To solve mean deviation for grouped data, you must calculate the mean of the data, find the absolute deviations from the mean, and then find the average of these deviations. Here's a step-by-step guide:
Step-by-Step Guide to Calculating Mean Deviation for Grouped Data
1. Calculate the Midpoints of Each Class Interval
- When dealing with grouped data, you're given class intervals (e.g., 0-10, 10-20).
- To begin, find the midpoint (x) of each interval using the formula: Midpoint (x) = (Lower Limit + Upper Limit) / 2.
- Example: For the class interval 10-20, the midpoint would be (10 + 20) / 2 = 15.
2. Calculate the Mean (X̄)
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The mean (average) of the grouped data is not simply the sum of the midpoints divided by the number of intervals.
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The mean is calculated as: X̄ = ∑fx / ∑f, where:
- f is the frequency (number of occurrences) of each class interval.
- x is the midpoint of each class interval (calculated in step 1).
- ∑fx is the sum of each midpoint multiplied by its corresponding frequency.
- ∑f is the sum of all the frequencies.
Example:
Class Interval Frequency (f) Midpoint (x) fx 0-10 5 5 25 10-20 10 15 150 20-30 15 25 375 Total 30 550 X̄ = 550 / 30 = 18.33
3. Calculate the Absolute Deviations from the Mean (|X-X̄|)
- For each class interval, subtract the calculated mean (X̄) from the midpoint (x), and take the absolute value: |X - X̄|.
- Example: In our example, if the mean is 18.33. for the first class midpoint of 5, the absolute deviation is |5-18.33|= 13.33.
4. Multiply Absolute Deviations by the Frequency
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Multiply each absolute deviation (|X - X̄|) by its corresponding frequency (f): f|X - X̄|
Example:
| Class Interval | Frequency (f) | Midpoint (x) | |x - X̄| | f|x - X̄| |
|---|---|---|---|---|---|
| 0-10 | 5 | 5 | |5-18.33| = 13.33 | 513.33= 66.65 |
| 10-20 | 10 | 15 | |15-18.33| = 3.33 | 103.33 = 33.3|
| 20-30 | 15 | 25 | |25-18.33| = 6.67 | 15*6.67 = 100.05 |
| Total | 30 | | | |∑ f|x-X̄|= 200|
5. Calculate the Mean Deviation
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Finally, divide the sum of all f|X - X̄| by the sum of all the frequencies (∑f):
Mean Deviation = ∑ f|X - X̄| / ∑ f
Example: Using the sample data, mean deviation is 200/30= 6.67.
Summary of Formulae
Here's a summary of the formulas used:
- Midpoint (x) = (Lower Limit + Upper Limit) / 2
- Mean (X̄) = ∑fx / ∑f
- Absolute Deviation = |x - X̄|
- Mean Deviation = ∑f|x - X̄| / ∑f
By following these steps, you can accurately calculate the mean deviation for grouped data. This measures the average amount by which the values deviate from the mean, providing insight into the data's dispersion or variability.
