Consider a sample with 10 observations of −3, 0, 5, 10, 12, −3, −2, 0, −3, and 8. 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-score for each observation. The z-score measures how many standard deviations an observation is from the mean.
First, we calculate the mean of the data:
Mean = (−3 + 0 + 5 + 10 + 12 + −3 + −2 + 0 + −3 + 8) / 10 = 4.4
Next, we calculate the standard deviation of the data:
Step 1: Calculate the squared difference between each observation and the mean:
(-3 - 4.4)^2 = 57.76
(0 - 4.4)^2 = 19.36
(5 - 4.4)^2 = 0.36
(10 - 4.4)^2 = 31.36
(12 - 4.4)^2 = 57.76
(-3 - 4.4)^2 = 57.76
(-2 - 4.4)^2 = 47.04
(0 - 4.4)^2 = 19.36
(-3 - 4.4)^2 = 57.76
(8 - 4.4)^2 = 12.96
Step 2: Calculate the average of the squared differences:
Average squared difference = (57.76 + 19.36 + 0.36 + 31.36 + 57.76 + 57.76 + 47.04 + 19.36 + 57.76 + 12.96) / 10 = 35.56
Step 3: Take the square root of the average squared difference to get the standard deviation:
Standard deviation = sqrt(35.56) = 5.96
Now, we calculate the z-score for each observation:
Z-score = (observation - mean) / standard deviation
Z-score for -3: (-3 - 4.4) / 5.96 = -1.56
Z-score for 0: (0 - 4.4) / 5.96 = -0.74
Z-score for 5: (5 - 4.4) / 5.96 = 0.10
Z-score for 10: (10 - 4.4) / 5.96 = 0.94
Z-score for 12: (12 - 4.4) / 5.96 = 1.27
Z-score for -3: (-3 - 4.4) / 5.96 = -1.56
Z-score for -2: (-2 - 4.4) / 5.96 = -1.39
Z-score for 0: (0 - 4.4) / 5.96 = -0.74
Z-score for -3: (-3 - 4.4) / 5.96 = -1.56
Z-score for 8: (8 - 4.4) / 5.96 = 0.60
By looking at the z-scores, we can determine if any of the observations are significantly different from the mean. Typically, a z-score greater than 2 or less than -2 is considered to be an outlier.
In this case, none of the z-scores are greater than 2 or less than -2. Therefore, there are no outliers in the data.
To determine if there are any outliers in the data, we can use z-scores.
Step 1: Calculate the mean and standard deviation of the data.
Mean (μ) = (−3 + 0 + 5 + 10 + 12 + −3 + −2 + 0 + −3 + 8) / 10 = 4.4
Standard Deviation (σ) = √( (∑ (x - μ)²) / n) = √( ((-3-4.4)² + (0-4.4)² + (5-4.4)² + (10-4.4)² + (12-4.4)² + (-3-4.4)² + (-2-4.4)² + (0-4.4)² + (-3-4.4)² + (8-4.4)² ) / 10 ) = 5.82
Step 2: Calculate the z-score for each observation.
Z-Score = (x - μ) / σ
Z-Score of -3 = (-3 - 4.4) / 5.82 = -1.55
Z-Score of 0 = (0 - 4.4) / 5.82 = -0.76
Z-Score of 5 = (5 - 4.4) / 5.82 = 0.10
Z-Score of 10 = (10 - 4.4) / 5.82 = 0.96
Z-Score of 12 = (12 - 4.4) / 5.82 = 1.31
Z-Score of -3 = (-3 - 4.4) / 5.82 = -1.55
Z-Score of -2 = (-2 - 4.4) / 5.82 = -1.24
Z-Score of 0 = (0 - 4.4) / 5.82 = -0.76
Z-Score of -3 = (-3 - 4.4) / 5.82 = -1.55
Z-Score of 8 = (8 - 4.4) / 5.82 = 0.62
Step 3: Identify any observations with z-scores outside the acceptable range.
Generally, outliers can be identified as any observations with z-scores greater than 3 or less than -3.
In this case, none of the observations have a z-score greater than 3 or less than -3. Therefore, there are no outliers in the data.