Understanding Measures of Dispersion: Mean Deviation and Standard Deviation

In statistics, measures of dispersion help us understand how spread out the data points are in a dataset. Two common measures are Mean Deviation and Standard Deviation. The question asks about the relationship between these two measures for a discrete series where not all values are the same.

What is Mean Deviation?

Mean Deviation is the average of the absolute deviations of data points from a central value (like the mean, median, or mode). When calculated about the mean, it measures the average distance of each data point from the mean, ignoring the direction (positive or negative). For a discrete series $x_1, x_2, ..., x_n$ with frequencies $f_1, f_2, ..., f_n$, the mean $\bar{x}$ is calculated as $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$. The Mean Deviation about the mean (let's call it $x$ as per the question) is given by the formula:

\( x = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i} \)

This formula takes the absolute value of the difference between each data point and the mean, sums these absolute differences (weighted by frequency), and then divides by the total number of data points.

What is Standard Deviation?

Standard Deviation is the square root of the variance. Variance is the average of the squared deviations of data points from the mean. Like mean deviation, it measures the spread of data, but it gives more weight to larger deviations by squaring them. For the same discrete series, the variance (\(\sigma^2\)) is:

\( \sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} \)

The Standard Deviation (let's call it $y$ as per the question) is the square root of the variance:

\( y = \sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}} \)

Comparing Mean Deviation and Standard Deviation

Both measures tell us about the spread of the data. However, they use different methods to handle the deviations from the mean:

• Mean Deviation uses the absolute values of deviations (\(|x_i - \bar{x}|\)).
• Standard Deviation uses the squares of deviations (\((x_i - \bar{x})^2\)) and then takes the square root of the average squared deviation.

The process of squaring deviations gives more importance to larger deviations compared to simply taking their absolute values. For example, a deviation of 5 contributes \(|5|=5\) to the sum for mean deviation, but \(5^2=25\) to the sum for variance (and indirectly to standard deviation). A deviation of 2 contributes \(|2|=2\) and \(2^2=4\). The squared deviation (25) is much larger relative to its absolute value (5) than the squared deviation (4) is relative to its absolute value (2).

Mathematical Relationship Between x (Mean Deviation) and y (Standard Deviation)

There is a known mathematical relationship between the mean deviation about the mean and the standard deviation for any dataset. Generally, for any distribution, the Standard Deviation is always greater than or equal to the Mean Deviation about the mean. That is, \(y \ge x\).

This relationship can be shown using mathematical inequalities like Cauchy-Schwarz inequality or by considering the convexity of the square function. The inequality \(y \ge x\) holds true for any dataset.

Considering the Condition "All Values Are Not Same"

The question specifies a discrete series where "all values are not same". This condition is important.

• If all values in the series were the same (e.g., 5, 5, 5, 5), then the mean would be 5. The deviation (\(x_i - \bar{x}\)) for every data point would be 0. In this case, both Mean Deviation \(x\) and Standard Deviation \(y\) would be 0, and we would have \(x = y\).
• However, since "all values are not same", there must be at least one data point that is different from the mean. This means there will be some non-zero deviations (\(x_i - \bar{x} \neq 0\)).

When there are non-zero deviations, the process of squaring these deviations in the standard deviation calculation (\((x_i - \bar{x})^2\)) gives more weight to larger deviations than taking their absolute values (\(|x_i - \bar{x}|\)) does for the mean deviation calculation. As a result, the standard deviation (\(y\)) will be strictly greater than the mean deviation about the mean (\(x\)).

Thus, when not all values are the same, the relationship is \(y > x\). This is equivalent to \(x < y\).

Conclusion

For a discrete series where all values are not the same, the standard deviation (y) is always strictly greater than the mean deviation about the mean (x). Therefore, the correct relationship is \(x < y\).

Revision Table: Key Statistical Concepts

Additional Information on Dispersion Measures

Measures of dispersion are crucial in statistics because the mean or other measures of central tendency alone do not fully describe a dataset. Two datasets can have the same mean but vastly different levels of spread.

• Range: The simplest measure, calculated as the difference between the maximum and minimum values. It's easy to compute but only uses two data points.
• Quartile Deviation: Half the difference between the upper and lower quartiles. It measures the spread of the middle 50% of the data and is not affected by extreme values.
• Mean Deviation: Provides the average absolute distance from a central point. It is relatively easy to understand but the use of absolute values can make it less convenient for further mathematical analysis compared to standard deviation.
• Standard Deviation: The most widely used measure of dispersion. It is sensitive to every value in the dataset and is particularly important in inferential statistics and hypothesis testing. Its square, the variance, has additive properties that are useful in statistical theory. Standard deviation is also in the same unit as the original data, unlike variance.

The condition "not all values are same" ensures that there is actual dispersion in the data (the data points are not all clustered at a single value). When there is dispersion, standard deviation (\(y\)) will be greater than mean deviation (\(x\)) because squaring deviations amplifies the effect of larger differences from the mean.