Key Information Extraction
The given sequence is: 1, x, x, x, y, y, 9, 16, 18. It is in increasing order.
The number of elements in the sequence is 9.
We are given that Mean = Median = 2 * Mode.
We need to find the value of y.
Determining Mode and Median
Since the sequence is increasing, we have 1 < x < y < 9.
- Mode: The mode is the most frequent number. In the sequence, 'x' appears 3 times and 'y' appears 2 times. Therefore, the mode is x.
- Median: For a sequence of 9 elements, the median is the 5th element. The 5th element is y.
Applying the Mean-Median-Mode Relationship
We are given that Median = 2 * Mode.
Substituting the values:
$y = 2 * x$ (Equation 1)
Calculating the Mean
The sum of the numbers is: $1 + x + x + x + y + y + 9 + 16 + 18 = 3x + 2y + 44$.
The mean is the sum divided by the count (9):
Mean = $ \frac{3x + 2y + 44}{9} $
Equating Mean and Median
We are given that Mean = Median.
Substituting the expressions:
$ \frac{3x + 2y + 44}{9} = y $
Multiply both sides by 9:
$ 3x + 2y + 44 = 9y $
Rearrange the terms:
$ 3x + 44 = 7y $ (Equation 2)
Solving for x and y
Substitute Equation 1 ($y = 2x$) into Equation 2:
$ 3x + 44 = 7(2x) $
$ 3x + 44 = 14x $
Subtract $3x$ from both sides:
$ 44 = 11x $
Solve for $x$:
$ x = \frac{44}{11} $
$ x = 4 $
Now substitute the value of $x$ back into Equation 1 ($y = 2x$) to find $y$:
$ y = 2 \times 4 $
$ y = 8 $
Verification
With x=4 and y=8, the sequence is 1, 4, 4, 4, 8, 8, 9, 16, 18.
- Mode = 4
- Median = 8
- Mean = $ \frac{1+4+4+4+8+8+9+16+18}{9} = \frac{72}{9} = 8 $
- Check: Mean (8) = Median (8) = 2 * Mode (2 * 4 = 8). The conditions are satisfied.
Therefore, the value of y is 8.
Important Questions from Mean Median Mode
- The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:
- Let $X$ be a continuous random variable whose cumulative distribution function (CDF)
$F_X(x) = \begin{cases} 0 & x < t \\ \frac{x-t}{4-t} & t \le x \le 4 \\ 1 & x \ge 4 \end{cases}$
If the median of $X$ is $3$, then what is the value of $t$? - The elements of the dataset $\{-5,1, a, 5, b\}$ are in ascending order. If both mean and median of the dataset are equal to 3, what is the value of $b$?
- The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ________.
The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam. From the data presented above, which one of the following is CORRECT?
