + x_6) + (x_6 + \dots + x_{11})\)
Notice that the sixth observation, \(x_6\), is included in both sums. So, this combined sum is actually the sum of all 11 observations plus the sixth observation (because it's counted twice):
\((\text{Sum of first 6}) + (\text{Sum of last 6}) = (\text{Sum of all 11}) + x_6\)
Now, we can substitute the values we calculated:
\(63 + 69 = 121 + x_6\)
\(132 = 121 + x_6\)
To find \(x_6\), we subtract 121 from 132:
\(x_6 = 132 - 121\)
\(x_6 = 11\)
Therefore, the sixth observation is 11.0.
Revision Table: Key Concepts
| Concept | Definition/Formula | Application in Problem |
|---|---|---|
| Arithmetic Mean | Sum of observations / Number of observations | Used to find total sum from given means and counts. |
| Sum of Observations | Mean × Number of observations | Key formula used to calculate total sums for different groups. |
| Overlapping Subsets | Groups sharing common elements (here, the 6th observation) | Used to form the equation where the common element is counted twice in the sum of subset sums. |
Additional Information: Mean and Data Analysis
The arithmetic mean is a fundamental measure of central tendency. It gives us a single value that represents the center of the data. However, the mean can be affected by extreme values (outliers).
Problems like this one, where subsets of data overlap, are common in statistics and data analysis. They help test the understanding of how means and sums relate to the individual data points and how summing overlapping sets works.
Other measures of central tendency include the median (the middle value) and the mode (the most frequent value). The choice of which measure to use depends on the nature of the data and the question being asked.
Similar Questions
If the average of 9 consecutive positive integers is 55, then what is the largest integer?
A cricketer has certain average of 10 innings. In the 11 th inning, he scored 108 runs, thereby increasing his average by 6 runs. What is his new average?
In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If average marks of the complete class is 72, then what is the average marks of the girls?
The average of 50 consecutive natural numbers is x. What will be the new average when the next four numbers are also included?
Consider two-digit numbers which remain the same when the digits interchange their positions. What is the average of such two-digit numbers?
Mean marks of 50 students were found to be 78.4. But later it was detected that 95 was misread as 59 and 25 was misread as 52. What is the difference between correct mean and incorrect mean?
What is the arithmetic mean of P and Q ?
The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?
A small company pays each of its 5 category ‘C’ workers Rs. 20,000, each of its 3 category ‘B’ workers Rs. 25,000 and a category ‘A’ worker Rs. 65,000. The number of workers earning less than the mean salary is
Consider the following distribution
Class
Frequency
0 - 20
17
20 - 40
28
40 - 60
32
60 - 80
F
80 - 100
19
If the mean of the above distribution is 50 then what is the value of f?
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.
Consider the following statements:
1. The average score of Class-B will definitely decrease.
2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct?
The average weight of A, B, Cis 40 kg, the average weight of B, D, Eis 42 kg and the weight of Fis equal to that of B. What is the average weight of A, B, C, D, Eand F?
A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?
