Understanding Mean Square Deviation in Statistics

In statistics, measures of dispersion help us understand the spread of data points in a distribution. Variance is a common measure, representing the mean square deviation from the arithmetic mean. The mean square deviation can also be calculated from other points, like the mode.

The question provides us with the following information about a frequency distribution:

• Variance (\(\sigma^2\)): 2
• Mode: 24
• Arithmetic Mean (\(\bar{x}\)): 25

We need to find the mean square deviation from the mode.

Calculating Mean Square Deviation from a Constant

The mean square deviation (MSD) from a constant 'a' for a frequency distribution \(x_i | f_i, i = 1, 2, \dots, n\) is given by the formula:

\[ \text{MSD}_a = \frac{\sum_{i=1}^{n} f_i (x_i - a)^2}{\sum_{i=1}^{n} f_i} \]The variance is specifically the mean square deviation from the arithmetic mean (\(\bar{x}\)). So, we are given \(\text{MSD}_{\bar{x}} = \sigma^2 = 2\).

We need to calculate \(\text{MSD}_{\text{Mode}}\), where the mode is 24.

Relationship Between MSD from Mean and MSD from Another Point

There is a general relationship between the mean square deviation from the mean (\(\bar{x}\)) and the mean square deviation from any other constant 'a'.

Let's start with the definition of \(\text{MSD}_a\):

\[ \text{MSD}_a = \frac{\sum f_i (x_i - a)^2}{\sum f_i} \]

We can rewrite \((x_i - a)\) as \((x_i - \bar{x} + \bar{x} - a)\). Let \(d = \bar{x} - a\). Then:

\[ (x_i - a)^2 = ((x_i - \bar{x}) + (\bar{x} - a))^2 = (x_i - \bar{x})^2 + 2(x_i - \bar{x})(\bar{x} - a) + (\bar{x} - a)^2 \]

Now substitute this back into the MSD formula:

\[ \text{MSD}_a = \frac{\sum f_i [(x_i - \bar{x})^2 + 2(x_i - \bar{x})(\bar{x} - a) + (\bar{x} - a)^2]}{\sum f_i} \]

We can split the summation:

\[ \text{MSD}_a = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} + \frac{\sum f_i [2(x_i - \bar{x})(\bar{x} - a)]}{\sum f_i} + \frac{\sum f_i (\bar{x} - a)^2}{\sum f_i} \]

Let's simplify each term:

1. The first term is the mean square deviation from the mean, which is the variance \(\sigma^2\). \[ \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} = \sigma^2 \]
2. The second term: \[ \frac{\sum f_i [2(x_i - \bar{x})(\bar{x} - a)]}{\sum f_i} = 2(\bar{x} - a) \frac{\sum f_i (x_i - \bar{x})}{\sum f_i} \] We know that the sum of deviations from the mean is always zero: \(\sum f_i (x_i - \bar{x}) = 0\). So, this term becomes \(2(\bar{x} - a) \times 0 = 0\).
3. The third term: \((\bar{x} - a)^2\) is a constant with respect to the summation variable \(i\). \[ \frac{\sum f_i (\bar{x} - a)^2}{\sum f_i} = \frac{(\bar{x} - a)^2 \sum f_i}{\sum f_i} = (\bar{x} - a)^2 \]

Combining these terms, we get the relationship:

\[ \text{MSD}_a = \sigma^2 + (\bar{x} - a)^2 \]

This formula states that the mean square deviation from any point 'a' is equal to the variance plus the square of the difference between the mean and 'a'.

Step-by-Step Calculation

We want to find the mean square deviation from the mode, which is our 'a'.

• Given Variance (\(\sigma^2\)) = 2
• Given Mean (\(\bar{x}\)) = 25
• Given Mode ('a') = 24

Using the derived formula:

\[ \text{MSD}_{\text{Mode}} = \sigma^2 + (\bar{x} - \text{Mode})^2 \]

Substitute the given values:

\[ \text{MSD}_{\text{Mode}} = 2 + (25 - 24)^2 \]

Calculate the difference and square it:

\[ \text{MSD}_{\text{Mode}} = 2 + (1)^2 \]

\[ \text{MSD}_{\text{Mode}} = 2 + 1 \]

\[ \text{MSD}_{\text{Mode}} = 3 \]

Thus, the mean square deviation from the mode is 3.

Revision Table: Key Statistical Measures

Additional Information: Properties of Mean Square Deviation

The mean square deviation is a measure of dispersion that squares the deviations from a reference point. This squaring makes all deviations positive and gives more weight to larger deviations.

• The mean square deviation is minimized when the reference point 'a' is the arithmetic mean (\(\bar{x}\)). This minimum value is the variance.
• The relationship \(\text{MSD}_a = \sigma^2 + (\bar{x} - a)^2\) shows that the MSD from any point 'a' is always greater than or equal to the variance, with equality holding only when \(a = \bar{x}\).
• Standard deviation (\(\sigma\)) is the square root of the variance, representing the typical deviation from the mean. While mode and mean are measures of central tendency, variance and mean square deviation are measures of dispersion.