Relative Frequency

Expressing a Count as a Proportion

Absolute frequency tells us how many times a value appears in a dataset.

For example, if we ask $30 \text{ students}$ about their phone brands, and $12 \text{ students}$ answer "Apple", then the frequency for the "Apple" brand is $12$.

Relative frequency is the number of observations in a value or group divided by the total number of observations. It gives the proportion of the whole that belongs to that value or group.

Uses of Relative Frequency

Relative frequency supports two kinds of comparison:

1. Compare Proportions: We can compare how large the proportion of one data group is compared to another group within the same dataset. For example, how large the proportion of Apple users is compared to Samsung users in one respondent group.
2. Compare Groups of Different Sizes: We can compare the percentage of students who like mathematics in a group of $30 \text{ students}$ with the percentage in a group of $40 \text{ students}$. Absolute counts alone do not account for the different group sizes.

Relative Frequency Formula

Divide the frequency of the group by the total frequency:

Where:

• Group Frequency: number of observations in the selected group.
• Total Frequency: number of all observations in the dataset.

Relative Frequency is usually expressed in the form of:

• Fraction: The ratio of the group frequency to the total frequency.
• Decimal: The result of the fraction division.
• Percentage: The decimal form multiplied by $100\%$. This form states how many observations occur per hundred.

Example of Calculating Relative Frequency

Consider the phone-brand data from the $30 \text{ students}$ above:

Total frequency is $12 + 10 + 8 = 30$ students.

Calculate the relative frequency for each brand:

1. Apple:

   $$\text{Relative Frequency Apple} = \frac{12}{30}$$

   Relative Frequency Apple is $0.4$

   In percent: $0.4 \times 100\% = 40\%$

2. Samsung:

   $$\text{Relative Frequency Samsung} = \frac{10}{30}$$

   Relative Frequency Samsung is $0.333\ldots$ (we round it to $0.33$)

   In percent: $0.33 \times 100\% = 33\%$

3. Xiaomi:

   $$\text{Relative Frequency Xiaomi} = \frac{8}{30}$$

   Relative Frequency Xiaomi is $0.266\ldots$ (we round it to $0.27$)

   In percent: $0.27 \times 100\% = 27\%$

Relative Frequency Table:

The sum of relative frequencies (in decimal form) should always be $1$, and in percent should be $100\%$. There might be slight differences due to rounding, but it should be very close to $1$ or $100\%$.

Interpreting Relative Frequency

From the relative frequency table above, we can say:

• $40\%$ of the students in that group use Apple brand phones.
• About $33\%$ students use Samsung.
• The rest, about $27\%$, use Xiaomi.

The table shows at a glance how the $30 \text{ students}$ are distributed among the three brands.

Relative frequency makes absolute counts comparable by relating each count to the size of the complete dataset.