Understanding Mean Deviation
Mean deviation is a statistical measure that calculates the average absolute difference between each data point in a set and the mean of that set. It helps understand the spread or dispersion of data around the central value (mean).
Identifying the First 10 Natural Numbers
Natural numbers are positive whole numbers starting from 1. The first 10 natural numbers are:
\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\)
Calculating the Mean of the Numbers
First, we need to find the mean (average) of these 10 numbers. The formula for the mean (\(\bar{x}\)) is the sum of all observations divided by the number of observations.
Sum of the first 10 natural numbers = \(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10\)
We can use the formula for the sum of the first \(n\) natural numbers, which is \(\frac{n(n+1)}{2}\). Here, \(n=10\).
Sum = \(\frac{10(10+1)}{2} = \frac{10 \times 11}{2} = \frac{110}{2} = 55\).
Mean (\(\bar{x}\)) = \(\frac{\text{Sum}}{\text{Number of observations}} = \frac{55}{10} = 5.5\).
Calculating Deviations from the Mean
Next, we find the absolute difference (deviation) between each number and the calculated mean (5.5). The formula for mean deviation is:
Mean Deviation (MD) = \(\frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}\)
Where \(x_i\) is each number, \(\bar{x}\) is the mean, and \(n\) is the total number of observations.
| Observation (\(x_i\)) | Absolute Deviation (\(|x_i - 5.5|\)) |
|---|---|
| 1 | \(|1 - 5.5| = |-4.5| = 4.5\) |
| 2 | \(|2 - 5.5| = |-3.5| = 3.5\) |
| 3 | \(|3 - 5.5| = |-2.5| = 2.5\) |
| 4 | \(|4 - 5.5| = |-1.5| = 1.5\) |
| 5 | \(|5 - 5.5| = |-0.5| = 0.5\) |
| 6 | \(|6 - 5.5| = |0.5| = 0.5\) |
| 7 | \(|7 - 5.5| = |1.5| = 1.5\) |
| 8 | \(|8 - 5.5| = |2.5| = 2.5\) |
| 9 | \(|9 - 5.5| = |3.5| = 3.5\) |
| 10 | \(|10 - 5.5| = |4.5| = 4.5\) |
Determining the Mean Deviation Value
Finally, we calculate the mean of these absolute deviations.
Sum of Absolute Deviations = \(4.5 + 3.5 + 2.5 + 1.5 + 0.5 + 0.5 + 1.5 + 2.5 + 3.5 + 4.5 = 25\).
Mean Deviation (MD) = \(\frac{25}{10} = 2.5\).
Therefore, the mean deviation of the first 10 natural numbers is 2.5.
Similar Questions
What is the mean deviation about the mean ?
What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)?
What is the mean deviation of first 10 even natural numbers?
The sum of deviations of n number of observations measured from 2.5 is 50. The sum of deviations of the same set of observations measured from 3.5 is -50. What is the value of n?
Important Questions from Mean Deviation
What is the mean deviation about the mean ?
- The mean deviation about median of 10 observations is 15. If each observation is multiplied by $-3$, then find the new mean deviation about median of resulting observations.
Let xi, i = 1, 2, ..., n be n observations and wi = pxi + k, i = 1, 2, ..., n where p and k are constants. If the mean of xi's is 48 and standard deviation is 12, whereas the mean of wi's is 55 and standard deviation is 15, then the value of p and k should be
If the mean deviation 1, 1 + d, 1 + 2d, ..., 1 + 100d from their mean is 255, then d is equal to
The mean of 5 observation is 5 and their variance is 124. If three of the observations are 1, 2, 6, then the mean deviation from the mean of the data is
