A frequency table shows each possible value and how many times it occurred (its frequency). The total of the frequency column tells you how many data values there are altogether.
Example 1 — Reading the table
Step 1
The table shows 20 matches were played (\(\sum f = 20\)).
Step 2
The most common number of goals was 2 (frequency 7) — this is the mode.
Step 3
In 2 matches, 0 goals were scored; in 7 matches, 2 goals were scored.
Tip: Always check \(\sum f\) equals the total number of observations before calculating any averages.
✓ Quick Check 1 — Reading frequency tables
Question 1 of 10
Mean from a Frequency Table — the fx Column
\[\text{Mean} = \frac{\sum fx}{\sum f}\]
Add an fx column (value × frequency). Sum it to get \(\sum fx\), then divide by \(\sum f\).
Example 2 — Number of siblings: 0(f=4), 1(f=8), 2(f=5), 3(f=3)
Common mistake: Dividing by the number of rows instead of \(\sum f\). Always use the total frequency column sum.
✓ Quick Check 2 — Mean from frequency tables
Question 1 of 10
Median and Mode from a Frequency Table
Use a cumulative frequency column to locate the median. The mode is simply the value with the highest frequency.
Example 3 — Finding the median using cumulative frequency
Step 1
Build cumulative frequency: running total of frequencies going down the table.
Step 2
\(n = 20\): median positions are 10th and 11th values.
Step 3
Cumulative frequency reaches 7 after x=1, and 14 after x=2. Both positions 10 and 11 fall in x=2.
Step 4
Median = \(\mathbf{2}\) goals.
As in Lesson 18.3 — Mode, the mode is just the x-value with the highest frequency — in the table above, x=2 has frequency 7 (the highest), so the mode is 2 goals.
Tip: For even \(n\): median position = between \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th. Find which row both positions fall in using cumulative frequency.
✓ Quick Check 3 — Median, mode, and cumulative frequency
Question 1 of 10
Try It: Frequency Table Summary
Table — Pets per household: 0(f=3), 1(f=6), 2(f=4), 3(f=2). Use this to answer the questions below.
#
Question
Answer
1
What is Σf?
15
2
What is Σfx?
0+6+8+6 = 20
3
What is the mean?
20 ÷ 15 ≈ 1.33
4
What is the mode?
1 pet (f=6)
5
What is the median? (n=15, position 8)
1 pet (cum.f reaches 9 at x=1)
Practice: Averages from Frequency Tables
Enter your answers as numbers.
#
Question
Answer
Result
1
Table: 2 (f=3), 3 (f=5), 4 (f=2). Find the mean.
2
Table: 1 (f=2), 2 (f=3), 3 (f=5). Find the mean.
3
Table: 5 (f=2), 6 (f=8), 7 (f=3). What is the modal value?
4
Table: 10 (f=4), 11 (f=7), 12 (f=9). What is the total frequency?
Table: 8 (f=6), 9 (f=3), 10 (f=1). What is the modal value?
8
Table: 2 (f=5), 3 (f=5). What is the total frequency?
9
Table: 1 (f=1), 2 (f=2), 3 (f=7). Find the mean.
10
Table: 20 (f=2), 21 (f=3), 22 (f=5). What is the modal value?
⚠ Things to Watch Out For
Dividing by number of rows instead of total frequency: Mean = Σfx ÷ Σf (total of all frequencies, not number of rows in the table).
Not creating the fx column: You must multiply each value by its frequency (fx) before summing. Adding up the values column directly ignores how many times each occurs.
Finding median — not using cumulative frequency: Build a cumulative frequency column to identify which group the median falls in. Don't guess from the frequencies alone.
Modal class vs modal value: From a frequency table, the mode is the value (or class) with the highest frequency — not the largest value or the middle row.
Σf ≠ number of rows: The total frequency Σf is the sum of all the frequency values. Always add the frequency column, not count the rows.