Consider a sample with 10 observations of 5, 11, 10, 6, 6, 15, 2, 15, 3, and 2. Use z-scores to determine if there are any outliers in the data; assume a bell-shaped distribution. (Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)
To determine if there are any outliers in the data, we need to calculate the z-scores for each observation.
The formula to calculate the z-score is:
z = (x - mean) / standard deviation
First, we need to calculate the mean:
mean = (5 + 11 + 10 + 6 + 6 + 15 + 2 + 15 + 3 + 2) / 10 = 7.5
Next, we need to calculate the standard deviation:
Step 1: Calculate the deviation for each observation:
(5 - 7.5) = -2.5
(11 - 7.5) = 3.5
(10 - 7.5) = 2.5
(6 - 7.5) = -1.5
(6 - 7.5) = -1.5
(15 - 7.5) = 7.5
(2 - 7.5) = -5.5
(15 - 7.5) = 7.5
(3 - 7.5) = -4.5
(2 - 7.5) = -5.5
Step 2: Square each deviation:
(-2.5)^2 = 6.25
(3.5)^2 = 12.25
(2.5)^2 = 6.25
(-1.5)^2 = 2.25
(-1.5)^2 = 2.25
(7.5)^2 = 56.25
(-5.5)^2 = 30.25
(7.5)^2 = 56.25
(-4.5)^2 = 20.25
(-5.5)^2 = 30.25
Step 3: Calculate the average of the squared deviations:
(6.25 + 12.25 + 6.25 + 2.25 + 2.25 + 56.25 + 30.25 + 56.25 + 20.25 + 30.25) / 10 = 20.5
Step 4: Take the square root of the average squared deviations:
sqrt(20.5) ≈ 4.53
Now we can calculate the z-scores for each observation:
z1 = (5 - 7.5) / 4.53 ≈ -0.55
z2 = (11 - 7.5) / 4.53 ≈ 0.77
z3 = (10 - 7.5) / 4.53 ≈ 0.55
z4 = (6 - 7.5) / 4.53 ≈ -0.33
z5 = (6 - 7.5) / 4.53 ≈ -0.33
z6 = (15 - 7.5) / 4.53 ≈ 1.65
z7 = (2 - 7.5) / 4.53 ≈ -1.22
z8 = (15 - 7.5) / 4.53 ≈ 1.65
z9 = (3 - 7.5) / 4.53 ≈ -0.99
z10 = (2 - 7.5) / 4.53 ≈ -1.22
We can consider any z-score that is greater than 3 or less than -3 as an outlier.
There are no z-scores in this data set that are greater than 3 or less than -3. Therefore, there are no outliers in the data.
To determine if there are any outliers in the data using z-scores, we need to calculate the z-score for each observation.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
- x is the value of the observation
- μ is the mean of the data
- σ is the standard deviation of the data
Let's calculate the z-scores for each observation step-by-step:
Step 1: Calculate the mean (μ) of the data
μ = (5 + 11 + 10 + 6 + 6 + 15 + 2 + 15 + 3 + 2) / 10
μ = 75 / 10
μ = 7.5
Step 2: Calculate the standard deviation (σ) of the data
To calculate the standard deviation, we first need to calculate the variance.
variance = (sum of (x - μ)^2) / n
variance = [(5 - 7.5)^2 + (11 - 7.5)^2 + (10 - 7.5)^2 + (6 - 7.5)^2 + (6 - 7.5)^2 + (15 - 7.5)^2 + (2 - 7.5)^2 + (15 - 7.5)^2 + (3 - 7.5)^2 + (2 - 7.5)^2] / 10
variance = [-2.5^2 + 3.5^2 + 2.5^2 + -1.5^2 + -1.5^2 + 7.5^2 + -5.5^2 + 7.5^2 + -4.5^2 + -5.5^2] / 10
variance = [6.25 + 12.25 + 6.25 + 2.25 + 2.25 + 56.25 + 30.25 + 56.25 + 20.25 + 30.25] / 10
variance = 222.5 / 10
variance = 22.25
Finally, calculate the standard deviation by taking the square root of the variance.
σ = sqrt(variance)
σ = sqrt(22.25)
σ ≈ 4.71
Step 3: Calculate the z-score for each observation
Now we can calculate the z-score for each observation using the formula z = (x - μ) / σ
For the given observations, the z-scores are as follows:
z1 = (5 - 7.5) / 4.71 ≈ -0.53
z2 = (11 - 7.5) / 4.71 ≈ 0.74
z3 = (10 - 7.5) / 4.71 ≈ 0.53
z4 = (6 - 7.5) / 4.71 ≈ -0.32
z5 = (6 - 7.5) / 4.71 ≈ -0.32
z6 = (15 - 7.5) / 4.71 ≈ 1.59
z7 = (2 - 7.5) / 4.71 ≈ -1.18
z8 = (15 - 7.5) / 4.71 ≈ 1.59
z9 = (3 - 7.5) / 4.71 ≈ -0.96
z10 = (2 - 7.5) / 4.71 ≈ -1.18
Step 4: Identify any outliers
Any observation with a z-score greater than or less than 3 would be considered an outlier.
In the given data, none of the z-scores are greater than 3 or less than -3. Therefore, there are no outliers in the data.