Let us start with the key concepts:

• Frequency — the number of times a particular value appears in the data.
• Relative frequency — the frequency divided by the total number of observations, \( \frac{f_i}{n} \). It shows what proportion of the data each value occupies, and is usually expressed as a decimal or a percentage.
• Frequency distribution — the complete table showing each value (or interval) alongside its frequency.

The type of data determines the appropriate chart:

• Discrete data — distinct, countable values (number of siblings, number of books). Best displayed with a bar chart — separate bars that do not touch, because there is no meaning between values (you cannot have 2.5 siblings).
• Continuous data — values within a range, measured (height, time, weight). Best displayed with a histogram — adjacent bars that touch, because the data flows continuously.

When there are many different values we group them into classes (intervals) such as 60–69, 70–79. In a histogram the height of each bar represents the frequency in that interval.

Example 1: Building a Frequency Count from Raw Data

Problem: The number of pets owned by 10 students is: 0, 1, 1, 2, 0, 3, 1, 2, 1, 0. What is the frequency of the value 1?

Solution:

1. Tally and count how many times each value appears.
2. The value 1 appears in students: 1, 1, 1, 1 — a total of four times.
3. Therefore the frequency of the value 1 is 4.

Answer: The frequency of 1 is \( 4 \).

Example 2: Complete Frequency Table and Sum Check

Problem: Students were asked how many siblings they have. The data collected: 0, 2, 1, 1, 3, 2, 0, 1, 2, 1, 4, 1. Build a frequency table and verify it.

Solution:

1. Possible values: 0, 1, 2, 3, 4. Count each one.
2. 0 appears twice; 1 appears five times; 2 appears three times; 3 appears once; 4 appears once.

3. Number of siblings	Frequency
   0	2
   1	5
   2	3
   3	1
   4	1

4. Check: \( 2 + 5 + 3 + 1 + 1 = 12 \), which equals exactly the number of students — the table is correct.
5. The most common number of siblings is 1 (frequency 5).

Answer: The table is valid (total frequency 12), and the modal value is \( 1 \).

Example 3: Identifying the Mode from a Given Table

Problem: A frequency table of test scores is given: 50→4, 60→6, 70→11, 80→5. Which score appears most often, and how many students were tested?

Solution:

1. The most frequent value is the one with the highest frequency.
2. The frequencies are 4, 6, 11, 5 — the highest is 11, belonging to the score 70.
3. The number of students is the sum of all frequencies: \( 4 + 6 + 11 + 5 = 26 \).

Answer: The most frequent score is \( 70 \), and \( 26 \) students were tested.

Example 4: Relative Frequency

Problem: In a sample of 40 families, 10 of them own exactly one car. What is the relative frequency of 'one car'?

Solution:

1. Relative frequency = frequency divided by total observations: \( \frac{f}{n} = \frac{10}{40} \).
2. \( \frac{10}{40} = \frac{1}{4} = 0.25 \).
3. As a percentage: \( 0.25 \times 100 = 25\% \).

Answer: The relative frequency is \( 0.25 \), i.e. \( 25\% \).

Example 5: Choosing the Appropriate Chart

Problem: A teacher recorded the running times (in seconds) of 50 students in a 100-metre race. Should the data be displayed as a bar chart or a histogram, and why?

Solution:

1. Identify the type of data: time is a measured quantity that can take any value within a range (12.3 seconds, 12.31 seconds, ...).
2. Therefore, the data are continuous — not discrete counted values.
3. Continuous data are grouped into intervals (e.g. 12–13 s, 13–14 s) and displayed in a histogram, where bars are adjacent to reflect the continuity.
4. A bar chart, by contrast, is appropriate for discrete data where gaps between values are meaningful.

Answer: A histogram is preferred, because running time is continuous data.

✗ Common mistake: Confusing the value with its frequency — when asked 'what is the most frequent value?' answering with the highest frequency count instead of the value itself.

✓ The correct way: The 'mode' is the value that appears most often, not the count. If 70 appears 11 times, the answer is 70, not 11.

✗ Common mistake: Drawing a histogram with adjacent bars for discrete data (such as number of siblings).

✓ The correct way: Discrete data require a bar chart with separate bars. The gaps between bars emphasise that there are no in-between values — 2.5 siblings is meaningless.

✗ Common mistake: Defining overlapping intervals, for example 60–70 and 70–80, so it is unclear where the value 70 belongs.

✓ The correct way: Define non-overlapping intervals, e.g. 60–69 and 70–79, or state a clear rule ('the upper boundary belongs to the next interval'). Every observation must fall into exactly one interval.

• Frequency = number of occurrences of a value; relative frequency \( = \frac{f_i}{n} \).
• The sum of all frequencies always equals the total number of observations \( n \).
• Discrete (counted) → bar chart (separate bars).
• Continuous (measured) → histogram (adjacent bars).
• The height of each bar = the frequency of that value or interval.