Q.The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is
(A) 2
(B) 2.57
(C) 3
(D) 3.75

The key idea here is the Mean Deviation About Mean. It measures the average of the absolute differences between each data point and the mean of the data set.

First, calculate the mean ($\bar{x}$) of the given data:

Next, find the absolute deviation of each data point from the mean, $|x_i - \bar{x}|$:

$|3 - 7| = 4$

$|10 - 7| = 3$

$|10 - 7| = 3$

$|4 - 7| = 3$

$|7 - 7| = 0$

$|10 - 7| = 3$

$|5 - 7| = 2$ …

When we talk about the "mean deviation about the mean," we are trying to quantify how much, on average, the individual data points in a set differ from their central value, which is the mean. It's a measure of dispersion or variability.

The key idea here is that we want to know the typical distance of data points from the mean. If we simply sum the differences $(x_i - \bar{x})$, the positive and negative deviations would cancel each other out, always resulting in zero. This is why we take the absolute value of each deviation, $|x_i - \bar{x}|$, before summing them. This ensures that all deviations contribute positively to the total spread, regardless of whether a data point is above or below the mean. We then average these absolute deviations to get the mean deviation.

Here's how to calculate it step-by-step:

1. List the data and count the observations.

   The given data set is $x = \{3, 10, 10, 4, 7, 10, 5\}$.

   The number of observations, $n$, is $7$.

2. Calculate the arithmetic mean ($\bar{x}$) of the data.

   The mean is the sum of all observations divided by the number of observations.

Sum of observations:

Now, calculate the mean:

So, the mean of the data is $7$.

3. Calculate the deviation of each data point from the mean.

For each $x_i$, we find the difference $x_i - \bar{x}$.

> [!WARNING]
> A common mistake is to stop here and sum these deviations. The sum of deviations from the mean, $\sum (x_i - \bar{x})$, is *always* zero. This is a fundamental property of the mean and why we need absolute values for mean deviation.

4. Calculate the absolute deviation of each data point from the mean.

We take the absolute value of each deviation, $|x_i - \bar{x}|$. This ensures that all differences contribute positively to the measure of spread.

Let's create a table to organize these calculations:

| $x_i$ | $x_i - \bar{x}$ | $|x_i - \bar{x}|$ |
| :---: | :-------------: | :---------------: |
| 3     | $3 - 7 = -4$    | 4                 |
| 10    | $10 - 7 = 3$    | 3                 | …