Measures of Central Tendency: Mean, Median, Mode, GM, and HM

Introduction to Central Tendency

In the hospitality industry, managers deal with vast amounts of data—daily room occupancy, food costs, guest satisfaction scores, and employee payroll. Analyzing every single data point is impossible. Therefore, we use Measures of Central Tendency, also known as "averages."

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.

flowchart TD
    A["Measures of Central Tendency"] --> B["Mathematical Averages"]
    A --> C["Positional Averages"]
    B --> B1["Arithmetic Mean"]
    B --> B2["Geometric Mean"]
    B --> B3["Harmonic Mean"]
    C --> C1["Median"]
    C --> C2["Mode"]

A hotel front desk where guest data is collected for statistical analysis. (Image: Basile Morin, CC BY-SA 4.0, via Wikimedia Commons)

1. Arithmetic Mean (AM)

The Arithmetic Mean is the sum of all observations divided by the total number of observations. It is the most widely used measure of central tendency.

Types of Mean

1. Simple Arithmetic Mean: Every observation is given equal importance.
2. Weighted Arithmetic Mean: Different observations are given different weights based on their importance.

Calculation Methods

A. Individual Series (Raw Data): $$\bar{x}=\frac{\sum x}{n}$$ Where $\sum x$ is the sum of values and $n$ is the number of items.

B. Discrete Series (Frequency Data): $$\bar{x}=\frac{\sum fx}{\sum f}$$ Where $f$ is the frequency of each value $x$.

C. Continuous Series (Grouped Data): $$\bar{x}=\frac{\sum fm}{\sum f}$$ Where $m$ is the midpoint of each class interval: $m=\frac{\text{Lower Limit}+\text{Upper Limit}}{2}$.

Worked Example: Average Daily Revenue

A boutique hotel in Pokhara records the daily room revenue (in NPR) for 5 days: 12,000, 15,000, 11,000, 18,000, and 14,000.

Step 1: Sum the values $$\sum x=12,000+15,000+11,000+18,000+14,000=70,000$$

Step 2: Divide by number of days (n=5) $$\bar{x}=\frac{70,000}{5}=14,000$$ The average daily revenue is NPR 14,000.

Weighted Arithmetic Mean

Used when some data points contribute more to the final average than others. $${\bar{x}}_w=\frac{\sum wx}{\sum w}$$

Example: A student's final grade is based on Assignments (20%), Mid-term (30%), and Final Exam (50%). If they score 80, 70, and 60 respectively: $${\bar{x}}_w=\frac{(0.2\times 80)+(0.3\times 70)+(0.5\times 60)}{0.2+0.3+0.5}=\frac{16+21+30}{1}=67$$

2. Median

The Median is the middle-most value of a data set when the observations are arranged in ascending or descending order. It divides the data into two equal halves.

Calculation Methods

A. Individual Series:

1. Arrange data in order.
2. If $n$ is odd: Median = Value of $(\frac{n+1}{2}{)}^{th}$ term.
3. If $n$ is even: Median = Average of $(\frac{n}{2}{)}^{th}$ and $(\frac{n}{2}+1{)}^{th}$ terms.

B. Continuous Series: $$\text{Median}=L+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h$$ Where:

• $L$: Lower limit of the median class.
• $n$: Total frequency.
• $cf$: Cumulative frequency of the class preceding the median class.
• $f$: Frequency of the median class.
• $h$: Class width.

Worked Example: Guest Age

A hotel collects the ages of 7 guests: 22, 45, 30, 22, 50, 65, 38.

Step 1: Arrange in ascending order 22, 22, 30, 38, 45, 50, 65

Step 2: Find the position Since $n=7$ (odd), position = $\frac{7+1}{2}=4^{th}$ term.

Step 3: Identify value The 4th term is 38. Median Age = 38 years.

3. Mode

The Mode is the value that appears most frequently in a data set. A distribution can be unimodal (one mode), bimodal (two modes), or multimodal.

Calculation Methods

A. Individual/Discrete Series: The value with the highest frequency is the mode.

B. Continuous Series: $$\text{Mode}=L+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h$$ Where:

• $L$: 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.
• $h$: Class width.

Worked Example: Popular Dish

A restaurant tracks the number of orders for 5 different dishes over a weekend:

• Momo: 120
• Chowmein: 80
• Thukpa: 40
• Pizza: 120
• Burger: 60

Since Momo and Pizza both have the highest frequency (120), this data is Bimodal. Modes = Momo and Pizza.

4. Geometric Mean (GM) and Harmonic Mean (HM)

Geometric Mean (GM)

The GM is the $n^{th}$ root of the product of $n$ observations. It is primarily used for calculating average growth rates, percentages, and ratios. $$\text{GM}=\sqrt[n]{x_1\cdot x_2\cdot \cdots \cdot x_n}$$

Example: If a hotel's occupancy grew by 5% in year one and 15% in year two, the average growth is found using GM.

Harmonic Mean (HM)

The HM is the reciprocal of the arithmetic mean of the reciprocals of the observations. It is used for calculating average speeds or average rates. $$\text{HM}=\frac{n}{\sum \frac{1}{x}}$$

Example: If a delivery bike travels from a hotel to a customer at 20 km/h and returns at 30 km/h, the average speed for the round trip is the HM. $$\text{HM}=\frac{2}{\frac{1}{20}+\frac{1}{30}}=\frac{2}{\frac{3+2}{60}}=\frac{2\times 60}{5}=24\text{ km/h}$$

A tool used to calculate roots for GM and reciprocals for HM. (Image: Matti Blume, CC BY-SA 2.0, via Wikimedia Commons)

Comparison of Mean, Median, and Mode

Relationship between Mean, Median, and Mode

For a moderately skewed distribution, the empirical relationship is: $$\text{Mode}=3\text{Median}-2\text{Mean}$$

Example: If the Mean of a hotel's guest rating is 4.2 and the Median is 4.4, find the Mode. $$\text{Mode}=3(4.4)-2(4.2)=13.2-8.4=4.8$$

In the real world

1. Pathao/InDrive (Mean): These apps calculate the "Average Estimated Time of Arrival (ETA)." They use the Arithmetic Mean of recent trips on the same route to give you a realistic waiting time.
2. Daraz (Median): When Daraz shows "Average Delivery Time," they often use the Median instead of the Mean. This is because a few extremely delayed packages (outliers) would make the Mean look very high, which would discourage customers. The Median provides a more "typical" experience.
3. Hotel Menu Planning (Mode): A hotel chef looks at the Mode of the "Daily Special" orders. If "Chicken Curry" is the mode (most ordered), the chef increases the procurement of chicken for the next week to avoid stock-outs.
4. NEPSE (Geometric Mean): Investors in the Nepal Stock Exchange use the Geometric Mean to calculate the Compound Annual Growth Rate (CAGR) of their portfolio over several years, as stock returns are multiplicative, not additive.

Exam tip

• Identify the Series: Before starting any calculation, check if the data is Individual (raw list), Discrete (value + frequency), or Continuous (class intervals). Using the wrong formula is the most common cause of marks loss.
• Median Class: In continuous series, always calculate the Cumulative Frequency (cf) first to find the Median Class.
• Modal Class: The modal class is simply the one with the highest frequency.
• Outliers: If the question asks "Which measure is best for this data?" and you see a value that is much larger or smaller than the rest (e.g., salaries: 20k, 22k, 21k, 500k), always choose the Median.
• Formula Precision: Write the general formula first before substituting values. TU examiners give step-marks for the correct formula even if the final calculation is wrong.