The median of the following incomplete frequency distribution is 10:
x= 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
f= 0 , 2 , 5 , 6, 3 , 1 , 2 , Y
Calculate the frequency of 8
To calculate the frequency of 8, we need to first determine the value of Y in the incomplete frequency distribution. Given that the median is 10 and represents the middle value in the data set, we know that the sum of the frequencies below the median must be equal to the sum of the frequencies above the median.
In this case, we can observe that the sum of the frequencies below the median is:
0 + 2 + 5 + 6 = 13
To find the value of Y, we need to subtract this sum from the total number of data points in the distribution, which is the sum of all frequencies given. In this case, that total is:
0 + 2 + 5 + 6 + 3 + 1 + 2 + Y
Using the properties of equality, we can set up an equation to solve for Y:
13 + Y = sum of all frequencies
Given the values of the frequencies in the incomplete distribution, we can substitute them into the equation:
13 + Y = 0 + 2 + 5 + 6 + 3 + 1 + 2 + Y
By adding the numbers on the right side of the equation, we have:
13 + Y = 19 + Y
To solve for Y, we can subtract Y from both sides of the equation:
13 = 19
This equation is inconsistent and cannot be solved. Therefore, there is no unique value for Y that satisfies the given incomplete frequency distribution.
To calculate the frequency of 8, we need to use the information given about the median of the distribution.
Since the median is given as 10, this means that when the data in the distribution is arranged in ascending order, the value at the middle position should be 10.
To find the middle position, we need to sum up the frequencies until we reach the middle position.
The total number of data points can be found by summing up the frequencies:
Total number of data points = 0 + 2 + 5 + 6 + 3 + 1 + 2 + Y
Since the median is the middle value, we need to find the position where half of the data points lie below it and half of the data points lie above it.
For example, if we have 19 data points, the median will be the value at the 10th position (since 9 values are below and 9 values are above). The exact position of the median depends on the total number of data points.
To find the position of the median, we can use the formula:
Position of the median = (Total number of data points + 1) / 2
So in this case, the position of the median would be (Total number of data points + 1) / 2 = (0 + 2 + 5 + 6 + 3 + 1 + 2 + Y + 1) / 2 = (20 + Y) / 2
Since the median is given as 10, the value at the middle position should be 10. Therefore, we can set up an equation:
Position of the median = Value at the middle position
(20 + Y) / 2 = 10
Next, we can solve this equation for Y:
20 + Y = 20
Y = 20 - 20
Y = 0
So the value of Y is 0.
Now that we know Y is 0, we can calculate the frequency of 8:
Frequency of 8 = 2 (from the given data)
Therefore, the frequency of 8 is 2.