Given the following values, which point would be considered an outlier?
x 1 2 3 4 5 6 7 8 9
y 0.8 2.1 3.1 3.8 5.2 8.4 6.9 8.1 9
A. (2, 2.1)
B. (4, 3.8)
C. (6, 8.4)
D. (9, 9)
C. (6, 8.4)
To determine which point would be considered an outlier, we need to analyze the data and look for any values that significantly deviate from the overall pattern.
One common method to identify outliers is by calculating the z-score for each data point. The z-score represents the number of standard deviations a data point is away from the mean. Generally, data points with a z-score greater than 3 or less than -3 are considered outliers.
To find the outlier in our given data, we need to calculate the z-scores for each y-value:
x 1 2 3 4 5 6 7 8 9
y 0.8 2.1 3.1 3.8 5.2 8.4 6.9 8.1 9
First, we need to calculate the mean (μ) and standard deviation (σ) of the y-values:
Mean (μ) = (0.8 + 2.1 + 3.1 + 3.8 + 5.2 + 8.4 + 6.9 + 8.1 + 9) / 9 = 5.4
Next, we calculate the standard deviation (σ):
Step 1: Subtract the mean from each y-value and square the result:
(0.8 - 5.4)^2 = 18.49
(2.1 - 5.4)^2 = 11.71
(3.1 - 5.4)^2 = 5.29
(3.8 - 5.4)^2 = 2.56
(5.2 - 5.4)^2 = 0.04
(8.4 - 5.4)^2 = 9
(6.9 - 5.4)^2 = 2.29
(8.1 - 5.4)^2 = 7.29
(9 - 5.4)^2 = 12.96
Step 2: Sum up the squared differences:
18.49 + 11.71 + 5.29 + 2.56 + 0.04 + 9 + 2.29 + 7.29 + 12.96 = 69.63
Step 3: Divide the sum by (n-1) where n is the number of data points (in this case, 9):
69.63 / 8 = 8.70
Step 4: Take the square root of the result to find the standard deviation (σ):
σ = √8.70 ≈ 2.95
Now, let's calculate the z-score for each y-value using the formula:
z = (y - μ) / σ
For (2, 2.1):
z = (2.1 - 5.4) / 2.95 ≈ -1.12
For (4, 3.8):
z = (3.8 - 5.4) / 2.95 ≈ -0.54
For (6, 8.4):
z = (8.4 - 5.4) / 2.95 ≈ 1.02
For (9, 9):
z = (9 - 5.4) / 2.95 ≈ 1.22
Based on the z-scores, none of the y-values have z-scores greater than 3 or less than -3. Therefore, none of the points are considered outliers.
To summarize, there are no outliers in the given data set.
To determine which point is considered an outlier, we can look for the point that is significantly different from the other points. One common way to identify outliers is by using the z-score.
To calculate the z-score for each point, we can use the formula:
z = (x - mean(x)) / std deviation(x)
Let's calculate the z-scores for each point:
x-values:
1, 2, 3, 4, 5, 6, 7, 8, 9
mean(x) = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 9 = 5
std deviation(x) = sqrt(((1 - 5)^2 + (2 - 5)^2 + (3 - 5)^2 + (4 - 5)^2 + (5 - 5)^2 + (6 - 5)^2 + (7 - 5)^2 + (8 - 5)^2 + (9 - 5)^2) / 9) = sqrt(10)
Using these values, we can calculate the z-scores for each point:
z(1) = (1 - 5) / sqrt(10)
z(2) = (2 - 5) / sqrt(10)
z(3) = (3 - 5) / sqrt(10)
z(4) = (4 - 5) / sqrt(10)
z(5) = (5 - 5) / sqrt(10)
z(6) = (6 - 5) / sqrt(10)
z(7) = (7 - 5) / sqrt(10)
z(8) = (8 - 5) / sqrt(10)
z(9) = (9 - 5) / sqrt(10)
Now, let's calculate the z-scores for each point:
z(1) = -2.82
z(2) = -1.41
z(3) = -0.71
z(4) = -0.14
z(5) = 0
z(6) = 0.71
z(7) = 1.41
z(8) = 2.12
z(9) = 2.82
We can see that the z-score is highest for (9, 9) with a z-score of 2.82. Therefore, point (9, 9) would be considered an outlier. The answer is option D.