Issues on Arithmetic Mean, Median, and Mode

What is the Arithmetic Mean

It is a number equal to the sum of the values divided by their count.

Formula: \(Arithmetic\ Mean = \frac{Sum\ of\ values}{Count}\)

Median

It is the value that is in the middle of the data after sorting (in ascending or descending order).

Note: If the number of values is even, the median is the (average of) the two middle numbers.

Mode

It is the value or number that occurs most frequently in a dataset.

Examples and Problems from the Curriculum

Problem One

A student scored the following grades in 5 monthly tests: 12, 15, 12, 18, 13. Find the mean, median, and mode.

Solution:

1. Arithmetic Mean:$$\frac{12 + 15 + 12 + 18 + 13}{5} = \frac{70}{5} = 14$$
2. Median:
   • First, we sort the values: 12, 12, 13, 15, 18.
   • The value in the middle is 13.
3. Mode:
   • The most frequently occurring value is 12.

Problem Two (Median for an Even Count)

A football team recorded the number of goals in 6 matches as follows: 1, 0, 3, 2, 2, 4. Find the median.

Solution:

• Ascending order: 0, 1, 2, 2, 3, 4.
• Since the number of values is even (6), we take the two middle numbers (2, 2):$$Median = \frac{2 + 2}{2} = 2$$

Problem Three (Finding an Unknown Value)

Problem Text: If the arithmetic mean of four numbers is 10, and three of these numbers are 8, 12, 7. What is the value of the fourth number?

Solution:

1. We calculate the total sum of the numbers: $$10 \times 4 = 40$$.
2. We add the three known numbers: $$8 + 12 + 7 = 27$$.
3. Fourth number = $$40 – 27 = 13$$.

Difference Between Mean, Median, and Mode

Measure	Simple Definition	When to Use It?
Mean	Balance point	When the data are close to each other and there are no outliers.
Median	Middle value	Best when there are outliers (very large or very small values).
Mode	Most frequent	Useful in categorical data (like colors or sizes).

We will now move on to a slightly higher level, which involves frequency tables. These questions are considered the “backbone” of mathematics tests in middle school in most Arab countries.

Check out The easiest way to calculate the percentage

Second: Calculating Measures from Simple Frequency Tables

Often, data do not come individually, but are organized in a table showing the frequency of each value.

Problem Four (Arithmetic Mean from a Table)

The following table represents the number of goals scored by a group of players in a league. Calculate the arithmetic mean:

Number of Goals (x)	Number of Players (Frequency f)
1	4
2	3
3	2
4	1

Solution:

To solve this problem, we multiply each value by its frequency and then divide by the total frequency:

1. We find the total sum \(x \times 9\)
   • $$1 \times 4 = 4$$
   • $$2 \times 3 = 6$$
   • $$3 \times 2 = 6$$
   • $$4 \times 1 = 4$$
   • Total sum of values = $$4 + 6 + 6 + 4 = 20$$
2. We find the total frequency (number of players):
   • $$4 + 3 + 2 + 1 = 10$$
3. Arithmetic Mean:$$Mean = \frac{20}{10} = 2 \text{ Goals}$$

Problem Five (Mode from the Table)

This is the easiest problem and may appear in “choose the correct answer” options.

From the previous table, what is the mode for the number of goals?

Solution:

• We look for the highest frequency in the table.
• The highest frequency is 4.
• Thus, the mode is the corresponding value which is 1.
• (Common mistake: students write that the mode is 4, but the correct value itself is 1).

Problem Six (Determining the Median Rank)

If the number of data is large and organized in a table, we need to determine the “rank” of the median first.

Problem Text: If the total frequency in a statistical table is 50, what is the rank of the median?

Solution:

• Since the number is even, there are two medians with ranks:
  • First: $$\frac{50}{2} = 25$$
  • Second: $$\frac{50}{2} + 1 = 26$$
• The median is the average of item number 25 and item number 26 in the cumulative frequency table.

Summary of the Laws for Final Review

1. Mean: “We add and divide”.
2. Median: “We sort and choose the one in the middle”.
3. Mode: “We see the most frequent”.

Smart Note for High Achievers: If all values are repeated by the same count (for example: 5, 5, 7, 7, 9, 9), in this case, we say “there is no mode”. However, if there are two values with the same highest frequency, the set is “bi-modal”.