What is the frequency distribution below and determine the range variance and standard deviation of the population function v=1-2-3-4-6-0
F=1-2-3-4-3-2
The frequency distribution shows that the population function has a range of 6, meaning that the values range from 0 to 6.
To find the variance of the population function, we need to find the mean of the values, which is:
(1+2+3+4+6+0)/6 = 2.67
Then we subtract the mean from each value, square the result, and multiply by its frequency:
(1-2.67)^2 * 1 = 2.11
(2-2.67)^2 * 2 = 1.78
(3-2.67)^2 * 3 = 0.30
(4-2.67)^2 * 1 = 1.11
(6-2.67)^2 * 1 = 9.16
(0-2.67)^2 * 2 = 7.12
Adding these values together gives us:
2.11 + 1.78 + 0.30 + 1.11 + 9.16 + 7.12 = 21.58
To find the variance, we divide this sum by the total frequency (10 in this case):
21.58/10 = 2.16
The standard deviation is the square root of the variance, so:
√2.16 = 1.47
Therefore, the range of the population function is 6, the variance is 2.16, and the standard deviation is 1.47.
To determine the frequency distribution, first, we can list the unique values in the population function v=1-2-3-4-6-0:
1, 2, 3, 4, 6, 0
Next, we count the frequency of each value in the population function F=1-2-3-4-3-2:
For value 1, the frequency is 1.
For value 2, the frequency is 2.
For value 3, the frequency is 2.
For value 4, the frequency is 1.
Now we can represent the frequency distribution in a table format:
Value | Frequency
-------------------
1 | 1
2 | 2
3 | 2
4 | 1
6 | 0
0 | 1
To determine the range of the population function, we subtract the minimum value from the maximum value:
Range = Maximum value - Minimum value
= 6 - 0
= 6
To calculate the variance and standard deviation, we need to calculate the mean of the population function.
Mean (μ) = (1 + 2 + 3 + 4 + 6 + 0) / 6
= 16 / 6
= 2.67 (approximately)
To calculate variance, we follow these steps:
1. Subtract the mean from each data point and square the result.
2. Multiply each squared difference by its corresponding frequency.
3. Sum all the multiplied values obtained from step 2.
4. Divide the sum obtained from step 3 by the total frequency.
Let's perform these calculations:
For value 1: (1 - 2.67)^2 * 1 = 2.1129
For value 2: (2 - 2.67)^2 * 2 = 4.8884
For value 3: (3 - 2.67)^2 * 2 = 0.3536
For value 4: (4 - 2.67)^2 * 1 = 1.7689
For value 6: (6 - 2.67)^2 * 0 = 0
For value 0: (0 - 2.67)^2 * 1 = 7.1489
Sum: 2.1129 + 4.8884 + 0.3536 + 1.7689 + 0 + 7.1489 = 16.2727
Variance = Sum / Total frequency = 16.2727 / 6 = 2.7121 (approximately)
Finally, to calculate the standard deviation, we take the square root of the variance:
Standard Deviation = sqrt(Variance) = sqrt(2.7121) = 1.6470 (approximately)
So, the frequency distribution is:
Value | Frequency
-------------------
1 | 1
2 | 2
3 | 2
4 | 1
6 | 0
0 | 1
The range is 6, the variance is 2.7121 (approximately), and the standard deviation is 1.6470 (approximately).