From a frequency table, mean = Σfx / Σf (sum of each value times its frequency, divided by total frequency), mode = the value with the highest frequency (read directly off the table, no calculation), and median = the middle value once all data is arranged in order, found using cumulative frequency to locate the (n/2)th or ((n+1)/2)th position. The three measures answer different questions, and mixing up their methods is the most common source of lost marks.
Mean = Σfx / Σf — The "average" — affected by every value, including extremes
Median = Middle value — Found via cumulative frequency, not affected by extremes
Mode = Highest frequency — Read directly from the table — no arithmetic needed
Students who understand each measure individually still lose marks by applying the wrong method to a frequency table — treating median like mean, or hunting for mode with a calculation instead of just reading it off. Each of the three needs a distinct approach.
The table shows the number of pets owned by 30 households. Find the mean, median, and mode.
| Number of pets (x) | Frequency (f) | fx | Cumulative frequency |
|---|---|---|---|
| 1 | 2 | 2 | 2 |
| 2 | 6 | 12 | 8 |
| 3 | 10 | 30 | 18 |
| 4 | 8 | 32 | 26 |
| 5 | 4 | 20 | 30 |
| Total | Σf = 30 | Σfx = 96 |
Mean = Σfx / Σf = 96 / 30 = 3.2Median = (3 + 3) / 2 = 3All three answers here — mean 3.2, median 3, mode 3 — are close but not identical. This is normal and expected. They only match exactly for perfectly symmetric data, which is rare in real O-Level questions. Don't assume you've made an error just because your three answers differ slightly.
n is odd: (n+1)/2 th value
n is even: average of (n/2)th and (n/2+1)th values
Common mistake
Students often use (n+1)/2 regardless of whether n is odd or even. When n is even, (n+1)/2 gives a non-integer position (e.g. n=30 gives position 15.5), which doesn't correspond to a real position in the list. Always check whether n is odd or even before choosing the formula.
When data is given in class intervals (grouped data) rather than exact values, the methods adjust slightly.
Example — estimating the mean from grouped data
| Height (cm) | Frequency (f) | Midpoint (x) | fx |
|---|---|---|---|
| 140 ≤ h < 150 | 5 | 145 | 725 |
| 150 ≤ h < 160 | 12 | 155 | 1860 |
| 160 ≤ h < 170 | 8 | 165 | 1320 |
For grouped data, mode becomes "modal class" — simply the class interval with the highest frequency, since you cannot identify one single mode value from grouped data. Always write "modal class" (not "mode") and give the full interval, e.g. "150 ≤ h < 160", when working with grouped data.
Some O-Level questions ask which average is "most appropriate" for a given situation, testing understanding rather than calculation, and the same judgement call comes up directly when comparing two data sets.
Can a frequency table have more than one mode?
Yes — if two or more values share the highest frequency, the data is called bimodal (two modes) or multimodal (more than two). State all values that share the highest frequency as the mode(s), rather than arbitrarily picking one.
Why is the mean from grouped data called an "estimate"?
Because grouped data only tells you how many values fall within each interval, not their exact values. Using the midpoint assumes the values are evenly distributed within each class, which may not be exactly true — so the resulting mean is described as an estimate, not an exact value.
Is the median always one of the actual data values?
For ungrouped data with odd n, yes — the median is exactly one of the listed values. For ungrouped data with even n, the median is the average of two middle values, which may not itself appear in the data set (e.g. if the two middle values are 3 and 4, the median is 3.5). For grouped data, the median is typically estimated from a cumulative frequency graph and also may not be an exact data value.
Why do I need a cumulative frequency column to find the median, but not for mean or mode?
Mean and mode can be read directly from the frequency column — mean through simple arithmetic (Σfx/Σf), mode by spotting the highest value. Median depends on position within an ordered list, and cumulative frequency is what tells you which data value occupies any given position (like the 15th or 16th value) without having to write out all 30 individual values by hand.
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