Introduction

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation, and presentation. In this chapter, we focus on methods to analyze grouped data using measures of central tendency: Mean, Median, and Mode.

1. Key Terms and Definitions

• Data: Information collected in numerical form.
• Ungrouped Data: Raw data not organized into groups.
• Grouped Data: Data organized into intervals (classes) with corresponding frequencies.
• Class Interval: A group into which data is divided.
• Class Limits: The smallest and largest data values in a class.
• Class Width: Difference between upper and lower class limits. Class Width = Upper Class Limit − Lower Class Limit
• Class Mark: Average of the class limits. Class Mark (xi) = (Lower Class Limit + Upper Class Limit) / 2
• Frequency (fi): Number of observations in a class.
• Cumulative Frequency: Running total of frequencies.

2. Measures of Central Tendency

2.1 Mean (Average)

The mean is the sum of all data values divided by the number of values.

a) Direct Method

b) Assumed Mean Method

c) Step-Deviation Method

2.2 Median

The median is the value that divides the data into two equal parts.

2.3 Mode

The mode is the value that appears most frequently in a data set.

3. Empirical Relationship

4. Graphical Representation: Ogives

• Less Than Ogive: Uses upper class boundaries and cumulative frequencies.
• More Than Ogive: Uses lower class boundaries and cumulative frequencies.

Median (Graphical): The x-coordinate of the point of intersection of the two ogives gives the median.

5. Practice Example

Problem: Calculate the mean, median, and mode for the following data:

Class Interval	Frequency
0–10	5
10–20	8
20–30	15
30–40	20
40–50	12

Step-by-step:

• Class Marks: 5, 15, 25, 35, 45
• ∑fi xi = 1760, ∑fi = 60 ⇒ Mean = 1760 / 60 = 29.33
• Cumulative frequency before median class (30–40) = 28, N/2 = 30 ⇒ Median = 30 + [(30−28)/20] × 10 = 31
• Modal class = 30–40 ⇒ Mode = 30 + [(20−15)/(2×20−15−12)] × 10 ≈ 31.67