What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)?
Calculating the Coefficient of Mean Deviation from Mean
The question asks us to find the coefficient of mean deviation for a given set of data points, using the mean as the reference point for deviation.
The data set is: 21, 34, 23, 39, 26, 37, 40, 20, 33, 27.
There are 10 data points, so $n = 10$.
Understanding Coefficient of Mean Deviation
The coefficient of mean deviation (CMD) is a relative measure of dispersion. It is used to compare the variability or dispersion of different data sets, especially when their means are significantly different or they are measured in different units. It is calculated as the ratio of the Mean Deviation (MD) to an average (like mean, median, or mode).
The formula for the coefficient of mean deviation from the mean is:
$$ \text{Coefficient of Mean Deviation (CMD)} = \frac{\text{Mean Deviation (MD)}}{\text{Mean} (\bar{x})} $$
The Mean Deviation (MD) from the mean is the average of the absolute deviations of each data point from the mean:
$$ \text{Mean Deviation (MD)} = \frac{\sum |x - \bar{x}|}{n} $$
Where:
- $x$ represents each data point.
- $\bar{x}$ represents the mean of the data.
- $|x - \bar{x}|$ is the absolute deviation of $x$ from the mean.
- $\sum |x - \bar{x}|$ is the sum of the absolute deviations.
- $n$ is the number of data points.
Step-by-Step Calculation
Step 1: Calculate the Mean ($\bar{x}$)
First, we need to find the mean of the given data points. The mean is the sum of all data points divided by the number of data points.
Sum of data points ($\sum x$) = $21 + 34 + 23 + 39 + 26 + 37 + 40 + 20 + 33 + 27 = 300$
Number of data points ($n$) = 10
Mean ($\bar{x}$) = $$ \frac{\sum x}{n} = \frac{300}{10} = 30 $$
The mean of the data is 30.
Step 2: Calculate the Absolute Deviations from the Mean
Next, we calculate the absolute difference between each data point and the mean (30). We can organize this in a table.
| Data Point (x) | Mean ($\bar{x}$) | Deviation ($x - \bar{x}$) | Absolute Deviation ($|x - \bar{x}|$) |
|---|---|---|---|
| 21 | 30 | 21 - 30 = -9 | |-9| = 9 |
| 34 | 30 | 34 - 30 = 4 | |4| = 4 |
| 23 | 30 | 23 - 30 = -7 | |-7| = 7 |
| 39 | 30 | 39 - 30 = 9 | |9| = 9 |
| 26 | 30 | 26 - 30 = -4 | |-4| = 4 |
| 37 | 30 | 37 - 30 = 7 | |7| = 7 |
| 40 | 30 | 40 - 30 = 10 | |10| = 10 |
| 20 | 30 | 20 - 30 = -10 | |-10| = 10 |
| 33 | 30 | 33 - 30 = 3 | |3| = 3 |
| 27 | 30 | 27 - 30 = -3 | |-3| = 3 |
Step 3: Calculate the Sum of Absolute Deviations
Sum of the absolute deviations ($\sum |x - \bar{x}|$) = $9 + 4 + 7 + 9 + 4 + 7 + 10 + 10 + 3 + 3 = 66$
Step 4: Calculate the Mean Deviation (MD) from the Mean
MD = $$ \frac{\sum |x - \bar{x}|}{n} = \frac{66}{10} = 6.6 $$
The mean deviation from the mean is 6.6.
Step 5: Calculate the Coefficient of Mean Deviation (CMD) from the Mean
Now, we can calculate the coefficient of mean deviation using the MD and the mean.
CMD = $$ \frac{\text{MD}}{\bar{x}} = \frac{6.6}{30} $$
CMD = 0.22
The coefficient of mean deviation of the given data from the mean is 0.22.
Final Answer
The calculated coefficient of mean deviation is 0.22.
Revision Table: Measures of Dispersion
| Measure | Type | Definition/Formula Example (from Mean) | Use Case |
|---|---|---|---|
| Range | Absolute | Maximum value - Minimum value | Quickest, but uses only two values. |
| Quartile Deviation | Absolute | $Q_3 - Q_1 / 2$ | Used for skewed data, less affected by extreme values. |
| Mean Deviation | Absolute | $\frac{\sum |x - \text{Average}|}{n}$ (Average can be Mean, Median, Mode) | Considers all values, but ignores sign of deviations. |
| Standard Deviation | Absolute | $\sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$ (Sample) or $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ (Population) | Most common, used in many statistical tests, considers squared deviations. |
| Coefficient of Range | Relative | $\frac{\text{Max} - \text{Min}}{\text{Max} + \text{Min}}$ | Comparing variability of different datasets. |
| Coefficient of Quartile Deviation | Relative | $\frac{Q_3 - Q_1}{Q_3 + Q_1}$ | Comparing variability of different datasets (skewed). |
| Coefficient of Mean Deviation | Relative | $\frac{\text{MD}}{\text{Average}}$ (Average can be Mean, Median, Mode) | Comparing variability of different datasets. |
| Coefficient of Variation | Relative | $\frac{\text{Standard Deviation}}{\text{Mean}} \times 100$ | Comparing variability relative to the mean, expressed as percentage. |
Additional Information on Measures of Dispersion
Measures of dispersion, also known as measures of variability, describe the spread or scatter of data points in a distribution. They tell us how much individual data points differ from the average and from each other.
- Absolute Measures: These measures are expressed in the same units as the data. They are useful for comparing the dispersion within a single dataset or between datasets with similar averages and units. Examples include Range, Quartile Deviation, Mean Deviation, and Standard Deviation.
- Relative Measures: These measures are unitless and are expressed as a ratio or percentage. They are useful for comparing the dispersion of two or more datasets that have different means or are measured in different units. Examples include Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, and Coefficient of Variation.
The choice of which measure of dispersion to use depends on the nature of the data and the purpose of the analysis. For example, the mean deviation is simple to calculate but is not suitable for further mathematical analysis because it ignores the signs of deviations. Standard deviation is widely used because it is amenable to algebraic treatment and forms the basis for many advanced statistical concepts.
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