What is the mean deviation of first 10 even natural numbers?
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Calculating Mean Deviation of Even Natural Numbers
This problem asks us to find the mean deviation of the first 10 even natural numbers. The mean deviation is a measure of dispersion that tells us the average distance of each data point from the mean (or median, but commonly the mean). To find the mean deviation, we first need to calculate the mean of the numbers and then the average of the absolute differences between each number and the mean.
Understanding Mean Deviation
The mean deviation is defined as the average of the absolute deviations of the data points from a central value (usually the mean or median). The formula for mean deviation from the mean is:
Mean Deviation = $\frac{\sum |x_i - \bar{x}|}{n}$
Where:
- $x_i$ represents each individual data point.
- $\bar{x}$ represents the mean of the data.
- $|x_i - \bar{x}|$ represents the absolute deviation of each data point from the mean.
- $\sum |x_i - \bar{x}|$ represents the sum of all absolute deviations.
- $n$ represents the number of data points.
Identifying the First 10 Even Natural Numbers
Natural numbers are positive integers starting from 1 (1, 2, 3, ...). Even natural numbers are natural numbers that are divisible by 2. The first 10 even natural numbers are:
- 2
- 4
- 6
- 8
- 10
- 12
- 14
- 16
- 18
- 20
So, the set of data points is $\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}$. The number of data points, $n$, is 10.
Calculating the Mean of the Even Numbers
The mean ($\bar{x}$) is the sum of all data points divided by the number of data points.
Sum of the first 10 even natural numbers = $2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110$.
Number of data points ($n$) = 10.
Mean ($\bar{x}$) = $\frac{\text{Sum of numbers}}{n} = \frac{110}{10} = 11$.
The mean of the first 10 even natural numbers is 11.
Calculating Absolute Deviations from the Mean
Next, we find the absolute difference between each number and the mean (11). We take the absolute value to ensure the deviations are positive.
| Even Natural Number ($x_i$) | Mean ($\bar{x}$) | Deviation ($x_i - \bar{x}$) | Absolute Deviation $|x_i - \bar{x}|$ |
|---|---|---|---|
| 2 | 11 | 2 - 11 = -9 | $|-9| = 9$ |
| 4 | 11 | 4 - 11 = -7 | $|-7| = 7$ |
| 6 | 11 | 6 - 11 = -5 | $|-5| = 5$ |
| 8 | 11 | 8 - 11 = -3 | $|-3| = 3$ |
| 10 | 11 | 10 - 11 = -1 | $|-1| = 1$ |
| 12 | 11 | 12 - 11 = 1 | $|1| = 1$ |
| 14 | 11 | 14 - 11 = 3 | $|3| = 3$ |
| 16 | 11 | 16 - 11 = 5 | $|5| = 5$ |
| 18 | 11 | 18 - 11 = 7 | $|7| = 7$ |
| 20 | 11 | 20 - 11 = 9 | $|9| = 9$ |
The sum of the absolute deviations is $\sum |x_i - \bar{x}| = 9 + 7 + 5 + 3 + 1 + 1 + 3 + 5 + 7 + 9 = 50$.
Calculating the Mean Deviation
Now, we calculate the mean deviation using the formula:
Mean Deviation = $\frac{\sum |x_i - \bar{x}|}{n} = \frac{50}{10} = 5$.
The mean deviation of the first 10 even natural numbers is 5.
Revision Table: Key Statistics Concepts
| Concept | Definition | Calculation (from mean) | Purpose |
|---|---|---|---|
| Mean ($\bar{x}$) | Average of a dataset | Sum of values / Number of values | Measure of central tendency |
| Deviation ($x_i - \bar{x}$) | Difference between a data point and the mean | $x_i - \bar{x}$ | Shows how far a point is from the mean (signed) |
| Absolute Deviation ($|x_i - \bar{x}|$) | Positive difference between a data point and the mean | $|x_i - \bar{x}|$ | Shows the distance from the mean (unsigned) |
| Mean Deviation | Average of absolute deviations from the mean | $\frac{\sum |x_i - \bar{x}|}{n}$ | Measure of dispersion, average distance from the mean |
Additional Information: Measures of Dispersion
Mean deviation is one type of measure of dispersion. Measures of dispersion describe how spread out the data points are in a dataset. Other common measures of dispersion include:
- Range: The difference between the highest and lowest values in the dataset.
- Variance: The average of the squared deviations from the mean. This is calculated as $\frac{\sum (x_i - \bar{x})^2}{n}$ (for population) or $\frac{\sum (x_i - \bar{x})^2}{n-1}$ (for sample).
- Standard Deviation: The square root of the variance. This is the most commonly used measure of dispersion. It is calculated as $\sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$ or $\sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$.
Mean deviation is relatively easy to calculate but is less commonly used than standard deviation because the absolute value function can make further mathematical analysis more difficult compared to the squared deviations used in variance and standard deviation.
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