Measures of Central Tendency (Grade 12 NSC Matric Mathematical Literacy): Revision Notes
Measures of Central Tendency
What are measures of central tendency?
A measure of central tendency is a single number that represents the typical or central value of a data set. These measures help us summarise large amounts of data using just one value, making it easier to understand and compare different data sets.
There are three main measures of central tendency:
- Mean (average)
- Median (middle value)
- Mode (most frequent value)
Each measure gives us different information about the same data set, and understanding when to use each one is crucial for data analysis.
The mean
The mean is the most commonly used measure of central tendency. It is calculated by adding all values in the data set together and dividing by the total number of values.
Formula:
Key points about the mean:
- Also known as the average
- Can be easily influenced by very high or very low values (outliers)
- Useful when data is evenly distributed
- Always include all values in your calculation
The median
The median is the middle value when all data values are arranged in ascending order (from smallest to largest).
Steps to find the median:
- Arrange all values from smallest to largest
- If there is an odd number of values: the median is the middle value
- If there is an even number of values: add the two middle values and divide by 2
Key points about the median:
- Not affected by extreme values (outliers)
- Good measure for skewed data
- Always arrange data in order first
The mode
The mode is the value that appears most frequently in a data set.
Key points about the mode:
- No calculation needed - just find the most frequent value
- A data set can have no mode if all values appear equally
- A data set can have more than one mode if multiple values appear most frequently
- Useful for categorical data
Worked example 1: School data
Worked Example: School Social Grant Data
A school principal collected data on the number of learners receiving social grants in each class. The data arranged in ascending order is:
0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7
Solutions:
- Count: 21 values in total
- Mode: 3 (appears 4 times, more than any other value)
- Median: 3 (the middle value in position 11)
- Mean:
Worked example 2: Comparing salaries
Worked Example: Call Centre Salary Comparison
Two call centres were compared using their monthly salary data:
Greytown salaries (in rand): 4200, 4320, 4500, 4650, 4650, 4650, 5500, 5650, 7250
Johannesburg salaries (in rand): 5500, 5525, 5980, 6250, 6250, 6250, 6300, 7800, 8200, 8900
Solutions for Greytown:
- Mean:
- Median: Since there are 9 values, the median is the 5th value = R4650
- Mode: R4650 (appears 3 times)
Solutions for Johannesburg:
- Median: Since there are 10 values, take the average of the 5th and 6th values =
Combined data median: When all 19 salary values are arranged in order, the median is R5650.
Worked example 3: Student performance comparison
Worked Example: Student Performance Analysis
Two internship students were compared across seven subjects:
| Subject | Student A | Student B |
|---|---|---|
| Mathematics | 95 | 95 |
| Physical Science | 93 | 93 |
| Life Sciences | 69 | 72 |
| Life Orientation | 87 | 87 |
| English | 90 | 90 |
| Home Language | 92 | 89 |
| Geography | 90 | 90 |
Calculations for both students:
- Mean: Both students = 88
- Median: Both students = 90
- Mode: Both students = 90
In this case, all three measures of central tendency are identical for both students. When this happens, other measures (like measures of spread) are needed to compare performance.
When to use each measure
Understanding when to apply each measure is crucial for accurate data analysis:
Use the mean when:
- Data is evenly distributed
- You want to include all values in your analysis
- Performing further calculations
Use the median when:
- Data contains outliers (extreme values)
- Data is skewed
- You want a measure that represents the typical middle value
Use the mode when:
- Working with categorical data
- You want to know the most common value
- The data has clear peaks or popular values
Exam Tips:
- Read questions carefully - underline key words like "Greytown" or "Johannesburg" when dealing with multiple data sets
- Always arrange data in ascending order before finding the median
- Show all working for mean calculations
- Count carefully - double-check the number of values in your data set
- State units in your final answers when given (e.g., rand, years, points)
Key Points to Remember:
- Mean = (the average)
- Median = Middle value when data is arranged in ascending order
- Mode = Most frequently occurring value in the data set
- Each measure provides different information about the same data set
- The mean is affected by outliers, but the median and mode are not