Measures of Dispersion – Range, Variance, Standard Deviation, IQR

Measures of Dispersion

Dispersion measures describe how data values spread around a central point. In hotel management, they help assess variability in room rates, occupancy, revenue, and guest satisfaction scores.

1. Range

The range is the simplest dispersion metric:

$$\text{Range}=\max (X)-\min (X)$$

It uses only the extreme values, so it is easy to compute but can be misleading if outliers exist.

Worked Example – Room Rates
Suppose a hotel records the following nightly rates (in NPR) for a week:

Range $=10,000-7,500=2,500$ NPR.

Figure: Bar chart of weekly room rates

The range tells us the spread is 2,500 NPR, but it ignores the distribution of the middle values.

2. Variance and Standard Deviation

Variance measures average squared deviation from the mean. Standard deviation is its square root, giving a dispersion metric in the same units as the data.

$$\bar{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$$ $$s^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\bar{x}{)}^2$$ $$s=\sqrt{s^2}$$

The denominator $n-1$ (Bessel’s correction) provides an unbiased estimate for a sample.

Worked Example – Same Room Rates

1. Compute mean:

$$\bar{x}=\frac{8,000+7,500+9,200+8,500+10,000+7,800+9,500}{7}=8,500\text{ NPR}$$

2. Compute squared deviations:

Sum of squared deviations $=5,480,000$.

3. Variance:

$$s^2=\frac{5,480,000}{7-1}=\frac{5,480,000}{6}=913,333.33{\text{ NPR}}^2$$

4. Standard deviation:

$$s=\sqrt{913,333.33}\approx 955.5\text{ NPR}$$

Figure: Histogram of room rates with mean and standard deviation

The standard deviation of ~956 NPR indicates typical deviation from the mean rate.

3. Inter‑Quartile Range (IQR)

IQR captures the spread of the middle 50 % of data, making it robust to outliers.

$$\text{IQR}=Q_3-Q_1$$

Where $Q_1$ is the 25th percentile and $Q_3$ is the 75th percentile.

Worked Example – Guest Satisfaction Scores
Assume a hotel collects guest satisfaction scores (1–5) from 15 guests:

$$1,2,2,3,3,3,4,4,4,4,5,5,5,5,5$$

1. Order the data (already ordered).
2. $Q_1$ is the 4th value (since $n=15$, $Q_1$ at position $0.25(15+1)=4$): $Q_1=3$.
3. $Q_3$ is the 12th value: $Q_3=5$.
4. IQR $=5-3=2$.

Figure: Box plot of guest satisfaction scores

The IQR of 2 shows that the middle 50 % of scores lie within a range of 2 points, unaffected by the two extreme 1s.

4. Comparison of Dispersion Measures

Mermaid diagram: Decision flow for choosing a dispersion metric

flowchart TD  
  A["Data size"] --> B{"Large sample?"}  
  B -->|"Yes"| C["Use SD for inference"]  
  B -->|"No"| D["Check for outliers"]  
  D -->|"Outliers present"| E["Use IQR"]  
  D -->|"No outliers"| F["Use Range or SD"]

5. Advantages and Disadvantages

6. Applications in Hotel Management

7. In the real world

1. eSewa – Transaction Amounts
   eSewa monitors daily transaction amounts. The standard deviation of transaction values is used to set fraud detection thresholds; a sudden spike beyond 3 SD indicates potential fraud.

2. Daraz – Delivery Time
   Daraz calculates the IQR of delivery times for each city. A narrow IQR signals reliable delivery, while a wide IQR prompts investigation into logistics bottlenecks.

3. Ncell – Network Latency
   Ncell measures variance of packet latency across its network. High variance triggers network optimization, ensuring consistent user experience.

8. Exam tip

• Understand definitions: Know the formulas for range, variance, standard deviation, and IQR.
• Practice calculations: Work through at least two full examples (one with mean/SD, one with IQR).
• Interpret results: Be ready to explain what a high or low dispersion value implies in a hotel context.
• Compare metrics: When asked to choose a dispersion measure, justify your choice based on data characteristics (size, outliers).
• Use tables: Summarize key differences in a concise table; this often earns partial marks.

Front desk where reservations are taken (Image: Architect: George B. Post & Son, Public domain, via Wikimedia Commons) Standard guest room used for data collection (Image: Jumpy542, CC0, via Wikimedia Commons)