consider the data set shown how does the outlier affect the mean median and mode? 7,11,10,8,10,23,8
it only affects the mean, shifting it toward the oulier
To understand how an outlier affects the measures of central tendency (mean, median, and mode) in a given dataset, let's begin by calculating these measures with and without the outlier.
First, let's compute the measures of central tendency without considering the outlier.
- Mean: To find the mean, sum up all the values in the dataset and divide by the total number of values. Without the outlier, the sum of the dataset is 7 + 11 + 10 + 8 + 10 + 8 = 54. Dividing this sum by the total number of values (6), we get a mean of 54 / 6 = 9.
- Median: The median is the middle value in the dataset when it is arranged in ascending order. Without the outlier, the dataset arranged in ascending order is 7, 8, 8, 10, 10, 11. The middle value is the average of the two central values since we have an even number of values. Therefore, the median is (8 + 10) / 2 = 9.
- Mode: The mode is the value that appears most frequently in the dataset. Without the outlier, the modes are 8 and 10 (each appeared twice).
Now, let's consider the impact of the outlier by including it in the calculations.
- Mean: The sum of the dataset with the outlier is 7 + 11 + 10 + 8 + 10 + 23 + 8 = 77. Dividing this sum by the total number of values (7), we get a mean of 77 / 7 ≈ 11.
- Median: When the dataset is arranged in ascending order with the outlier, we have: 7, 8, 8, 10, 10, 11, 23. The median remains the same since it is still the average of the two central values, which is (8 + 10) / 2 = 9.
- Mode: The mode remains the same since the outlier value (23) doesn't affect the modes, which are still 8 and 10 (each appeared twice).
In conclusion, the outlier (23) mainly affects the mean by pulling it away from the other values. However, the median and mode remain unaffected by the presence of the outlier in this dataset.