Measures of Central Tendency

The Arithmetic Mean ($\bar{X}$) is the most widely used mathematical average, defined as the quotient obtained by dividing the sum of all observations by the total number of items:

Calculation Formulas across Continuous Series:

• 1. Direct Method: $$\bar{X} = \frac{\Sigma f m}{N}$$ Where $m$ is the class mid-value $(L_1 + L_2)/2$, $f$ is frequency, and $N = \Sigma f$.
• 2. Assumed Mean (Shortcut) Method: $$\bar{X} = A + \frac{\Sigma f d}{N}$$ Where $A$ is an assumed mean chosen from $m$, and $d = m - A$.
• 3. Step-Deviation Method (Simplest Computation): $$\bar{X} = A + \left(\frac{\Sigma f d'}{N}\right) \times c$$ Where $d' = \frac{m - A}{c}$, and $c$ is the common class width factor.

Key Mathematical Properties of Arithmetic Mean:

1. The algebraic sum of deviations of all values from their arithmetic mean is always zero: $$\Sigma (X - \bar{X}) = 0$$
2. The sum of squared deviations from the mean is a minimum: $$\Sigma (X - \bar{X})^2 < \Sigma (X - A)^2 \quad (\text{for any } A \ne \bar{X})$$
3. Combined Mean Formula: For two groups of sizes $N_1, N_2$ with means $\bar{X}_1, \bar{X}_2$: $$\bar{X}_{12} = \frac{N_1 \bar{X}_1 + N_2 \bar{X}_2}{N_1 + N_2}$$

The Median ($M$) is the positional middle-most value of a dataset arranged in ascending or descending order of magnitude, dividing the distribution into two equal halves (50% below, 50% above):

Calculation in Continuous Frequency Distributions:

1. Find the Median item position: $m_{\text{pos}} = \frac{N}{2}$.
2. Locate the Median Class from the Cumulative Frequency ($c.f.$) column.
3. Apply the interpolation formula: $$M = L_1 + \frac{\frac{N}{2} - c.f.}{f} \times i$$ Where $L_1$ is lower limit of median class, $c.f.$ is cumulative frequency of the preceding class, $f$ is frequency of the median class, and $i$ is class width.

Partition Values: Lower Quartile ($Q_1$) and Upper Quartile ($Q_3$):

• Lower Quartile ($Q_1$): Divides the lowest 25% of values: $$Q_1 = L_1 + \frac{\frac{N}{4} - c.f.}{f} \times i$$
• Upper Quartile ($Q_3$): Divides the lowest 75% of values (top 25%): $$Q_3 = L_1 + \frac{\frac{3N}{4} - c.f.}{f} \times i$$

The Mode ($Z$) is the value which occurs with the maximum frequency in a distribution (the most fashionable or typical value):

When Inspection Fails: The Grouping Method

If frequencies are irregular or the maximum frequency appears multiple times, use a 6-Column Grouping Table and an Analysis Table to identify the true modal class.

Calculation of Mode in Continuous Series:

Where:

• $L_1$ = Lower limit of the modal class
• $f_1$ = Frequency of the modal class
• $f_0$ = Frequency of the class preceding the modal class
• $f_2$ = Frequency of the class succeeding the modal class
• $i$ = Class interval width

The Empirical Relationship (Karl Pearson):

In a moderately skewed, asymmetric unimodal distribution, the Mean, Median, and Mode satisfy the empirical equation:

Comparative Suitability Matrix: