Mode: Calculation, Grouping Method & Empirical Relationship

1. What Is the Mode?

The word Mode is derived from the French phrase La Mode, meaning “fashion” or “the prevailing style”.

In statistics, the Mode (denoted by Z or Mo) is defined as the value of the variable which occurs most frequently in a distribution — that is, the point of maximum frequency density.

Real-World Business Intuition:

• If an Indore mobile shop sells 100 phones in a week: 10 iPhones, 15 Samsungs, and 75 OnePlus models, the modal phone is OnePlus!

2. Calculation of Mode Across Data Series

A. Individual & Discrete Series by Inspection

• Simply identify the observation that repeats the maximum number of times.
• Example: Data: 12, 18, 24, 18, 30, 18, 40 → Mode = 18 (repeats 3 times).
• Caution: If maximum frequencies are nearly identical (e.g., frequencies are 45, 46, 44), simple inspection fails and the Grouping Method must be used!

B. Continuous Series Formula

When the modal class is identified, the exact Mode is determined using the interpolation formula:

Solved Numerical Problem:

Find the Mode from the following sales distribution:

Solution:

1. Identify Modal Class: The highest frequency is 25, which lies in the class 30 - 40.
2. List Variables:
   • Lower limit of modal class (L1) = 30
   • Frequency of modal class (f1) = 25
   • Frequency of preceding class (f0) = 12
   • Frequency of succeeding class (f2) = 14
   • Class interval width (i) = 40 - 30 = 10
3. Calculate:
   Z = 30 + [ (25 - 12) ÷ (2(25) - 12 - 14) ] × 10
Z = 30 + [ 13 ÷ (50 - 26) ] × 10
Z = 30 + [ 13 ÷ 24 ] × 10
Z = 30 + [ 0.5417 × 10 ] = 30 + 5.42 = ₹35.42 (in thousands) = ₹35,420

3. When Inspection Fails: The Grouping & Analysis Method

If two classes have identical top frequencies, or if frequencies rise and fall irregularly, a Grouping Table (6 columns) and an Analysis Table are constructed:

1. Column 1: Original frequencies (mark highest).
2. Column 2: Sum of frequencies in pairs (1+2, 3+4, 5+6…).
3. Column 3: Sum of frequencies in pairs starting from 2nd item (2+3, 4+5…).
4. Column 4: Sum of frequencies in threes (1+2+3, 4+5+6…).
5. Column 5: Sum of frequencies in threes starting from 2nd item (2+3+4, 5+6+7…).
6. Column 6: Sum of frequencies in threes starting from 3rd item (3+4+5, 6+7+8…).

In the Analysis Table, tick marks are allocated to the classes contributing to the highest sum in each column. The class with the maximum tally marks is declared the true Modal Class!

4. The Empirical Relationship Between Mean, Median & Mode

In a perfectly symmetrical (bell-shaped) distribution, all three central measures coincide at the exact center:
Mean = Median = Mode

However, in real-world business data, distributions are usually moderately asymmetrical (skewed). Renowned statistician Karl Pearson proved that the distance between the Mean and the Mode is approximately three times the distance between the Mean and the Median:

Solved Exam Example:

In a factory, the Mean wage is ₹450 and the Median wage is ₹470. Find the Mode.

• Mode = 3(Median) - 2(Mean)
• Mode = 3(470) - 2(450)
• Mode = 1410 - 900 = ₹510

5. Merits, Demerits & Real-Life Business Utility of Mode

Merits:

• Easy to understand; represents the most common practical reality.
• Completely unaffected by extreme high or low outlier values.
• Can be determined graphically from a Histogram.

Demerits:

• Ill-defined in bimodal (two modes) or multimodal datasets.
• Not based on all items in the series.
• Incapable of algebraic manipulation (cannot calculate combined mode).

Dominant Business Applications:

• Footwear & Apparel Manufacturing: Bata, Nike, and Zara determine production volumes of shirt collars and shoe sizes based on the Mode.
• Public Transport Planning: City buses schedule peak fleet frequency according to modal commuter rush hours.