The Frequency distribution of a histogram is the following scores:
X F
1 1
2 1
3 2
4 1
1) what is the total area in the histogram?
2) what is half the area in the histogram?
3) Assume that the scores are measurements of a discrete variable. Find the median.
4) Assume the the scores are mesaurements of a
continuous variable. Find the median?
To answer these questions, let's go step by step:
1) To find the total area in the histogram, you need to multiply each score (X) by its corresponding frequency (F) and then sum them up. So in this case, you would have:
Area = (X1 * F1) + (X2 * F2) + (X3 * F3) + (X4 * F4)
= (1 * 1) + (2 * 1) + (3 * 2) + (4 * 1)
= 1 + 2 + 6 + 4
= 13
Therefore, the total area in the histogram is 13.
2) Half the area in the histogram means finding the area that represents half of the total area. To do this, you need to divide the total area by 2. In this case, half the area would be:
Half Area = Total Area / 2
= 13 / 2
= 6.5
So half the area in the histogram is 6.5.
3) Assuming the scores are measurements of a discrete variable, the median is the middle value when the scores are ordered from smallest to largest. In this case, the scores are already given in ascending order, so we just need to find the middle value.
The total frequency is the sum of all frequencies in the histogram, which is 1 + 1 + 2 + 1 = 5. Since 5 is an odd number, we can directly find the median by locating the middle position, which can be calculated using the formula:
Median Position = (Total Frequency + 1) / 2
Median Position = (5 + 1) / 2
= 6 / 2
= 3
So the median is the value at the third position, which is 3.
4) Assuming the scores are measurements of a continuous variable, finding the median is slightly different. In this case, you need to calculate the median position using the formula mentioned earlier, and then locate the corresponding value in the cumulative frequency distribution.
To find the cumulative frequency distribution, you need to sum up the frequencies cumulatively. Using the given frequency distribution:
Cumulative Frequency:
X F Cumulative Frequency
1 1 1
2 1 2
3 2 4
4 1 5
Now, find the median position:
Median Position = (Total Frequency + 1) / 2
= (5 + 1) / 2
= 6 / 2
= 3
From the cumulative frequency distribution, you can see that the third position falls between the values of 2 and 3 (inclusive). Therefore, the median for a continuous variable is estimated to be in between these two scores.
1) To find the total area in the histogram, we need to calculate the sum of the products of each X value and its corresponding frequency (F).
1 x 1 + 2 x 1 + 3 x 2 + 4 x 1 = 1 + 2 + 6 + 4 = 13
Therefore, the total area in the histogram is 13 units.
2) To find half the area in the histogram, we need to divide the total area by 2.
Half the area = 13 / 2 = 6.5 units
3) To find the median for a discrete variable, we need to determine the value that separates the data into two equal parts.
In this case, the median would be the middle value in the cumulative frequency distribution. To calculate the cumulative frequency, we can sum up the frequencies (F) as we go through the X values.
Cumulative frequency:
1 + 1 = 2 (less than or equal to 1)
2 + 1 = 3 (less than or equal to 2)
3 + 2 = 5 (less than or equal to 3)
5 + 1 = 6 (less than or equal to 4)
Since the total number of scores is 6, the median is found at the 3rd score, which is the value 3.
4) To find the median for a continuous variable, we need to calculate the cumulative frequency distribution and determine the X value that corresponds to the middle cumulative frequency.
Using the same cumulative frequency calculations as in question 3:
Cumulative frequency:
1 + 1 = 2 (less than or equal to 1)
2 + 1 = 3 (less than or equal to 2)
3 + 2 = 5 (less than or equal to 3)
5 + 1 = 6 (less than or equal to 4)
The median for a continuous variable is found using interpolation:
Median = L + ((N/2 - CF) / f) * w
Where:
L = Lower class boundary of the median class
N = Total number of scores
CF = Cumulative frequency of the class before the median class
f = Frequency of the median class
w = Width of the median class interval
In this case, the median class is the interval 2-3, with a frequency of 2.
L = 2 (lower boundary of the median class)
N = 6 (total number of scores)
CF = 2 (cumulative frequency before the median class)
f = 2 (frequency of the median class)
w = 1 (width of the median class interval)
Plugging the values into the formula:
Median = 2 + ((6/2 - 2) / 2) * 1
Median = 2 + ((3 - 2) / 2) * 1
Median = 2 + (1 / 2)
Median = 2.5
Therefore, for a continuous variable, the median is 2.5.