Statistics: Mean, Median, and Mode
Understanding the measures of central tendency for ungrouped data
Introduction to Ungrouped Data
Ungrouped data is raw data that has not been organized into groups or classes. It is the simplest form of data directly collected from observations or experiments.
Key Vocabulary:
- Data: A collection of facts or information
- Ungrouped Data: Raw data that hasn't been organized into groups
- Mean: The average of a set of numbers
- Median: The middle value in an ordered set of numbers
- Mode: The value that appears most frequently in a data set
- Central Tendency: A measure that represents the center of a data set
Mean: The Average Value
The mean is the most commonly used measure of central tendency. It is calculated by adding all values and dividing by the number of values.
Example:
Find the mean of: 4, 1, 3, 5, 3, 2, 3
Step 1: Add all values → 4 + 1 + 3 + 5 + 3 + 2 + 3 = 21
Step 2: Divide by number of values → 21 ÷ 7 = 3
Mean = 3
Median: The Middle Value
The median is the middle value when data is arranged in order. For an even number of values, it's the average of the two middle numbers.
Example:
Find the median of: 4, 1, 3, 5, 3, 2, 3
Step 1: Arrange in order → 1, 2, 3, 3, 3, 4, 5
Step 2: Find the middle value → The 4th value is 3
Median = 3
Mode: The Most Frequent Value
The mode is the value that appears most frequently in a data set. A data set may have one mode, more than one mode, or no mode at all.
Example:
Find the mode of: 4, 1, 3, 5, 3, 2, 3
Step 1: Count frequency of each value → 1(1), 2(1), 3(3), 4(1), 5(1)
Step 2: Identify the value with highest frequency → 3 appears 3 times
Mode = 3
Hook Activity: Let's Get Started!
Imagine you're the coach of a basketball team. Here are the points scored by your players in the last game:
12, 15, 22, 8, 15, 10, 18, 15, 20, 12
Quick Questions:
1. What's the average score per player?
2. What's the middle score if you arrange them in order?
3. Which score appears most frequently?
These questions represent the three measures of central tendency we'll learn about!
Explicit Teaching (I Do)
Let me demonstrate how to calculate mean, median, and mode step by step with a complete example.
Example Data: 8, 3, 5, 7, 6, 5, 4, 5, 2
Mean Calculation:
Sum = 8 + 3 + 5 + 7 + 6 + 5 + 4 + 5 + 2 = 45
Number of values = 9
Mean = 45 ÷ 9 = 5
Median Calculation:
Ordered data: 2, 3, 4, 5, 5, 5, 6, 7, 8
Middle value (5th term) = 5
Mode Calculation:
Value frequencies: 2(1), 3(1), 4(1), 5(3), 6(1), 7(1), 8(1)
Most frequent value = 5 (appears 3 times)
Results: Mean = 5, Median = 5, Mode = 5
Group Work (We Do)
Let's work together on these problems. Calculate the mean, median, and mode for each data set.
Problem 1: Apple Picking
The number of apples picked by 7 workers: 20, 25, 30, 25, 35, 20, 40
Calculate:
Mean:
Median:
Mode:
Problem 2: Test Scores
Marks obtained by 12 students: 78, 85, 92, 88, 76, 85, 91, 85, 80, 79, 88, 86
Calculate:
Mean:
Median:
Mode:
Independent Work (You Do)
Now try these problems on your own. Complete the table with the correct mean, median, and mode.
| Data Set | Mean | Median | Mode |
|---|---|---|---|
| 1, 18, 12, 15, 9, 6, 17, 13 | |||
| 2, 20, 22, 24, 24, 28, 32 | |||
| 42, 24, 36, 50, 42, 48, 36, 50 |
Assessment
Test your understanding with these challenge questions.
Problem:
The table shows scores of 40 pupils in a Mathematics test:
| Scores | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| Number of pupils | 4 | 6 | 7 | 9 | 6 | 5 | 3 |
Calculate the mean, median, and mode of the scores.
Mean:
Median:
Mode:
Homework
Complete the following exercises to reinforce your learning:
Exercise 1:
The daily temperatures (in °C) recorded in a week: 25, 27, 26, 24, 28, 27, 26
Find the mean, median, and mode.
Exercise 2:
The number of cars sold by a dealership in 8 consecutive months: 12, 15, 14, 18, 12, 20, 15, 17
Determine the mean, median, and mode.
Exercise 3:
Create your own data set with at least 10 values where:
- The mean is greater than the median
- The mode is different from both mean and median
- Explain why this occurs
Key Points to Remember
Mean
Use when data has no extreme values and you need the mathematical average
Median
Best when data has extreme values (outliers) that could skew the mean
Mode
Use when you need to know the most common or popular value