Measures of Central Tendency Revision Notes for Grade 10 NSC Matric Mathematics

Measures of Central Tendency

Measures of central tendency help us find a single value that represents the centre or typical value of a data set. There are three main measures you need to know for your NSC exam: mean, median, and mode.

Mean

Definition: The mean is the sum of all values in a data set divided by the number of values. The mean is also called the average or arithmetic mean.

The formula for calculating the mean is:

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + ... + x_n}{n}$

Where:

$\bar{x}$ represents the mean
$n$ is the number of values in the data set
$x_i$ represents each individual value

How to calculate the mean

Follow these simple steps:

Add up all the values in your data set
Count how many values you have
Divide the sum by the number of values

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Worked Example: Calculating the mean

Question: What is the mean of the data set {10; 20; 30; 40; 50}?

Solution:

Step 1: Calculate the sum of the data

10 + 20 + 30 + 40 + 50 = 150

Step 2: Divide by the number of values in the data set

Since there are 5 values in the data set: mean = $\frac{150}{5} = 30$

Median

Definition: The median is the value in the central position when the data set has been arranged from lowest to highest value.

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An important property of the median is that exactly half of the values are less than the median and the other half are greater than the median.

How to find the median

Sort the data from smallest to largest value
Find the middle value:
- If there is an odd number of values: the median equals the middle value
- If there is an even number of values: the median is halfway between the two middle values

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Worked Example: Median with odd number of values

Question: What is the median of {10; 14; 86; 2; 68; 99; 1}?

Solution:

Step 1: Sort the values

Arranged from smallest to largest: 1; 2; 10; 14; 68; 86; 99

Step 2: Find the number in the middle

There are 7 values in the data set. Since this is an odd number, the median equals the value in the middle position (4th position).

Therefore, the median = 14

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Worked Example: Median with even number of values

Question: What is the median of {11; 10; 14; 86; 2; 68; 99; 1}?

Solution:

Step 1: Sort the values

Arranged from smallest to largest: 1; 2; 10; 11; 14; 68; 86; 99

Step 2: Find the number in the middle

There are 8 values in the data set. Since this is an even number, the median lies halfway between the two middle values (4th and 5th positions).

The 4th position is 11 and the 5th position is 14.

Median = $\frac{11 + 14}{2} = 12.5$

Mode

Definition: The mode is the value that occurs most often in the data set. It can also be described as the most frequent or most common value.

How to find the mode

Count how many times each value appears in the data set
Identify the value that appears most often

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Important notes about mode:

A data set can have more than one mode if multiple values have the highest frequency
If all values occur with equal frequency, the data set has no mode
The mode is particularly useful for qualitative data (categories)

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Worked Example: Finding the mode

Question: Find the mode of {2; 2; 3; 4; 4; 4; 6; 6; 7; 8; 8; 10; 10}

Solution:

Step 1: Count how many times each value appears

Table

Step 2: Find the value that appears most often

From counting, we can see that 4 appears 3 times, which is more than any other value.

Therefore, the mode = 4

Comparison of the three measures

When comparing mean, median, and mode, it's important to understand their different characteristics:

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Key Characteristics:

Mean: Uses all values in the calculation, so it's affected by extremely high or low values (outliers)
Median: Only depends on the middle value(s), so it's less affected by outliers
Mode: Shows the most common value, useful for understanding what's typical

Real-world example

Consider this data showing bread masses measured over a week:

For each day, we can calculate all three measures:

Notice how the mean and median values are close to each other but not exactly equal. This is common with real data sets.

Effect of outliers

Definition: An outlier is a value in the data set that is not typical of the rest of the set. It is usually much greater or much less than all other values.

How outliers affect measures of central tendency

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Critical Impact of Outliers:

The mean is significantly affected by outliers because it includes all values in its calculation.

The median is less affected by outliers because it only depends on the middle position(s).

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Worked Example: Effect of outliers

Question: Compare the mean and median of learners' heights before and after including an exceptionally tall learner.

Original data (10 learners): {150; 172; 153; 156; 146; 157; 157; 143; 168; 157}

After adding outlier: {150; 172; 153; 156; 146; 157; 157; 143; 168; 157; 181}

Solution:

Original mean = $\frac{1559}{10} = 155.9$ cm

New mean = $\frac{1740}{11} = 158.2$ cm

Change in mean = 158.2 - 155.9 = 2.3 cm

Original median = 156.5 cm (average of 5th and 6th values when sorted)

New median = 157 cm (6th value when 11 values are sorted)

Change in median = 157 - 156.5 = 0.5 cm

Conclusion: The mean changed by 2.3 cm while the median only changed by 0.5 cm. This shows that the median is less affected by outliers than the mean.

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Exam Tips:

Always sort your data before finding the median
Check your arithmetic when calculating the mean
For mode questions, create a frequency table to count values systematically
Remember that continuous data (like measurements) rarely has a clear mode
When data contains outliers, the median often gives a better representation of the centre than the mean

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Key Points to Remember:

Mean: Add all values and divide by the count - affected by outliers
Median: The middle value when data is sorted - less affected by outliers
Mode: The most frequently occurring value - useful for categorical data
Always sort data before finding the median
The mean uses all data values, while median only uses position

Measures of Central Tendency (Grade 10 NSC Matric Mathematics): Revision Notes