6, 9, 15, x + 4, x + 8, x + 12, 30, 32
The problem asks us to find the value of x given a dataset arranged in ascending order and its median.
Understanding Median Calculation
The dataset provided is: 6, 9, 15, x + 4, x + 8, x + 12, 30, 32. There are 8 observations in total (n=8). Since 'n' is an even number, the median is calculated as the average of the two middle observations.
- The middle positions are the $\frac{n}{2}$th and $(\frac{n}{2} + 1)$th observations.
- For n=8, these are the $\frac{8}{2} = 4$th and $(\frac{8}{2} + 1) = 5$th observations.
Calculating the Value of x
The 4th observation is x + 4 and the 5th observation is x + 8. The median is given as 19.
Using the median formula:
$ \text{Median} = \frac{(\text{4th observation}) + (\text{5th observation})}{2} $Substitute the values:
$ 19 = \frac{(x + 4) + (x + 8)}{2} $Now, solve for x:
- Simplify the numerator: $19 = \frac{2x + 12}{2}$
- Multiply both sides by 2: $19 \times 2 = 2x + 12$
- $38 = 2x + 12$
- Subtract 12 from both sides: $38 - 12 = 2x$
- $26 = 2x$
- Divide by 2: $x = \frac{26}{2}$
- $x = 13$
Therefore, the value of x is 13.
Verification
If x=13, the data becomes: 6, 9, 15, (13+4), (13+8), (13+12), 30, 32, which is 6, 9, 15, 17, 21, 25, 30, 32. The median is $\frac{17 + 21}{2} = \frac{38}{2} = 19$. This confirms our result.
Similar Questions
- 40 % is the passing criteria in an examination. Out of 9 students who appeared, 4 failed and the remaining received 80 %, 55 %, 52 %, 66 % and 81% marks. The median of the percentage marks is equal to:
- Find the median and the mode of the following data:
2, 3, 5, 7, 2, 3, 3, 5, 7 and 9. - The marks scored by 10 students are given below.
17, 19, 12, 20, 19, 14, 15, 18, 19, 14
The mode of the given data is: - The marks scored by 10 students are given below.
18, 13, 14, 14, 19, 14, 17, 17, 14, 20
The mode of the data is: - In the given data if 30 is replaced by 100 then find the difference of the two medians.
80, 90, 40, 30, 20, 10, 70, 60, 50 - Find the mean height of persons from the following data.
Height (cm) Number of persons 120 3 130 4 140 5 150 6 160 2 - If the mean of the following data is k, then find the value of k.
Value 3 2 k 4 5 Frequency k 2k 3k 4k 5k - Let a set S = {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}. Then the value of $4 \times \text{mean} + 2 \times \text{mode} - 8 \times \text{median}$ is:
- The weights (in kg) of 7 men are 64, 63, 62, 65, 67, 66 and 61. The median weight is:
- Find the sum of the mean and median of the given data.
12, 10, 16, 18, 20, 26, 14, 28
Important Questions from Elementary Statistics
Demand for seats in a university is at its highest in the fall; demand also trends to grow and fall off in 25 year waves. In time service forecasting, the former demand characteristic would be called ______ and the latter would be called _______.
The system of combining two or more overlapping series of index numbers to obtain a single continuous series is called
The rise in the number of patients due to heatstroke is an example of:
According to government data, 24 percent of teenagers in India under the age of 18 years live in households with incomes that are classified at a particular income level. A simple random sample of 400 teenagers in India under the age of 18 years was selected for a study of learning. If the government data is correct, which of the following best approximates the probability that at least 27 per cent of the teenagers in the sample live in households that are classified at a particular income level?
Which index satisfies the factor reversal test?
