Which set of population data is the least dispersed from its mean?
2, 3, 2, 9
4, 0, 4, 0
6, 2, 2, 2
9, 3, 5, 3
Calculate the mean for each, then get the total of the differences between the mean and the individual scores. Which sum is smallest?
4,0,4,0
To determine which set of population data is the least dispersed from its mean, we need to calculate the measures of dispersion for each set and compare them.
The measure of dispersion commonly used to assess the spread of data around the mean is the standard deviation. The lower the standard deviation, the less dispersed the data is from the mean.
Let's calculate the standard deviations for each set of data:
1) Set 1: 2, 3, 2, 9
To find the mean, we add up all the numbers and divide by the total number of values: (2 + 3 + 2 + 9) / 4 = 4.
Next, we find the deviations from the mean for each value: (2-4), (3-4), (2-4), (9-4) = -2, -1, -2, 5.
Then, we square each deviation: (-2)^2, (-1)^2, (-2)^2, 5^2 = 4, 1, 4, 25.
The variance is found by taking the average of the squared deviations: (4 + 1 + 4 + 25) / 4 = 34 / 4 = 8.5.
Finally, the standard deviation is the square root of the variance: sqrt(8.5) ≈ 2.92.
2) Set 2: 4, 0, 4, 0
Mean = (4 + 0 + 4 + 0) / 4 = 2.
Deviations from the mean: (4-2), (0-2), (4-2), (0-2) = 2, -2, 2, -2.
Squared deviations: 2^2, (-2)^2, 2^2, (-2)^2 = 4, 4, 4, 4.
Variance = (4 + 4 + 4 + 4) / 4 = 16 / 4 = 4.
Standard deviation = sqrt(4) = 2.
3) Set 3: 6, 2, 2, 2
Mean = (6 + 2 + 2 + 2) / 4 = 3.
Deviations from the mean: (6-3), (2-3), (2-3), (2-3) = 3, -1, -1, -1.
Squared deviations: 3^2, (-1)^2, (-1)^2, (-1)^2 = 9, 1, 1, 1.
Variance = (9 + 1 + 1 + 1) / 4 = 12 / 4 = 3.
Standard deviation = sqrt(3) ≈ 1.73.
4) Set 4: 9, 3, 5, 3
Mean = (9 + 3 + 5 + 3) / 4 = 5.
Deviations from the mean: (9-5), (3-5), (5-5), (3-5) = 4, -2, 0, -2.
Squared deviations: 4^2, (-2)^2, 0^2, (-2)^2 = 16, 4, 0, 4.
Variance = (16 + 4 + 0 + 4) / 4 = 24 / 4 = 6.
Standard deviation = sqrt(6) ≈ 2.45.
Comparing the standard deviations, we can see that:
Set 1 has a standard deviation of 2.92
Set 2 has a standard deviation of 2
Set 3 has a standard deviation of 1.73
Set 4 has a standard deviation of 2.45
Therefore, set 3: 6, 2, 2, 2 is the least dispersed from its mean, as it has the smallest standard deviation.