Mode, Median, and Mean Revision Notes for Leaving Cert Mathematics

Mode, Median, and Mean

What is an average?

The word average is commonly used in everyday language to describe a typical value that represents a whole set of data. We often hear about the average weekly wage, average daily temperature, or average score in competitions.

An average allows us to represent all values in a dataset with a single, representative value. This makes it extremely useful for comparing different sets of data by simply comparing their averages rather than looking at every individual value.

There are three main types of averages used in statistics: mode, median, and mean.

The Mode

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

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The mode is particularly useful when:

One value appears much more often than others
Working with non-numerical data (like colours of cars)
You want to identify the most common occurrence

Finding the mode

To find the mode, look for the value that appears most frequently in your dataset. It's often helpful to arrange the data in order first.

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Worked Example 1: Finding the mode

The ages of students on a school bus are: 12, 15, 12, 13, 14, 16, 15, 11, 12, 16, 15, 16, 14, 10, 13, 17, 15, 17

Step 1: Arrange the data in order 10, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17

Step 2: Count the frequency of each value Looking at the ordered list, we can see that 15 appears most frequently (4 times).

Step 3: Identify the mode The number in this list with the greatest frequency is 15.

Therefore, the mode = 15

The Median

The median is the middle value when all the data values are arranged in order from smallest to largest.

Finding the median

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Step 1: Always arrange the numbers in order of size, starting with the smallest Step 2: Find the middle position

For lists with an odd number of values:

If there are $n$ numbers in a list, the middle term is $\frac{1}{2}(n + 1)$

For lists with an even number of values:

Take the average of the two middle values

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Worked Example 2: Finding the median

Find the median of these numbers: 5, 8, 12, 4, 9, 3, 7, 2

Step 1: Write the numbers in order of size 2, 3, 4, 5, 7, 8, 9, 12

Step 2: Find the middle position There are 8 numbers, so we need the average of the 4th and 5th values.

Step 3: Calculate the median The 4th value is 5 and the 5th value is 7. The median is $\frac{5 + 7}{2} = \frac{12}{2} = 6$

Therefore, the median = 6

The Mean

The mean is the most commonly used type of average. It is calculated by adding all the values together and dividing by the number of values.

The mean is important because it considers every piece of data. However, it can be affected by extreme values.

Finding the mean formula

$\text{Mean} = \frac{\text{sum of the numbers}}{\text{number of numbers}}$

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Step 1: Find the sum of all the numbers Step 2: Divide this sum by the number of numbers

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Worked Example 3: Finding the mean

Find the mean of these numbers: 12, 14, 10, 17, 21, 22

Step 1: Add all the numbers together $12 + 14 + 10 + 17 + 21 + 22 = 96$

Step 2: Divide by the number of values There are 6 numbers, so:

$\text{Mean} = \frac{96}{6} = 16$

Therefore, the mean = 16

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Worked Example 4: Combined groups

Five girls and three boys took part in a quiz. The mean mark for the girls was 54. The mean mark for the boys was 62. Find the mean mark for the whole group.

Step 1: Calculate total marks for each group To find the mean, add the total of the marks for the girls to the total of the marks for the boys and divide the result by 8.

Total marks for girls = $54 \times 5 = 270$
Total marks for boys = $62 \times 3 = 186$

Step 2: Find the combined total Total for all 8 students = $270 + 186 = 456$

Step 3: Calculate the overall mean $\text{Mean for whole group} = \frac{456}{8} = 57$

Therefore, the mean mark for the whole group = 57

Exam tips

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Essential exam strategies:

Always arrange data in order when finding the median - this is the most important step
Show all working clearly including the sum calculation for the mean
Double-check your arithmetic when adding numbers for the mean
Remember the mode can be non-numerical (like favourite colours or sports)
The mean uses every data value, making it sensitive to unusually high or low values
For combined group problems, multiply each group's mean by the number in that group first

Remember!

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Key Points to Remember:

Mode = Most frequently occurring value in the dataset
Median = Middle value when all data is arranged in ascending order
Mean = Sum of all values divided by the total number of values
Always arrange data in size order before finding the median
The mean considers all data points but can be affected by extreme values

Mode, Median, and Mean (Leaving Cert Mathematics): Revision Notes