Learning Objectives

By the end of this chapter, you will be able to:

What is Central Tendency?

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. It represents the “typical” or “average” value.

mindmap
  root("(Measures of Central Tendency"))
    Arithmetic Mean
      Simple Mean
      Weighted Mean
    Median
      Middle Value
      Position Based
    Mode
      Most Frequent
      Can be Multiple

1. Arithmetic Mean

The arithmetic mean (commonly called “average”) is the sum of all values divided by the number of values.

Formula for Individual Data (Ungrouped)

\[\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n}\]

Where:

Step-by-Step Example 1: Simple Mean

Problem: Calculate the mean salary (in thousands) of 8 government officers: 45, 52, 48, 55, 50, 47, 53, 50

Solution:

Step 1: List all values

Step 2: Calculate the sum \(\sum x = 45 + 52 + 48 + 55 + 50 + 47 + 53 + 50 = 400\)

Step 3: Count observations \(n = 8\)

Step 4: Apply the formula \(\bar{x} = \frac{400}{8} = 50\)

Answer: The mean salary is 50 thousand (NPR 50,000)

Mean for Discrete Frequency Distribution

When data is presented with frequencies, use:

\[\bar{x} = \frac{\sum fx}{n} = \frac{\sum fx}{\sum f}\]

Where:

Step-by-Step Example 2: Discrete Data

Problem: Calculate the mean number of training programs attended by employees:

Training Programs (x) Number of Employees (f)
0 5
1 12
2 18
3 10
4 5

Solution:

Step 1: Create a calculation table

$x$ $f$ $fx$
0 5 0
1 12 12
2 18 36
3 10 30
4 5 20
Total $\sum f = 50$ $\sum fx = 98$

Step 2: Apply the formula

\[\bar{x} = \frac{\sum fx}{\sum f} = \frac{98}{50} = 1.96\]

Answer: The mean number of training programs is 1.96 ≈ 2 programs

Mean for Continuous (Grouped) Frequency Distribution

For grouped data with class intervals:

\[\bar{x} = \frac{\sum fm}{n}\]

Where $m$ = mid-point of each class = $\frac{\text{Lower limit + Upper limit}}{2}$

Step-by-Step Example 3: Grouped Data

Problem: Calculate the mean monthly income of government employees:

Monthly Income (NPR ‘000) Number of Employees
20-30 8
30-40 15
40-50 25
50-60 18
60-70 10
70-80 4

Solution:

Step 1: Find mid-points and create calculation table

Class $f$ Mid-point ($m$) $fm$
20-30 8 25 200
30-40 15 35 525
40-50 25 45 1125
50-60 18 55 990
60-70 10 65 650
70-80 4 75 300
Total 80   3790

Step 2: Calculate mid-points

Step 3: Apply the formula

\[\bar{x} = \frac{\sum fm}{\sum f} = \frac{3790}{80} = 47.375\]

Answer: The mean monthly income is NPR 47,375

2. Weighted Mean

The weighted mean is used when different values have different levels of importance (weights).

Formula

\[\bar{x}_w = \frac{\sum wx}{\sum w}\]

Where:

When to Use Weighted Mean

flowchart TD
    A{"Are all items<br/>equally important?"}
    A -->|Yes| B[Use Simple Mean]
    A -->|No| C[Use Weighted Mean]

    C --> D["Examples:"]
    D --> D1["GPA calculation"]
    D --> D2["Price indices"]
    D --> D3["Composite scores"]
    D --> D4["Performance ratings"]

Step-by-Step Example 4: Weighted Mean

Problem: Calculate the weighted average performance score for a department:

Performance Area Score (x) Weight (w)
Work Quality 85 3
Punctuality 90 2
Teamwork 75 2
Initiative 80 1
Communication 78 2

Solution:

Step 1: Create calculation table

Area $x$ $w$ $wx$
Work Quality 85 3 255
Punctuality 90 2 180
Teamwork 75 2 150
Initiative 80 1 80
Communication 78 2 156
Total   $\sum w = 10$ $\sum wx = 821$

Step 2: Apply the formula

\[\bar{x}_w = \frac{\sum wx}{\sum w} = \frac{821}{10} = 82.1\]

Answer: The weighted average performance score is 82.1

Note: Simple mean would be $\frac{85+90+75+80+78}{5} = 81.6$, which doesn’t account for the higher importance of Work Quality.

3. Median

The median is the middle value when data is arranged in order. It divides the distribution into two equal halves.

Key Properties

Formula for Ungrouped Data

Step 1: Arrange data in ascending order

Step 2: Find position of median

\[\text{Median Position} = \frac{n+1}{2}\]

Step 3:

Step-by-Step Example 5: Median (Odd n)

Problem: Find the median of these efficiency scores: 72, 85, 68, 91, 76, 80, 74

Solution:

Step 1: Arrange in ascending order \(68, 72, 74, 76, 80, 85, 91\)

Step 2: Find position ($n = 7$, odd) \(\text{Position} = \frac{7+1}{2} = 4^{th} \text{ position}\)

Step 3: Identify the median Counting: 68(1), 72(2), 74(3), 76(4), 80(5), 85(6), 91(7)

Answer: Median = 76

Step-by-Step Example 6: Median (Even n)

Problem: Find the median: 45, 52, 38, 61, 55, 49, 58, 42

Solution:

Step 1: Arrange in ascending order \(38, 42, 45, 49, 52, 55, 58, 61\)

Step 2: Find position ($n = 8$, even) \(\text{Position} = \frac{8+1}{2} = 4.5^{th} \text{ position}\)

Step 3: Average of 4th and 5th values

\[\text{Median} = \frac{49 + 52}{2} = 50.5\]

Answer: Median = 50.5

Median for Grouped Data

\[\text{Median} = L + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h\]

Where:

Step-by-Step Example 7: Median for Grouped Data

Problem: Find median from this salary distribution:

Salary (NPR ‘000) Frequency
20-30 5
30-40 10
40-50 20
50-60 15
60-70 10

Solution:

Step 1: Calculate cumulative frequency

Class $f$ Cumulative Frequency ($cf$)
20-30 5 5
30-40 10 15
40-50 20 35
50-60 15 50
60-70 10 60
Total 60  

Step 2: Find $\frac{n}{2}$ \(\frac{n}{2} = \frac{60}{2} = 30\)

Step 3: Identify median class

Step 4: Identify values

Step 5: Apply the formula

\[\text{Median} = 40 + \left(\frac{30 - 15}{20}\right) \times 10\] \[= 40 + \left(\frac{15}{20}\right) \times 10\] \[= 40 + 0.75 \times 10 = 40 + 7.5 = 47.5\]

Answer: Median salary = NPR 47,500

4. Mode

The mode is the value that occurs most frequently in a dataset.

Types of Modal Distributions

flowchart LR
    A[Mode Types] --> B["Unimodal<br/>One mode"]
    A --> C["Bimodal<br/>Two modes"]
    A --> D["Multimodal<br/>More than two"]
    A --> E["No Mode<br/>All values equal frequency"]

Step-by-Step Example 8: Mode (Ungrouped Data)

Problem: Find the mode: 5, 7, 8, 5, 9, 5, 10, 7, 5, 8

Solution:

Step 1: Count frequency of each value

Value Frequency
5 4
7 2
8 2
9 1
10 1

Step 2: Identify highest frequency

Answer: Mode = 5

Mode for Grouped Data

\[\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h\]

Where:

Step-by-Step Example 9: Mode for Grouped Data

Problem: Find the mode from this age distribution of employees:

Age Group Frequency
20-25 5
25-30 12
30-35 28
35-40 20
40-45 10
45-50 5

Solution:

Step 1: Identify modal class

Step 2: Identify values

Step 3: Apply the formula

\[\text{Mode} = 30 + \left(\frac{28 - 12}{2(28) - 12 - 20}\right) \times 5\] \[= 30 + \left(\frac{16}{56 - 32}\right) \times 5\] \[= 30 + \left(\frac{16}{24}\right) \times 5\] \[= 30 + 0.667 \times 5 = 30 + 3.33 = 33.33\]

Answer: Mode = 33.33 years

Relationship Between Mean, Median, and Mode

In a Symmetrical Distribution

\[\text{Mean} = \text{Median} = \text{Mode}\]

In a Skewed Distribution

flowchart LR
    subgraph "Positively Skewed (Right)"
        A1[Mode] --> A2[Median] --> A3[Mean]
    end

    subgraph "Negatively Skewed (Left)"
        B1[Mean] --> B2[Median] --> B3[Mode]
    end

Empirical Relationship (for moderately skewed distributions):

\[\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}\]

Or equivalently:

\[\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})\]

Choosing the Right Measure

Situation Best Measure Reason
Symmetrical distribution Mean Uses all data
Skewed distribution Median Not affected by outliers
Categorical/nominal data Mode Only applicable measure
Presence of extreme values Median Robust to outliers
Need for further calculations Mean Algebraic properties
Open-ended classes Median or Mode Mean cannot be calculated precisely

Summary of Formulas

Mean Formulas

Data Type Formula
Individual $\bar{x} = \frac{\sum x}{n}$
Discrete $\bar{x} = \frac{\sum fx}{\sum f}$
Grouped $\bar{x} = \frac{\sum fm}{\sum f}$
Weighted $\bar{x}_w = \frac{\sum wx}{\sum w}$

Median Formula (Grouped)

\[\text{Median} = L + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h\]

Mode Formula (Grouped)

\[\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h\]

Practice Problems

Problem 1

Calculate the mean, median, and mode for: 15, 18, 22, 18, 25, 18, 30, 28, 22

Problem 2

The following table shows weekly wages of workers. Calculate mean and median:

Wages (NPR) Workers
100-200 8
200-300 14
300-400 23
400-500 20
500-600 10

Problem 3

A civil servant’s performance is evaluated on three criteria with different weights. Calculate the weighted mean:

Criterion Score Weight
Technical Skills 88 40%
Leadership 75 35%
Communication 82 25%

Next Chapter

In the next chapter, we will study Measures of Dispersion - understanding how spread out our data is using variance, standard deviation, and coefficient of variation.