Calculating the mean deviation about the mean for ungrouped data involves finding the average of the absolute deviations from the mean of the dataset. Here's a step-by-step guide:
-
Calculate the Mean (Average):
- Add up all the values in the data set.
- Divide the sum by the total number of values (n).
Formula: Mean (μ) = (∑xi) / n
Where:
- ∑xi represents the sum of all data points.
- n is the number of data points.
-
Calculate the Deviations:
- For each data point, subtract the mean (μ) from the data point (xi).
- This gives you the deviation of each data point from the mean.
Formula: Deviation = xi - μ
-
Calculate the Absolute Deviations:
- Take the absolute value of each deviation calculated in the previous step. This means ignoring any negative signs. Absolute deviations are always positive or zero.
Formula: |xi - μ|
-
Calculate the Mean Deviation:
- Add up all the absolute deviations.
- Divide the sum of the absolute deviations by the total number of data points (n).
Formula: Mean Deviation (MD) = (∑|xi - μ|) / n
Example:
Let's say you have the following data set: 5, 8, 10, 12, 15
-
Calculate the Mean:
μ = (5 + 8 + 10 + 12 + 15) / 5 = 50 / 5 = 10
-
Calculate the Deviations:
- 5 - 10 = -5
- 8 - 10 = -2
- 10 - 10 = 0
- 12 - 10 = 2
- 15 - 10 = 5
-
Calculate the Absolute Deviations:
- |-5| = 5
- |-2| = 2
- |0| = 0
- |2| = 2
- |5| = 5
-
Calculate the Mean Deviation:
MD = (5 + 2 + 0 + 2 + 5) / 5 = 14 / 5 = 2.8
Therefore, the mean deviation about the mean for this data set is 2.8.
In summary, the mean deviation about the mean provides a measure of the average distance of each data point from the center of the data (mean), giving an indication of the data's variability.
