Measures of Central Tendency

Master mean, median, and mode for ungrouped and grouped data

CAPS Grade 10 Mathematics

This document explores the Measures of Central Tendency as outlined in the Grade 10 CAPS curriculum. These measures are essential for understanding the "center" or typical value of a dataset.

Understanding the Mean (Average)

The mean is the average value - the point where the data balances.

Understanding the Median

The median is the middle value - it splits the data into two equal halves.

Understanding the Mode

The mode is the most common value - the highest bar in the frequency graph.

Learning Outcomes

Calculate the mean for ungrouped and grouped data
Find the median for ungrouped and grouped data
Determine the mode for ungrouped data and modal class for grouped data
Understand the differences between mean, median, and mode
Apply measures of central tendency to real-world datasets

1. Mean (x̄)

The mean, often referred to as the "average," is a fundamental measure of central tendency.

Ungrouped Data

x̄ = (Σxᵢ) / n

Sum all values, divide by number of values.

Data: 3, 5, 7, 9, 11

x̄ = (3+5+7+9+11)/5 = 35/5 = 7

Grouped Data

x̄ = Σ(fᵢ·xᵢ) / Σfᵢ

Use class midpoints (xᵢ) and frequencies (fᵢ).

Grouped Mean Example

Data

Interval	Frequency
1-3	2
4-6	3
7-9	5

Solution

Midpoints: 2, 5, 8

x̄ = (2×2 + 3×5 + 5×8)/(2+3+5) = (4+15+40)/10 = 59/10 = 5.9

Quiz 1 - Mean

Find the mean of: 4, 8, 12, 16, 20

A) 10

B) 12

C) 14

D) 16

2. Median (M)

The median is the middle value of a dataset when arranged in ascending order.

Position Formula

Position = (n + 1)/2

Ungrouped Example

Data: 3, 5, 7, 9, 11

n = 5, position = 3 → Median = 7

Even Number of Values

Data: 3, 5, 7, 9

n = 4, position between 2nd and 3rd

Median = (5+7)/2 = 6

Grouped Median

Data

Interval	Frequency	Cumulative Freq
1-3	2	2
4-6	3	5
7-9	5	10

Solution

n = 10, n/2 = 5

Median class: 7-9 (cumulative >5)

Quiz 2 - Median

Find the median of: 8, 12, 15, 19, 22, 25

A) 15

B) 17

C) 19

D) 22

3. Mode (Mo)

The mode is the value (or class) that appears most frequently.

Ungrouped Example

Data: 3, 5, 7, 5, 11

Mode = 5

Dataset can be bimodal or have no mode

Grouped (Modal Class)

Modal class = interval with highest frequency

From earlier table: modal class = 7-9 (frequency 5)

Quiz 3 - Mode

Find the mode: 2, 4, 6, 4, 8, 4, 10

A) 2

B) 4

C) 6

D) 8

Comparison of Measures

Measure	What it tells us	Best used when	Affected by outliers?
Mean	Average value	Symmetric data	Yes
Median	Middle value	Skewed data	No
Mode	Most common value	Categorical data	No

Practice & Assess

Test your knowledge with these interactive games.

Match - Measure to Formula

Mean (ungrouped)

x̄ = Σxᵢ/n

Median position

(n+1)/2

Mean (grouped)

x̄ = Σfᵢxᵢ/Σfᵢ

Mode

Most frequent

Fill - Mean Formula

x̄ = Σxᵢ / ___

Practice Questions

Find mean: 10, 20, 30, 40, 50

Answer: 30

Find median: 12, 15, 18, 21, 24, 27

Answer: 19.5

Find mode: 5, 8, 5, 9, 5, 10, 8

Answer: 5

Summary of Key Concepts

Mean: Average = sum ÷ n (ungrouped); Σfᵢxᵢ/Σfᵢ (grouped)

Median: Middle value; position = (n+1)/2 for ungrouped; median class for grouped

Mode: Most frequent value; modal class for grouped data

Key Terms

Mean Median Mode Average Midpoint Frequency Ungrouped data Grouped data Cumulative frequency Modal class Median class Central tendency

Data Handling Statistics